There is also work on strategyproof linear regression [7, 10, 13]. The setup of these models is also quite different from ours – typically, the strategic agents submit (x, y) pairs where x is fixed and y can be chosen strategically, and the evaluator’s goal is to perform linear regression in a way that incentivizes truthful reporting of y. In our setting, on the other hand, agents strategically generate their features x, and the evaluator rewards them in some way based on those features. Work exploring other aspects of how evaluation rules lead to investment of effort can be found in the economics literature, particularly. \n\nA helpful academic PDF source to read so you can continue writing the above:
Hu; Lily; Chen; Yiling
2017-11-30
<div><div>
 
 
 
 
 
 
 <p><a href="/pdf/1712.00064">Download PDF</a></p><blockquote>
 Abstract: The persistence of racial inequality in the U.S. labor market against a
general backdrop of formal equality of opportunity is a troubling phenomenon
that has significant ramifications on the design of hiring policies. In this
paper, we show that current group disparate outcomes may be immovable even when
hiring decisions are bound by an input-output notion of "individual fairness."
Instead, we construct a dynamic reputational model of the labor market that
illustrates the reinforcing nature of asymmetric outcomes resulting from
groups' divergent accesses to resources and as a result, investment choices. To
address these disparities, we adopt a dual labor market composed of a Temporary
Labor Market (TLM), in which firms' hiring strategies are constrained to ensure
statistical parity of workers granted entry into the pipeline, and a Permanent
Labor Market (PLM), in which firms hire top performers as desired. Individual
worker reputations produce externalities for their group; the corresponding
feedback loop raises the collective reputation of the initially disadvantaged
group via a TLM fairness intervention that need not be permanent. We show that
such a restriction on hiring practices induces an equilibrium that, under
particular market conditions, Pareto-dominates those arising from strategies
that statistically discriminate or employ a "group-blind" criterion. The
enduring nature of equilibria that are both inequitable and Pareto suboptimal
suggests that fairness interventions beyond procedural checks of hiring
decisions will be of critical importance in a world where machines play a
greater role in the employment process.
 </blockquote>
 
 
 </div><div>
 <h2>Submission history</h2><p> From: Lily Hu [<a href="/show-email/218153b2/1712.00064">view email</a>]
 <br />
 <strong><a href="/abs/1712.00064v1">[v1]</a></strong>
 Thu, 30 Nov 2017 20:24:30 UTC (181 KB)<br /><strong>[v2]</strong>
Wed, 21 Feb 2018 16:33:06 UTC (307 KB)<br /></p></div>||||I|||| Skip to main content
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 arXiv:1712.00064 (cs)
 [Submitted on 30 Nov 2017 (v1), last revised 21 Feb 2018 (this version, v2)]

 Title: A Short-term Intervention for Long-term Fairness in the Labor Market

 Authors: Lily Hu, Yiling Chen
 Download PDF
 Abstract: The persistence of racial inequality in the U.S. labor market against a general backdrop of formal equality of opportunity is a troubling phenomenon that has significant ramifications on the design of hiring policies. In this paper, we show that current group disparate outcomes may be immovable even when hiring decisions are bound by an input-output notion of "individual fairness." Instead, we construct a dynamic reputational model of the labor market that illustrates the reinforcing nature of asymmetric outcomes resulting from groups' divergent accesses to resources and as a result, investment choices. To address these disparities, we adopt a dual labor market composed of a Temporary Labor Market (TLM), in which firms' hiring strategies are constrained to ensure statistical parity of workers granted entry into the pipeline, and a Permanent Labor Market (PLM), in which firms hire top performers as desired. Individual worker reputations produce externalities for their group; the corresponding feedback loop raises the collective reputation of the initially disadvantaged group via a TLM fairness intervention that need not be permanent. We show that such a restriction on hiring practices induces an equilibrium that, under particular market conditions, Pareto-dominates those arising from strategies that statistically discriminate or employ a "group-blind" criterion. The enduring nature of equilibria that are both inequitable and Pareto suboptimal suggests that fairness interventions beyond procedural checks of hiring decisions will be of critical importance in a world where machines play a greater role in the employment process.
 Comments: 10 pages 
 Subjects: Computer Science and Game Theory (cs.GT) ; General Finance (q-fin.GN)
 Cite as: arXiv:1712.00064 [cs.GT] 
 (or arXiv:1712.00064v2 [cs.GT] for this version) 
 https://doi.org/10.48550/arXiv.1712.00064 
 Focus to learn more 
 arXiv-issued DOI via DataCite 
 https://doi.org/10.1145/3178876.3186044 
 Related DOI: Focus to learn more 
 DOI(s) linking to related resources 
 

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 From: Lily Hu [view email]
 [v1] Thu, 30 Nov 2017 20:24:30 UTC (181 KB)
 [v2] Wed, 21 Feb 2018 16:33:06 UTC (307 KB)
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A Short-term Intervention for Long-term Fairness in the Labor Market
https://arxiv.org/abs/1712.00064
Stephen Coate; Glenn C Loury
2022-11-02
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 Abstract
 
 <p>A key question concerning affirmative action is whether the labor-market gains it brings to minorities can continue without it becoming a permanent fixture in the labor market. We argue that this depends on how the policy affects employers' beliefs about the productivity of minority workers. We study the joint determination of employer beliefs and worker productivity in a model of statistical discrimination in job assignments. We prove that, even when identifiable groups are equally endowed ex ante, affirmative action can bring about a situation in which employers (correctly) perceive the groups to be unequally productive, ex post.</p>
 
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 journal article
 Will Affirmative-Action Policies Eliminate Negative Stereotypes?
 Stephen Coate and Glenn C. Loury
 The American Economic Review
 Vol. 83, No. 5 (Dec., 1993), pp. 1220-1240 (21 pages)
 Published By: American Economic Association
 https://www. jstor .org /stable/2117558
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 Abstract

 A key question concerning affirmative action is whether the labor-market gains it brings to minorities can continue without it becoming a permanent fixture in the labor market. We argue that this depends on how the policy affects employers' beliefs about the productivity of minority workers. We study the joint determination of employer beliefs and worker productivity in a model of statistical discrimination in job assignments. We prove that, even when identifiable groups are equally endowed ex ante, affirmative action can bring about a situation in which employers (correctly) perceive the groups to be unequally productive, ex post.

 Journal Information

 The American Economic Review is a general-interest economics journal. Established in 1911, the AER is among the nation's oldest and most respected scholarly journals in the economics profession and is celebrating over 100 years of publishing. The journal publishes 11 issues containing articles on a broad range of topics.

 Publisher Information

 Once composed primarily of college and university professors in economics, the American Economic Association (AEA) now attracts 20,000+ members from academe, business, government, and consulting groups within diverse disciplines from multi-cultural backgrounds. All are professionals or graduate-level students dedicated to economics research and teaching.

 Rights & Usage

 This item is part of a JSTOR Collection.
 For terms and use, please refer to our Terms and Conditions
 The American Economic Review © 1993 American Economic Association
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Original text
https://www.jstor.org/stable/2117558
None
None
Optimal Scoring for Dynamic Information
Acquisition∗
Yingkai Li† Jonathan Libgober‡
This version: November 12, 2023
newest version
Abstract
A principal seeks to learn about a binary state and can do so by enlisting
an agent to acquire information over time using a Poisson information arrival
technology. The agent learns about this state privately, and his effort choices
are unobserved by the principal. The principal can reward the agent with a
prize of fixed value as a function of the agent’s sequence of reports and the
realized state. We identify conditions that each individually ensure that the
principal cannot do better than by eliciting a single report from the agent after
all information has been acquired. We also show that such a static contract is
suboptimal under sufficiently strong violations of these conditions. We contrast
our solution to the case where the agent acquires information “all at once;”
notably, the optimal contract in the dynamic environment may provide strictly
positive base rewards to the agent even if his prediction about the state is
incorrect.
Keywords— Scoring rules, dynamic contracts, information acquisition, budgets, Poisson signals.
∗Part of this work was completed while Jonathan Libgober was visiting Yale University. The
authors thank Dirk Bergemann, Tan Gan, Marina Halac, Jason Hartline, David Kempe, Nicolas
Lambert, Elliot Lipnowski, Mallesh Pai, Harry Pei, Larry Samuelson, Philipp Strack and Kai Hao
Yang for helpful comments and suggestions. Yingkai Li thanks Sloan Research Fellowship FG-2019-
12378 for financial support.
†Cowles Foundation for Research in Economics, Yale University. Email: yingkai.li@yale.edu.
‡Department of Economics, University of Southern California. Email: libgober@usc.edu.
11 Introduction
This paper concerns the design of contracts to incentivize experts to acquire information for
predicting a future state when doing so is costly and when this information acquisition is
private. This problem may arise in a variety of situations, often when the contract designer
expects to make a high-stakes decision—for instance, whether to launch a military attack in
response to a perceived threat, whether to make a sizable investment, or whether to approve
a risky vaccine to fight a budding pandemic. In these cases, whether a given action is best
(e.g., invasion, investment, approval, respectively, for the above examples) may depend on
the underlying state, which we will take as binary in this paper for simplicity. Notice that
this problem features the combination of moral hazard (since the principal cannot monitor
the expert’s effort) and endogenous adverse selection (since the expert’s actions generate
private information).
In our model, while the agent has no direct interest in the state or the principal’s
action, he can be rewarded with a prize on which (i) he places bounded value, but (ii) the
principal places no intrinsic value. For example, this reward could be a recommendation or
a grade, often much more valuable for early-career workers than any feasible direct payment.
Alternatively, this could also reflect budget constraints on the principal, where the residual
value of payments is second-order relative to the value of information. For instance, this
situation applies to non-profit organizations that subject to budget constraints based on
external funding in the form of grants.
Crucially, the rewards given to the expert can depend on the state of interest, where
this state is unaffected by the principal’s actions. For instance, suppose the principal tasks
the agent with determining whether or not the economy will be in a recession on some
given date, as this may determine the relative profitability of an investment. In that case,
while the principal’s investment decision would have a negligible influence on the state of
the economy, rewards could still vary depending on market outcomes. Allowing the reward
to depend on the state incentivizes the agent to exert effort to learn about it.1
And lastly, we consider an information acquisition environment for the expert where
learning takes time. Thus, the principal must provide incentives for effort to be exerted
repeatedly, with possible interactions between the incentives provided at earlier times and
later times. The dynamic nature of the agent’s information acquisition problem is a significant point of departure from past work on incentivizing information acquisition (e.g.,
1As we point out in Section 2.4, our analysis also applies to the case where the principal can
condition rewards on a binary signal
https://yingkai-li.github.io/homepage/Optimal_Scoring.pdf
None
2006-06-14
Econometric Tools for Analyzing Market Outcomes.
Daniel Ackerberg, C. Lanier Benkard,
Steven Berry, and Ariel Pakes.∗
June 14, 2006
Abstract
This paper outlines recently developed techniques for estimating the primitives
needed to empirically analyze equilibrium interactions and their implications in
oligopolistic markets. It is divided into an introduction and three sections; a section
on estimating demand functions, a section on estimating production functions, and
a section on estimating “dynamic” parameters (parameters estimated through their
implications on the choice of controls which determine the distribution of future
profits).
The introduction provides an overview of how these primitives are used in typical
I.O. applications, and explains how the individual sections are structured. The topics
of the three sections have all been addressed in prior literature. Consequently each
section begins with a review of the problems I.O. researchers encountered in using the
prior approaches. The sections then continue with a fairly detailed explanation of the
recent techniques and their relationship to the problems with the prior approaches.
Hopefully the detail is rich enough to enable the reader to actually program up a
version of the techniques and use them to analyze data. We conclude each section
with a brief discussion of some of the problems with the more recent techniques. Here
the emphasis is on when those problems are likely to be particularly important, and
on recent research designed to overcome them when they are.
Keywords; Demand Estimation, Production Function Estimation, Dynamic Estimation, Strategic Interactions, Equilibrium Outcomes.
JEL Classification; C1, C3, C5, C7, L1, L4, L5.
∗The authors are from the UCLA, Stanford University, Yale University, and Harvard University,
respectively, and thank their NSF grants for financial support. This paper was prepared for the Handbook
of Econometrics, J.J. Heckman (ed.)
1
Recent complementary developments in computing power, data availability, and econometric technique have lead to rather dramatic changes in the way we do empirical analysis of market interactions. This paper reviews a subset of the econometric techniques
that have been developed. The first section considers developments in the estimation
of demand systems, the second considers developments in the estimation of production
functions, and the third is on dynamic estimation, in particular on estimating the costs
of investment decisions (where investments are broadly interpreted as any decision which
affects future, as well as perhaps current, profits).
These are three of the primitives that are typically needed to analyze market interactions
in imperfectly competitive industries. To actually do the analysis, that is to actually
unravel the causes of historical events or predict the impact of possible policy changes,
we need more information than is contained in these three primitives. We would also
need to know the appropriate notion of equilibrium for the market being analyzed, and
provide a method of selecting among equilibria if more than one of them were consistent
with our primitives and the equilibrium assumptions. Though we will sometimes use
familiar notions of equilibrium to develop our estimators, this paper does not explicitly
consider either the testing of alternative equilibrium assumptions or the issue of how one
selects among multiple equilibria. These are challenging tasks which the profession is
just now turning to.
For each of the three primitives we do analyze, we begin with a brief review of the
dominant analytic frameworks circa 1990 and an explanation of why those frameworks
did not suffice for the needs of modern Industrial Organization. We then move on to
recent developments. Our goal here is to explain how to use the recently developed
techniques and to help the reader identify problems that might arise when they are
used. Each of the three sections have a different concluding subsection.
There have been a number of recent papers which push the demand estimation literature
in different directions, so we conclude that section with a brief review of those articles
and why one might be interested in them. The section on production function concludes
with a discussion of the problems with the approach we outline, and some suggestions for
overcoming them (much of this material is new). The section on the costs of investments,
which is our section on “dynamics”, is largely a summary and integration of articles that
are still in various stages of peer review; so we conclude here with some caveats to the
new approaches.
We end this introduction with an indication of the ways Industrial Organization makes
use of the developments outlined in each of the three sections of the paper. This should
direct the researcher who is motivated by particular substantive issues to the appropriate
section of the paper. Each section is self-contained, so the reader ought to be able to
2
read any one of them in isolation.
Demand systems are used in several contexts. First demand systems are the major tool
for comparative static analysis of any change in a market that does not have an immediate
impact on costs (examples include the likely effects of mergers, tax changes, etc.). The
static analysis of the change usually assumes a mode of competition (almost always either
Nash in prices or in quantities) and either has cost data, or more frequently estimates
costs from the first order conditions for a Nash equilibrium. For example in a Nash
pricing (or Bertrand) equilibrium with single products firm, price equals marginal cost
plus a markup. The markup can be computed as a function of the estimated demand
parameters, so marginal costs can be estimated as price minus this markup. Given
marginal costs, demand, and the Nash pricing assumption the analyst can compute
an equilibrium under post change conditions (after the tax or the merger). Assuming
the computed equilibrium is the equilibrium that would be selected, this generates the
predictions for market outcomes after the change. If the analyst uses the pre-change
data on prices to estimate costs, the only primitive required for this analysis is the
demand function and the ownership pattern of the competing products (which is usually
observed).
A second use of demand systems is to analyze the effect of either price changes or new
goods on consumer welfare. This is particularly important for the analysis of markets that are either wholly or partially regulated (water, telecommunications, electricity,
postage, medicare and medicaid, ....). In this context we should keep in mind that
many regulatory decisions are either motivated by non-market factors (such as equity
considerations), or are politically sensitive (i.e. usually either the regulators or those
who appointed them are elected). As a result the analyst often is requested to provide a
distribution of predicted demand and welfare impacts across demographic, income and
location groups. For this reason a “representative agent” demand system simply will not
do.
The use of demand systems to analyze welfare changes is also important in several other
contexts. The “exact” consumer price index is a transform of the demand system. Thus
ideally we would be using demand systems to construct price indices also (and there
is some attempt by the BLS research staff to construct experimental indexes in this
way). Similarly the social returns to (either publicly or privately funded) research or
infrastructure investments are often also measured with the help of demand systems.
Yet a third way in which demand systems are important to the analysis of I.O. problems
is that some of them can be used to approximate the likely returns to potential new
products. Demand systems are therefore an integral part of the analysis of product
placement decisions, and more generally, for the analysis of the dynamic responses to
3
any policy or environmental change. Finally the way in which tastes are formed, and
the impacts of advertising on that process, are problems of fundamental interest to
I.O. Unfortunately these are topics we will not address in the demand section of this
paper. Our only consolation is the hope is that the techniques summarized here will
open windows that lead to a deeper understanding of these phenomena.
Production or cost functions are a second primitive needed for comparative static analysis. However partly because product specific cost data are not available for many markets, the direct estimation of cost functions has not been an active area of research
lately. There are exceptions, notably some illuminating studies of learning by doing (see
Benkard (2000) and the literature cited there), but not many of them.
What has changed in the past decade and a half is that researchers have gained access
to a large number of plant (sometimes firm) level data sets on production inputs and
outputs (usually the market value of outputs rather than some measure of the physical
quantity of the output). This data, often from various census offices, has stimulated
renewed interest in production function estimation and the analysis of productivity. The
data sets are typically (though not always) panels, and the availability of the data has
focused attention on a particular set of substantive and technical issues.
Substantively, there has been a renewal of interest in measuring productivity and gauging how some of the major changes in the economic environment that we have witnessed
over the past few decades affect it. This includes studies of the productivity impacts
of; deregulation, changes in tariff barriers, privatization, and broad changes in the institutional environment (e.g. changes in the legal system, in health care delivery, etc.).
The micro data has enabled this literature to distinguish between the impacts of these
changes on two sources of growth in aggregate productivity; (i) growth in the productivity of individual establishments, and (ii) growth in industry productivity resulting from
a reallocating more of the output to the more productive establishments (both among
continuing incumbents, and between exitors and new entrants). Interestingly, the prior
literature on productivity was also divided in this way. One part focused on the impacts of investments, in particular of research and infrastructure investments, on the
productive efficiency of plants. The other focused on the allocative efficiency of different
market structures and the impacts of alternative policies on that allocation (in particular
of merger and monopoly policy).
From an estimation point of view, the availability of large firm or plant level panels and
the desire to use them to analyze the impacts of major changes in the environment has
renewed interest in the analysis of simultaneity (endogeneity of inputs) and selection
(endogeneity of attrition) on parameter estimates. The data made clear that there are
both; (i) large differences in measured “productivity” across plants (no matter how
4
one measures productivity) and that these differences are serially correlated (and hence
likely to effect input choices), and (ii) large sample attrition and addition rates in these
panels (see Dunne, Roberts and Samuelson (1988) and Davis and Haltwinger (1992) for
some of the original work on U.S. manufacturing data). Moreover, the changes in the
economic environment that we typically analyze had different impacts on different firms.
Not surprisingly, the firms that were positively impacted by the changes tended to have
disproportionate growth in their inputs, while those that it affected negatively tended
to exhibit falling input demand, and not infrequently, to exit.
The traditional corrections for both simultaneity and selection, corrections based largely
on simple statistical models (e.g., use of fixed effect and related estimators for simultaneity, and the use of the propensity score for selection) were simply not rich enough to
account for the impacts of such major environmental changes. So the literature turned
to simultaneity and selection corrections based on economic models of input and exit
choices. The section of this chapter on production functions deals largely with these
latter models. We first review the new procedures emphasizing the assumptions they
use, and then provide suggestions for amending the estimators for cases where those
assumptions are suspect.
The last section of the paper deals explicitly with dynamic models. Despite a blossoming
empirical literature on the empirical analysis of static equilibrium models, there has
been very little empirical work based on dynamic equilibrium models to date. The I.O.
literature’s focus on static settings came about not because dynamics were thought to
be unimportant to the outcomes of interest. Indeed it is easy to take any one of the
changes typically analyzed in static models and make the argument that the dynamic
implications of the change might well overturn their static effects. Moreover, there was
a reasonable amount of agreement among applied researchers that the notion of Markov
Perfect equilibrium provided a rich enough framework for the analysis of dynamics in
oligopolistic settings.
The problem was that even given this framework the empirical analysis of the dynamic
consequences of the changes being examined was seen as too difficult a task to undertake.
In particular, while some of the parameters needed to use the Markov Perfect framework
to analyze dynamic games could be estimated without imposing the dynamic equilibrium
conditions, some could not. Moreover until very recently the only available methods for
estimating these remaining parameters were extremely burdensome, in terms of both
computation and researcher time.
The computational complexity resulted from the need to compute the continuation values to the dynamic game in order to estimate the model. The direct way of obtaining
continuation values was to compute them as the fixed point to a functional equation, a
5
high order computational problem. Parameter values were inferred from observed behavior by computing the fixed point that determines continuation values at different trial
parameter values, and then searching for the parameter value that makes the behavior
implied by the continuation values “as close as possible” to the observed data. This
“nested fixed point” algorithm is extremely computationally burdensome; the continuation values need to be computed many times and each time they are computed we need
to solve the fixed point.
A recent literature in industrial organization has developed techniques that substantially
reduce the computational and programming burdens of using the implications of dynamic
games to estimate the parameters needed for subsequent applied analysis. That literature
requires some strong assumptions, but delivers estimating equations which have simple
intuitive explanations and are easy to implement.
Essentially the alternative techniques deliver different semiparametric estimates of continuation values. Conditional on a value of the parameter vector, these estimated continuation values are treated as the true continuation values and used to determine optimal
policies (these can be entry and exit policies, investments of various forms, or bidding
strategies in dynamic auctions). The parameters are estimated by matching the policies
that are predicted in this way to the policies that are observed in the data. Note that
this process makes heavy use of nonparametric techniques; nonparametric estimates of
either policies or values must be estimated at every state observed in the data. Not
surprisingly then Monte Carlo evidence indicates that the small sample properties of the
estimators can be quite important in data sets of the size we currently use. This, in
turn, both generates preferences for some semiparametric estimators over others, and
makes obvious a need for small sample bias correction procedures which, for the most
part, have yet to be developed. We now move on to the body of the paper.
1 Demand Systems.
Demand systems are probably the most basic tool of empirical Industrial Organization.
They summarize the demand preferences that determines the incentives facing producers.
As a result some form of demand system has to be estimated before one can proceed with
a detailed empirical analysis of pricing (and/or production) decisions, and, consequently
of the profits and consumer welfare likely to be generated by the introduction of new
goods.
Not long ago graduate lectures on demand systems were largely based on “representative
agent” models in “product” space (i.e. the agent’s utility was defined on the product
6
per se rather than on the characteristics of the product). There were a number of
problems with this form of analysis that made if difficult to apply in the context of I.O.
problems. We begin with an overview of those problems, and the “solutions” that have
been proposed to deal with them.
Heterogeneous Agents and Simulation.
First almost all estimated demand system were based on market level data: they would
regress quantity purchased on (average) income and prices. There were theoretical papers
which investigated the properties of market level demand systems obtained by explicitly
aggregating up from micro models of consumer choices (including a seminal paper by
Houthakker (1955)). However we could not use their results to structure estimation on
market level data without imposing unrealistic a priori assumptions on the distribution
of income and “preferences” (or its determinants like size, age, location, etc.) across
consuming units.
Simulation estimators, which Pakes (1986) introduced for precisely this problem, i.e. to
enable one to use a micro behavioral model with heterogeneity among agents to structure
the empirical analysis of aggregate data, have changed what is feasible in this respect.
We can now aggregate up from the observed distribution of consumer characteristics and
any functional form that we might think relevant. That is we allow different consumers
to have different income, age, family size, and/or location of residence. We then formulate a demand system which is conditional on the consumer’s characteristics and a vector
of parameters which determines the relationship between those characteristics and preferences over products (or over product characteristics). To estimate those parameters
from market level data we simply
• draw vectors of consumer characteristics from the distribution of those characteristics in the market of interest (in the U.S., say from the March CPS),
• determine the choice that each of the households drawn would make for a given
value of the parameter vector,
• aggregate those choices into a prediction for aggregate demand conditional on the
parameter vector, and
• employ a search routine that finds the value of that parameter vector which makes
these aggregate quantities as close as possible to the observed market level demands.
7
The ability to obtain aggregate demand from a distribution of household preferences has
had at least two important impacts on demand analysis. First it has allowed us to use
the same framework to study demand in different markets, or in the same market at
different points in time. A representative agent framework might generate a reasonable
approximation to a demand surface in a particular market. However there are often
large differences in the distribution of income and other demographic characteristics
across markets, and these in turn make an approximation which fits well in one market
do poorly in others.
For example we all believe (and virtually all empirical work indicates) that the impact
of price depends on income. Our micro model will therefore imply that the price elasticity of a given good depends on the density of the income distribution among the
income/demographic groups attracted to that good. So if the income distribution differed across regional markets, and we used an aggregate framework to analyze demand,
we would require different price coefficients for each market. Table I provides some data
on the distribution of the income distribution across U.S. counties (there are about three
thousand counties in the U.S.). It is clear that the income distribution differs markedly
across these “markets”; the variance being especially large in the high income groups
(the groups which purchase a disproportionate share of goods sold).
Table I: Cross County Differences in Household Income∗
Income Fraction of U.S. Distribution of Fraction
Group Population in Over Counties
(thousands) Income Group Mean Std. Dev.
0-20 0.226 0.289 0.104
20-35 0.194 0.225 0.035
35-50 0.164 0.174 0.028
50-75 0.193 0.175 0.045
75-100 0.101 0.072 0.033
100-125 0.052 0.030 0.020
125-150 0.025 0.013 0.011
150-200 0.022 0.010 0.010
200 + 0.024 0.012 0.010
∗ From Pakes (2004, Review of Industrial Organization) “Common Sense and Simplicity
in Empirical Industrial Organization.”
8
A heterogenous agent demand model with an interaction between price and income
uses the available information on differences in the distribution of income to combine
the information from different markets. This both enables us to obtain more precise
parameter estimates, and provides a tool for making predictions of likely outcomes in
new markets.
The second aspect of the heterogenous agent based systems that is intensively used is its
ability to analyze the distributional impacts of policies or environmental changes that
effect prices and/or the goods marketed. These distributional effects are often of primary
concern to both policy makers and to the study of related fields (e.g. the study of voting
patterns in political economy, or the study of tax incidence in public finance).
The Too Many Parameters and New Goods Problems.
There were at least two other problems that appeared repeatedly when we used the earlier
models of demand to analyze Industrial Organization problems. They are both a direct
result of positing preferences directly on products, rather than on the characteristics of
products.
1. Many of the markets we wanted to analyze contained a large number of goods that
are substitutes for one another. As a result when we tried to estimate demand
systems in product space we quickly ran into the “too many parameters problem”.
Even a (log) linear demand system in product space for J products requires estimates of on the order of J
2 parameters (J price and one income coefficient in the
demand for every one of the J products). This was often just too many parameters
to estimate with the available data.
2. Demand systems in product space do not enable the researcher to analyze demand
for new goods prior to their introduction.
Gorman’s polar forms (Gorman (1959)) for multi-level budgeting were an ingenious attempt to mitigate the too many parameter problem. However they required assumptions
which were often unrealistic for the problem at hand. Indeed typically the grouping procedures used empirically paid little attention to accommodating Gorman’s conditions.
Rather they were determined by the policy issue of interest. As a result one would see
demand systems for the same good estimated in very different ways with results that
bore no relationship to each other1. Moreover the reduction in parameters obtained from
1For example it was not uncommon to see automobile demand systems that grouped goods into
imports and domestically produced in studies where the issue of interest involved tariffs of some form,
9
multilevel budgeting was not sharp enough to enable the kind of flexibility needed for
many I.O. applications (though it was for some, see for e.g. Hausman (1996) and the
literature cited there).
The new goods problem was central to the dynamics of analyzing market outcomes. That
is in order to get any sort of idea of the incentives for entry in differentiated product
markets, we need to be able to know something about the demand for a good which
had not yet been introduced. This is simply beyond the realm of what product based
demand systems can do. On the other hand entry is one of the basic dynamic adjustment
mechanisms in Industrial Organization, and it is hard to think of say, the likely price
effects of a merger2, or the longer run effects of an increase in gas prices, without some
way of evaluating the impacts of those events on the likelihood of entry.
The rest of this section of the paper will be based on models of demand that posit
preferences on the characteristics of products rather than on products themselves. We
do not, however, want to leave the reader with the impression that demand systems in
product based, in particular product space models that allow for consumer heterogeneity,
should not be used. If one is analyzing a market with a small number of products, and
if the issue of interest does not require an analysis of the potential for entry, then it
may well be preferable to use a product space system. Indeed all we do when we move
to characteristic space is to place restrictions on the demand systems which could, at
least in principle, be obtained from product space models. On the other hand these
restrictions provide a way of circumventing the “too many parameter” and “new goods”
problems which has turned out to be quite useful.
1.1 Characteristic Space: The Issues.
In characteristics space models
• Products are bundles of characteristics.
• Preferences are defined on those characteristics.
and alternatively by gas mileage in studies where the issue of interest was environmental or otherwise
related to fuel consumption. Also Gorman’s results were of the “if and only if” variety; one of his two
sets of conditions were necessary if one is to use multi-level budgeting. For more detail on multi-level
budgeting see Deaton and Muellbauer (1980).
2Not surprisingly, then, directly after explaining how they will analyze the price effects of mergers
among incumbent firms, the U.S. merger guidelines (DOJ (1992)) remind the reader that the outcome
of the analysis might be modified by an analysis of the likelihood of entry. Though they distinguish
between different types of potential entrants, their guidelines for evaluating the possibility of entry
remain distinctly more ad hoc then the procedures for analyzing the initial price changes.
10
• Each consumer chooses a bundle that maximizes its utility. Consumers have different relative preferences (usually just marginal preferences) for different characteristics, and hence make different choices.
• Simulation is used to obtain aggregate demand.
Note first that in these models the number of parameters required to determine aggregate demand is independent of the number of products per se; all we require is the
joint distribution of preferences over the characteristics. For example if there were five
important characteristics, and preferences over them distributed joint normally, twenty
parameters would determine the own and cross price elasticities for all products (no
matter the number of those products). Second, once we estimate those parameters, if
we specify a new good as a different bundle of characteristics then the bundles currently
in existence, we can predict the outcomes that would result from the entry of the new
good by simply giving each consumer an expanded choice set, one that includes the old
and the new good, and recomputing demand in exactly the same way as it was originally
computed3.
Having stated that, at least in principle, the characteristic space based systems solve
both the too many parameter and the new goods problems, we should now provide
some caveats. First what the system does is restrict preferences: it only allows two
products to be similar to one another through similarities in their characteristics. Below
we will introduce unmeasured characteristics into the analysis, but the extent to which
unmeasured characteristics have been used to pick up similarities in tastes for different
products is very limited. As a result if the researcher does not have measures of the
characteristics that consumers care about when making their purchase decisions, the
characteristic based models are unlikely to provide a very useful guide to which products
are good substitutes for one another. Moreover it is these substitution patterns that
determine pricing incentives in most I.O. models (and as a result profit margins and the
incentives to produce new goods).
As for new goods, there is a very real sense in which characteristic based systems can
only provide adequate predictions for goods that are not too “new”. That is, if we
formed the set of all tuples of characteristics which were convex combinations of the
characteristics of existing products, and considered a new product whose characteristics
are outside of this set, then we would not expect the estimated system to be able to
provide much information regarding preferences for the new good, as we would be “trying
3This assumes that there are no product specific unobservables. As noted below, it is typically
important to allow for such unobservables when analyzing demand for consumer products, and once one
allows for them we need to account for them in our predictions of demand for new goods. For an example
see Berry, Levinsohn and Pakes, 2004.
11
to predict behavior outside of the sample”. Moreover many of the most successful product
introductions are successful precisely because they consist of a tuple of characteristics
that is very different than any of the characteristic bundles that had been available before
it was marketed (think, for example, of the lap top computer, or the Mazda Miata4).
Some Background.
The theoretical and econometric groundwork for characteristic based demand systems
dates back at least to the seminal work of Lancaster (1971) and McFadden (1974, 1981)5.
Applications of the Lancaster/McFadden framework however, increased significantly after Berry, Levinsohn, and Pakes (1995) showed how to circumvent two problems that
had made it difficult to apply the early generation of characteristic based models in I.O.
contexts.
The problems were that
1. the early generation of models used functional forms which restricted cross and
own price elasticities in ways which brought into question the usefulness of the
whole exercise,
2. the early generation of models did not allow for unobserved product characteristics.
The second problem was first formulated in a clear way by Berry (1994), and is particularly important when studying demand for consumer goods. Typically these goods are
differentiated in many ways. As a result even if we measured all the relevant characteristics we could not expect to obtain precise estimates of their impacts. One solution
is to put in the “important” differentiating characteristics and an unobservable, say ξ,
which picks up the aggregate effect of the multitude of characteristics that are being
omitted. Of course, to the extent that producers know ξ when they set prices (and recall
ξ represents the effect of characteristics that are known to consumers), goods that have
high values for ξ will be priced higher in any reasonable notion of equilibrium.
4For more detail on just how our predictions would fail in this case see Pakes (1995).
5Actually characteristics based models have a much longer history in I.O. dating back at least to
Hotelling’s (1929) classic article, but the I.O. work on characteristic based models focused more on their
implications for product placement rather than on estimating demand systems per se. Characteristic
based models also had a history in the price index literature as a loose rational for the use of hedonic
price indices; see Court (1939), Griliches (1961), and the discussion of the relationship between hedonics
and I.O. equilibrium models in Pakes (2004).
12
This produces an analogue to the standard simultaneous equation problem in estimating
demand systems in the older demand literature; i.e. prices are correlated with the
disturbance term. However in the literature on characteristics based demand systems
the unobservable is buried deep inside a highly non-linear set of equations, and hence it
was not obvious how to proceed. Berry (1994) shows that there is a unique value for the
vector of unobservables that makes the predicted shares exactly equal to the observed
shares. Berry, Levinsohn, and Pakes (1995; henceforth BLP) provide a contraction
mapping which transforms the demand system into a system of equations that is linear
in these unobservables. The contraction mapping is easy to compute, and once we have
a system which is linear in the disturbances we can again use instruments, or any of
the other techniques used in more traditional endogeneity problems, to overcome this
“simultaneity problem.”
The first problem, that is the use of functional forms which restricted elasticities in
unacceptable ways, manifested itself differently in different models and data sets. The
theoretical I.O. literature focussed on the nature competition when there was one dimension of product competition. This could either be a “vertical” or quality dimension
as in Shaked and Sutton (1982) or a horizontal dimension, as in Salop (1979) (and in
Hotelling’s (1929) classic work). Bresnahan (1981), in his study of the automobile demand and prices, was the first to bring this class of models to data. One (of several)
conclusions of the paper was that a one-dimensional source of differentiation among products simply was not rich enough to provide a realistic picture of demand: in particular
it implied that a particular good only had a non-zero cross price elasticity with its two
immediate neighbors (for products at a corner of the quality space, there was only one
neighbor).
McFadden himself was quick to point out the “IIA” (or independence of irrelevant alternatives) problem of the logit model he used. The simplest logit model, and the one
that had been primarily used when only aggregate data was available (data on quantities, prices, and product characteristics), has the utility of the i
th consumer for the jth
product defined as
Ui,j = xjβ + i,j
where the xj are the characteristics of product j (including the unobserved characteristic
and price) and the {i,j} are independent (across both j for a given i and across i for
a given j) identically distributed random variables6. Thus xjβ is the mean utility of
6
In the pure logit, they have a double exponential distribution. Though this assumption was initially
quite important, it is neither essential for the argument that follows, nor of as much importance for
current applied work. Its original importance was due to the fact that it implied that the integral that
13
product j and i,j is the individual specific deviation from that mean.
There is a rather extreme form of the “IIA” problem in the demand generated by this
model. The model implies that the distribution of a consumer’s preferences over products
other than the product it bought, does not depend on the product it bought. One can
show that this implies the following.
• Two agents who buy different products are equally likely to switch to a particular
third product should the price of their product rise. As a result two goods with
the same shares have the same cross price elasticities with any other good (cross
price elasticities are a multiple of sj sk, where sj is the share of good j). Since both
very high quality goods with high prices and very low quality goods with low prices
have low shares, this implications is inconsistent with basic intuition.
• since there is no systematic difference in the price sensitivities of consumers attracted to the different goods, own price derivatives only depends on shares (∂s/∂p) =
−s(1−s). This implies that two goods with same share must have the same markup
in a single product firm “Nash in prices” equilibrium, and once again luxury and
low quality goods can easily have the same shares.
No data will ever change these implications of the two models. If your estimates do not
satisfy them, there is a programming error, and if your estimates do satisfy them, we are
unlikely to believe the results.
A way of ameliorating this problem is to allow the coefficients on x to be individualspecific. Then, when we increase the price of one good the consumers who leave that
good have very particular preferences, they were consumers who preferred the x’s of
that good. Consequently they will tend to switch to another good with similar x’s
generating exactly the kind of substitution patterns that we expect to see. Similarly,
now consumers who chose high priced cars will tend to be consumers who care less about
price. Consequently less of them will substitute from the good they purchase for any
given price increase, a fact which will generate lower price elasticities and a tendency for
higher markups on those goods.
This intuition also makes it clear how the IIA problem was ameliorated in the few studies
which had micro data (data which matched individual characteristics to the products
those individuals chose), and used it to estimate a micro choice model which was then
determined aggregate demand had a closed form, a feature which receded in importance as computers
and simulation techniques improved.
14
explicitly aggregated into an aggregate demand system. The micro choice model interacted observed individual and product characteristics, essentially producing individual
specific β’s in the logit model above. The IIA problem would then be ameliorated to
the extent that the individual characteristic data captured the differences in preferences
for different x-characteristics across households. Unfortunately many of the factors that
determine different households preferences for different characteristics are typically not
observed in our data sets, so without allowing for unobserved as well observed sources
of differences in the β, estimates of demand systems typically retain many reflections of
the IIA problem as noted above; see, in particular Berry, Levinsohn, and Pakes (2004;
henceforth MicroBLP) and the literature cited there.
The difficulty with allowing for individual specific coefficients on product characteristics
in the aggregate studies was that once we allowed for them the integral determining
aggregate shares was not analytic. This lead to a computational problem; it was difficult
to find the shares predicted by the model conditional on the model’s parameter vector.
This, in turn, made it difficult, if not impossible, to compute an estimator with desirable
properties. Similarly in micro studies the difficulty with allowing for unobserved individual specific characteristics that determined the sensitivity of individuals to different
product characteristics was that once we allowed for them the integral determining individual probabilities was not analytic. The literature circumvented these problems as did
Pakes (1986), i.e. by substituting simulation for integration, and then worried explicitly
about the impact of the simulation error on the properties of the estimators (see Berry,
Linton, and Pakes (2004) and the discussion below).
1.2 Characteristic Space; Details of a Simple Model.
The simplest characteristic based models assumes that each consumer buys at most one
unit of one of the differentiated goods. The utility from consuming good j depends on the
characteristics of good j, as well as on the tastes (interpreted broadly enough to include
income and demographic characteristics) of the household. Heterogenous households
have different tastes and so may choose different products.
The utility of consumer (or household) i for good j in market (or time period) t if it
purchases the j
th good is
uijt = U(˜xjt, ξjt, zit, νit, yit − pjt, θ), (1)
where ˜xjt is a K-dimensional vector of observed product characteristics other than price,
15
pjt is the price of the product, ξj,t represents product characteristics unobserved to the
econometrician, zit and νit are vectors of observed and unobserved (to the econometrician) sources of differences in consumer tastes, yit is the consumer’s income, and θ is a
vector of parameters to be estimated. When we discuss decisions within a single market,
we will often drop the t subscript.
Note that the “partial equilibrium” nature of the problem is incorporated into the model
by letting utility depend on the money available to spend outside of this market (yi −
pj ). In many applications, the expenditure in other markets is not explicitly modelled.
Instead, yiis subsumed into either νi or zi and utility is modelled as depending explicitly
on price, so that utility is
uij = U(˜xj , ξj , zi, νi, pj , θ). (2)
The consumer chooses one of j products and also has the j = 0 choice of not buying any
of the goods (i.e., choosing the “outside option”.) Denote the utility of outside good as
ui0 = U(˜x0, ξ0, zi, νi, θ), (3)
where ˜x0 could either be a vector of “characteristics” of the outside good, or else could
be an indicator for the outside good that shifts the functional form of U (because the
outside good may be difficult to place in the same space of product characteristics as
the “inside” goods.) The existence of the outside option allows us to model aggregate
demand for the market’s products; in particular it allows market demand to decline if
all within-market prices rise.
The consumer makes the choice that gives the highest utility. The probability of that
product j is chosen is then the probability that the unobservables ν are such that
uij > uir, ∀r 6= j. (4)
The demand system for the industry’s products is obtained by using the distribution of
the (zi, νi) to sum up over the values for these variables that satisfy the above condition
in the market of interest.
16
Note that, at least with sufficient information on the distribution of the (zi
, νi), the same
model can be applied when; only market level data are available, when we have micro
data which matches individuals to the choices they make, when we have strata samples
or information on the total purchases of particular strata, or with any combination of the
above types of data. In principal at least, this should make it easy to compare different
studies on the same market, or to use information from one study in another.
Henceforth we work with the linear case of the model in equations (2 and 3). Letting
xj = (˜xj , pj ), that model can be written as
Uij = Σkxjkθik + ξj + ij , (5)
with
θik = θk + θ
o
k
0zi + θ
u
k
0νi,
where the “o” and “u” superscripts designate the interactions of the product characteristic coefficients with the observed and the unobserved individual attributes, and it is
understood that xi0 ≡ 1.
We have not written down the equation for Ui,0, i.e. for the outside alternative, because
we can add an individual specific constant term to each choice without changing the
order of preferences over goods. This implies we need a normalization and we chose
Ui,0 = 0 (that is we subtract Ui,0 from each choice). Though this is notationally convenient we should keep in mind that the utilities from the various choices are now actually
the differences in utility between the choice of the particular good and the outside alternative7
.
Note also that we assume a single unobservable product characteristic, i.e. ξj ∈ R, and
its coefficient does not vary across consumers. That is, if there are multiple unobservable
characteristics then we are assuming they can be collapsed into a single index whose form
does not vary over consumers. This constraint is likely to be more binding were we to
have data that contained multiple choices per person (see, for example Heckman and
Snyder (1997))8. Keep in mind, however, that any reasonable notion of equilibrium
would make pj depend on ξj (as well as on the other product characteristics).
The only part of the specification in (5) we have not explained are the {ij}. They
represent unobserved sources of variation that are independent across individuals for a
7We could also multiply each utility by positive constant without changing the order, but we use this
normalization up by assuming that the i,j are i.i.d. extreme value deviates, see below.
8Attempts we have seen to model a random coefficient on the ξ have lead to results which indicate
that there was no need for one, see Das, Olley, and Pakes (1996).
17
given product, and across products for a given individual. In many situations it is hard
to think of such sources of variation, and as a result one might want to do away with the
{ij}. We show below that it is possible to do so, and that the model without the {ij}
has a number of desirable properties. On the other hand it is computationally convenient
to keep the {ij}, and the model without them is a limiting case of the model with them
(see below), so we start with the model in (5). As is customary in the literature, we will
assume that the {ij} are i.i.d. with the double exponential distribution.
Substituting the equation which determines θi,k into the utility function in (5) we have
Uij = δj + Σkrxjkzirθ
o
rk + Σklxjkνilθ
u
kl + ij , (6)
where
δj = Σkxjkθk + ξj .
Note that the model has two types of interaction terms between product and consumer
characteristics; (i) interactions between observed consumer characteristics (the zi) and
product characteristics (i.e. Σkrxjkzirθ
o
rk.), and interactions between unobserved consumer characteristics (the νi) and product characteristics (i.e. Σklxjkνilθ
u
kl.) It is these
interactions which generate reasonable own and cross price elasticities (i.e. they are
designed to do away with the IIA problem).
1.3 Steps in Estimation: Product Level Data.
There are many instances in which use of the model in (6) might be problematic, and we
come back to a discussion of them below. Before doing so, however, we want to consider
how to estimate that model. The appropriate estimation technique depends on the data
available and the market being modelled. We begin with the familiar case where only
product level demand data is available, and where we can assume that we have available
a set of variables w that satisfies E[ξ|w] = 0. This enables us to construct instruments
to separate out the effect of ξ from that of x in determining shares. The next section
considers additional sources of information, and shows how the additional sources of
information can be used to help estimate the parameters of the problem. In the section
that follows we come back to the “identifying” assumption, E[ξ|w] = 0, consider the
instruments it suggests, and discuss alternatives.
When we only have product level data all individual characteristics are unobserved, i.e.
zi ≡ 0. Typically some of the unobserved individual characteristics, the νi will have a
known distribution (e.g. income), while some will not. For those that do not we assume
that distribution up to a parameter to be estimated, and subsume those parameters
18
into the utility function specification (for example assume a normal distribution and
subsume the mean in θk and the standard deviation in θ
u
k
). The resultant known joint
distribution of unobserved characteristics is denoted by fν(·). We now describe the
estimation procedure.
The first two steps of this procedure are designed to obtain an estimate of ξ(·) as a
function of θ. We then require an identifying assumption that states that at θ = θ0,
the true value of θ, the distribution of ξ(·; θ) obeys some restriction. The third step is a
standard method of moments step that finds the value of θ that makes the distribution
of the estimated ξ(·, θ) obey that restriction to the extent possible.
Step I. We first find an approximation to the aggregate shares conditional on a particular
value of (δ, θ). As noted by McFadden (1974) the logit assumption implies that, when
we condition on the νi, we can find the choice probabilities implied by the model in (6)
analytically. Consequently the aggregate shares are given by
σj (θ, δ) = Z
exp[δj + Σklxjkνilθ
u
kl]
1 + P
q
exp[δq + Σklxqkνilθ
u
kl]
f(ν)d(ν). (7)
Typically this integral is intractable. Consequently we follow Pakes (1986) and use
simulation to obtain an approximation of it. I.e. we take ns pseudo random draws from
fν(·) and compute
σj (θ, δ, P ns) = Xns
r=1
exp[δj + Σklxjkνilrθ
u
kl]
1 + P
q
exp[δq + Σklxqkνilrθ
u
kl]
. (8)
where P
ns denotes the empirical distribution of the simulation draws. Note that the use
of simulation introduces simulation error. The variance of this error decreases with ns
but for given ns can be made smaller by using importance sampling or other variance
reduction techniques (for a good introduction to these techniques see Rubinstein (1981)).
Below we come back to the question of how the simulation error effects the precision of
the parameter estimates.
Step II. Let the vector of observed shares be s
n = [sn
1
, . . . , sn
J
], where n denotes the size
of the sample from which these shares are calculated (which is often very large). Step
19
II finds the unique values of δ that makes the predicted shares for a given θ and set of
simulation draws equal to s
n
. BLP show that iterating on the system of equations
δ
k
j
(θ) = δ
k−1
j
(θ) + ln[s
n
j
] − ln[σj (θ, δk−1, P ns)] (9)
leads to the unique δ that makes σj (θ, δ, P ns) = s
n
j
for all j
9
.
Call the fixed point obtained from the iterations δ(θ, sn, P ns). The model in (6) then
implies that
ξj (θ, sn, P ns) = δ(θ, sn, P ns) − Σkxjkθk. (10)
I.e. we have solved for the {ξj} as a function of the parameters, the data, and our
simulation draws.
“Identification.” An identifying restriction for our model will be a restriction on
the distribution of the true ξ, the ξ obtained when we evaluate the above equation at
n = ns = ∞, that will only be satisfied by ξj (θ, s∞, P ∞) when θ = θ0 (but not at other
values of θ). Different restrictions may well be appropriate in different applied cases,
and we come back to a discussion of possible restrictions below. For now, however, we
illustrate by assuming we have a set of instruments, say w that satisfy E[ξ(θ0)|w] = 0.
In that case the third and final step of the algorithm is as follows.
Step III. Interact ξj (θ, sn, P ns) with function of w and find that value of θ that makes
the sample moments as close as possible to zero. I.e. minimize kGJ,n,ns(θ)k where
GJ,n,ns(θ) = X
j
ξj (θ, sn, P ns)fj (w). (11)
9Note that one has to recompute the shares at the “new” δ at each iteration. The system of equations
is a mapping from possible values of δ into itself. BLP prove that the mapping is a contraction mapping
with modulus less that one. The iterations therefore converge geometrically to the unique fixed point of
the system.
20
Formal conditions for the consistency and asymptotic normality of this estimator are
given in Berry, Linton, and Pakes (2004), and provided one accounts for simulation and
sampling error in the estimate of the objective function, standard approximations to the
limit distribution work (see, for e.g. Pakes and Pollard, 1989). A few of the properties
of this limit distribution are discussed below. For now we want only to note that there
is an analytic form for the θ parameters conditional on the θ
u
; i.e. for the given θ
u
the
solution for θ is given by the standard instrumental variable formula. So the nonlinear
search is only over θ
u
.
1.4 Additional Sources of Information on Demand Parameters.
Often we find that there is not enough information in product level demand data to
estimate the entire distribution of preferences with sufficient precision. This should not
be surprising given that we are trying to estimate a whole distribution of preferences
from just aggregate choice probabilities. Other than functional form, the information
that is available for this purpose comes from differences in choice sets across markets or
time periods (this allows you to sweep out preferences for given characteristics), and differences in preferences across markets or over time for a fixed choice set (the preferences
differences are usually associated with known differences in demographic characteristics).
The literature has added information in two ways. One is to add an equilibrium assumption and work out its implications for the estimation of demand parameters, the other
is to add data. We now consider each of these in turn.
1.4.1 Adding the Pricing Equation.
There is a long tradition in economics of estimating “hedonic” or reduced form equations for price against product characteristics in differentiated product markets (see, in
particular Court (1939) and Griliches (1961)). Part of the reason those equations were
considered so useful, useful enough to be incorporated as correction procedures in the
construction of most countries’ Consumer Price Indices, was that they typically had
quite high R2’s10. Indeed, at least in the cross section, the standard pricing equations
estimated by I.O. economists have produced quite good fits (i.e. just as the model predicts, goods with similar characteristics tend to sell for similar prices, and goods in parts
of the characteristic space with lots of competitors tend to sell for lower prices). Perhaps
it is not surprising then that when the pricing system is added to the demand system
10For a recent discussion of the relationship between hedonic regressions and pricing equations with
special emphasis on implications for the use of hedonics in the CPI, see Pakes (2004).
21
the precision of the demand parameters estimates tends to improve noticeably (see, for
e.g. BLP).
Adding the pricing system from an oligopoly model to the demand system and estimating
the parameters of two systems jointly is the analogue of adding the supply equation to the
demand equation in a perfectly competitive model and estimating the parameters of those
systems jointly. So it should not be surprising that the empirical oligopoly literature itself
started by estimating the pricing and demand systems jointly (see Bresnahan (1981)).
On the other hand there is a cost of using the pricing equation. It requires two additional
assumptions; (i) an assumption on the nature of equilibrium, and (ii) an assumption on
the cost function.
The controversial assumption is the equilibrium assumption. Though there has been
some empirical work that tries a subset of the alternative equilibrium assumptions and
sees how they fit the data (see for e.g. Berry, Levinsohn, and Pakes (1999) or Nevo
(2001)), almost all of it has assumed static profit maximization, no uncertainty, and
that one side of the transaction has the power to set prices while the other can only
decide whether and what to buy conditional on those prices. There are many situations
in which we should expect current prices to depend on likely future profits (e.g.’s include
any situation in which demand or cost tomorrow depends on current sales, and/or where
there are collusive possibilities; for more discussion see the last section of this paper).
Additionally there are many situations, particularly in markets where vertical relationships are important, where there are a small number of sellers facing a small number of
buyers; situations where we do not expect one side to be able to dictate prices to another
(for an attempt to handle these situations see Pakes, Porter, Ho, and Ishii (2006)).
On the other hand many (though not all) of the implications of the results that are
of interest will require the pricing assumption anyway, so there might be an argument
for using it directly in estimation. Moreover, as we have noted, the cross-sectional
distribution of prices is often quite well approximated by our simple assumptions, and,
partly as a result, use of those assumptions is often quite helpful in sorting out the
relevance of alternative values of θ.
We work with a Nash in prices, or Bertrand, assumption. Assume that marginal cost,
to be denoted by mc, is log linear in a set of observables rkj and a disturbance which
determines productivity or ωj , i.e.
ln[mcj ] = Xrk,jθ
c
k + ωj . (12)
22
r will typically include product characteristics, input prices and, possibly the quantity
produced (if there are non-constant returns to scale). As a result our demand and cost
disturbances (i.e. ξ and ω) will typically be mean independent of some of the components
of r but not of others. Also we might expect a positive correlation between ξ and ω since
goods with a higher unobserved quality might well cost more to produce.
Since we characteristically deal with multiproduct firms, and our equilibrium assumption
is that each firm sets each of its price to maximize the profits from all of its products
conditional on the prices set by its competitors, we need notation for the set of products
owned by firm f, say Jf . Then the Nash condition is that firms set each of their prices
to maximize P
j∈Jf
(pj − Cj (·))Msj (·), where Cj is total costs. This implies that for
j = 1, . . . , J
σj (·) + X
l∈Jf
(pl − mcl)M
∂σl(·)
∂pj
= 0. (13)
Note that we have added a system of J equations (one for each price) and R = dim(r)
parameters to the demand system. So provided J > R we have added degrees of freedom.
To incorporate the information in (13) and (12) into the estimation algorithm rewrite
the first order condition as s + (p − mc)∆ = 0, where ∆i,j is nonzero for elements of a
row that are owned by the same firm as the row good. Then
p − mc = ∆−1σ(·).
Now substitute from (12) to obtain the cost disturbance as
ln(p − ∆−1σ) − r
0
θ
c = ω(θ), (14)
and impose the restrictions that
Efj (w)ωj (θ) = 0 at θ = θ0.
We add the empirical analogues of these moments to the demand side moments in (11)
and proceed as in any method of moments estimation algorithm. This entails one additional computational step. Before we added the pricing system every time we evaluated
a θ we had to simulate demand and do the contraction mapping for that θ. Now we also
have to calculate the markups for that θ.
23
1.4.2 Adding Micro Data.
There are a number of types of micro data that might be available. Sometimes we have
surveys that match individual characteristics to a product chosen by the individual. Less
frequently the survey also provides information on the consumer’s second choice (see for
e.g. MicroBLP), or is a panel which follows multiple choices of the same consuming
unit over time. Alternatively we might not have the original survey’s individual choice
data, but only summary statistics that provide information on the joint distribution of
consumer and product characteristics (for a good example of this see Petrin’s, 2002, use of
Consumer Expenditure Survey moments in his study of the benefits to the introduction of
the minivan). We should note that many of the micro data sets are choice based samples,
and the empirical model should be built with this in mind (see, for e.g. MicroBLP (2004);
for more on the literature on choice based sampling see Manski and Lerman (1977) and
Imbens and Lancaster (1994)).
Since the model in (6) is a model of individual choice, it contains all the detail needed
to incorporate the micro data into the estimation algorithm. Thus the probability of an
individual with observed characteristics zi choosing good j given (θ, δ) is given by
P r(j|zi, θ, δ) = Z
ν
exp[δj + Σklxjkzilθ
o
kl + Σklxjkνilθ
u
kl]
1 + P
q
exp[δq + Σklxqkzilθ
o
kl + Σklxjkνilθ
u
kl]
f(ν)d(ν). (15)
What Can be Learned from Micro Data. Assume temporarily that we can actually
compute the probabilities in (15) analytically. Then we can use maximum likelihood to
estimate (θ
o
, θu). These estimates do not depend on any restrictions on the distribution
of ξ. I.e. by estimating free δj coefficients, we are allowing for a free set of ξj .
On the other hand recall that
δj = Σkxjkθk + ξj .
So we cannot analyze many of the implications of the model (including own and cross
price elasticities) without a further assumption which enables us to separate out the
effect of ξ from the effect of the x on δ (i.e. without the identifying assumption referred
to above). The availability of micro data, then, does not solve the simultaneity problem.
24
In particular it does not enable us to separate out the effect of price from unobservable
characteristics in determining aggregate demand. On the other hand there are a few
implications of the model that can be analyzed from just the estimates of (δ, θo, θu).
In particular estimates of consumer surplus from the products currently marketed (and
hence “ideal” consumer price indices) depend only on these parameters, and hence do
not require the additional identifying assumption.
Now say we wanted to use the data to estimate θ. In order to do so we need a further
restriction so assume, as before, that we have instruments w, and can provide instrumental variable estimates of the θ. The number of observations for the instrumental variable
regressions is the number of products. That is, at least if we chose to estimate (θ
o
, θu)
without imposing any constraints on the distribution of ξ, the precision of the estimates
of θ will depend only on the richness of the product level data. Moreover IV regressions
from a single cross-section of products in a given market are not likely to produce very
precise results; in particular there is likely to be very little independent variance in prices.
Since additional market level data is often widely available, this argues for integrating it
with the micro data, and doing an integrated analysis of the two data sources.
One more conceptual point on survey data. What the survey data adds is information
on the joint distribution of observed product and consumer attributes. We would expect
this to be very helpful in estimating θ
o
, the parameters that determine the interactions
between z and x. There is a sense in which it also provides information on θ
u
, but that
information is likely to be much less precise. That is we can analyze the variance in
purchases among individuals with the same choice set and the same value of z and use
that, together with the i.i.d structure of the , to try and sort out the variance-covariance
of the ν. However this requires estimates of variances conditional on z, and in practice
such estimates are often quite imprecise. This is another reason for augmenting crosssectional survey data with aggregate data on multiple markets (or time periods) in an
integrated estimation routine; then the observed variance in z could determine the θ
o
and differences in choice sets could help sweep out the impact of the θ
u parameter.
When the data does have second choice information, or when we observe the same
consuming unit purchasing more than one product, there is likely to be much more
direct information on θ
u
. This because the correlation between the x − intensity of the
first choice and the second choice of a given individual is a function of both θ
o and the
θ
u
terms, and the θ
o
terms should be able to be estimated from only the first choice
data. A similar comment can be made for repeated choices, at least provided the utility
function of the consuming unit does not change from choice to choice.
Table 2 illustrates some of these points. It is taken from MicroBLP where the data
consisted of a single cross sectional survey of households, and the market level data from
25
the same year. The survey contained information on household income, the number of
adults, the number of children, the age (of the head) of household, and whether their
residence was rural, urban, or suburban (and all of these were used in the estimation).
That study had particularly rich information on vehicle preferences, as each household
reported its second as well as its first best choice.
Table 2 provides the best price substitutes for selected models from demand systems
for automobiles that were estimated in four different ways; (i) the full model allows
for both the zi and the νi (i.e. for interactions between both observed and unobserved
individual characteristics and product characteristics), (ii) the logit models that allow
for only the zi, and (iii) the σ
0
s only model allows for only the νi. The most important
point to note is that without allowing for the νi there is a clear IIA problem. The
prevalence of the Caravan and the FS pickups when we use the logit estimates (the
models without the νi) is a result of them being the vehicles with the largest market
shares and the apparent absence of the observed factors which cause different households
to prefer different product characteristics differentially. Comparing to column (iv) it is
clear that the extent of preference heterogeneity caused by household attributes not in
our data is large. MicroBLP also notes that when they tried to estimate the full model
without the second choice information their estimates of the θ
u parameters were very
imprecise; too imprecise to present. However when they added the second choice data
they obtained both rather precise estimates of the contributions of the unobserved factors
and substitution patterns that made quite a bit of sense. Finally we note that the fact
that there was only a single year’s worth of data made the estimates of θ quite imprecise,
and the paper uses other sources of information to estimate those parameters.
Computational and Estimation Issues: Micro Data. There are a number of
choices to make here. At least in principal we could (i) estimate (θ
o
, θu, δ) pointwise, or
(ii) make an assumption on the distribution of ξ (eg. E[ξ|w] = 0), and estimate (θ
o
, θu, θ)
instead of (θ
o
, θu, δ). However the fact that ξ is a determinant of price, and price is in the
x vector, makes it difficult to operationalize (ii). To do so it seems that one would have
to make an assumption on the primitive distribution of ξ, solve out for equilibrium prices
conditional on (θ, ξ, x), substitute that solution into the choice probabilities in (15), and
then use simulation to integrate out the ξ and ν in the formula for those probabilities.
This both involves additional assumptions and is extremely demanding computationally.
The first procedure also has the advantage that its estimates of (θ
o
, θu) are independent
of the identifying restriction use to separate out the effect of ξ from the effect of x on θ.
Assume that we do estimate (θ
o
, θu, δ). If there are a large number of products or J, this
will be a large dimensional search (recall that there are J components of δ), and large
dimensional searches are difficult computationally. One way to overcome this problem is
26
Table 2: Price Substitutes for Selected Vehicles,
A Comparison Among Models∗.
Vehicle Full Model Logit 1st Logit 1st & 2nd Sigma Only
Metro Tercel Caravan Ford FS PU Civic
Cavalier Escort Caravan Ford FS PU Escort
Escort Tempo Caravan Ford FS PU Ranger
Corolla Escort Caravan Ford FS PU Civic
Sentra Civic Caravan Ford FS PU Civic
Accord Camry Caravan Ford FS PU Camry
Taurus Accord Caravan Ford FS PU Accord
Legend Town Car Caravan Ford FS PU LinTnc
Seville Deville Caravan Ford FS PU Deville
Lex LS400 MB 300 Econovan Ford FS PU Seville
Caravan Voyager Voyager Voyager Voyager
Quest Aerostar Caravan Caravan Aerostar
G Cherokee Explorer Caravan Chv FS PU Explorer
Trooper Explorer Caravan Chv FS PU Rodeo
GMC FS PU Chv FS PU Caravan Chv FS PU Chv FS PU
Toyota PU Ranger Caravan Chv FS PU Ranger
Econovan Dodge Van Caravan Ford FS PU Dodge Van
From Berry, Levinsohn, and Pakes (2004, JPE)“Estimating Differentiated Product Models from a Combination of Micro and Macro Data.”
27
to use the aggregate data to estimate δ conditional on θ from the contraction mapping
in (9), and restrict the non-linear search to searching for (θ
o
, θu).
Finally since the probabilities in (15) are not analytic, either they, or some transform
of them (like the score), will have to be simulated. There is now quite a bit of work on
simulating the probabilities of a random coefficient logit model (see Train (2003) and the
literature cited there). Here we only want to remind the reader that in the applications
we have in mind it is likely to be difficult to use the log (or a related) function of the
simulated probabilities in the objective function. Recall that if p
ns(θ) is the simulated
probability, and p
ns(θ) = p(θ) + ens, where ens is a zero mean simulation error, then
log[p
ns(θ) ≈ log[p(θ)] + e
ns
p(θ)
−
(e
ns)2
2 × p(θ)
2
.
So if the simulated probabilities are based on ns independent simulation draws each
of which has variance V (p(θ)) the bias in the estimate of the log probability will be
approximately
Elog[p
ns(θ)] − log[p(θ)] ≈ −
1
2 × ns × p(θ)
,
and ns must be large relative to p(θ) for this bias to go away (this uses the fact that
V ar(p
ns(θ)) ≈ p(θ)/ns).
In many Industrial Organization problems the majority of the population do not purchase
the good in a given period, and the probabilities of the inside goods are formed by
distributing the remainder of the population among a very large number of goods. For
example in MicroBLP’s auto example, only ten per cent of household’s purchase a car
in the survey year, and that ten percent is distributed among more than two hundred
models of cars. So it was common to have probabilities on the order of 10−4. It should
not be a surprise then that they chose to fit moments which were linear functions of
the error in estimating the probabilities (they fit the covariances of car characteristics
and household characteristics predicted by the model to those in the data) rather than
maximizing a simulated likelihood.
28
1.4.3 Identifying Restrictions.
Recall that the source of the endogeneity problem in the demand estimates is the correlation of the product specific unobservable, our ξ, with some of the observable characteristics of the product; in particular we are worried about a correlation of ξ with price. The
contraction mapping in (9) is helpful in this respect as it delivers ξ as a linear function
of observables. As a result, any of the standard ways of solving endogeneity problems in
linear models can be employed here.
The most familiar way of dealing with endogeneity problems in linear models is to use
instruments. The question then becomes what is an appropriate instrument for x
0
s in
the demand system, a question which has been discussed extensively in the context of
perfectly competitive models of supply and demand. As in those models cost shifters
that are excluded from demand and uncorrelated with the demand error are available
as instruments. The familiar problem here is that input prices typically do not vary
much; at least not within a single market. There are a couple of important exceptions.
One is when production takes place in different locations even though the products are
all sold in one market (as is common when investigating trade related issues, see Berry,
Levinsohn, and Pakes (1999)). Another is when a subset of the x
0
s are exogenous, the
cost factors are differentially related to different x
0
s, and the x−intensity of different
product varies. In this case interactions between the cost factors and those x’s should
be useful instruments.
In addition to cost instruments, Nevo (2001) uses an idea from Hausman (1996) marketequilibrium version of the AIDS model, applied to a time-series/cross-section panel of
geographically dispersed set of markets. The underlying assumption is that demand
shocks are not correlated across markets while cost shocks are correlated across markets.
The prices of goods in other markets then become instruments for the price of goods in
a given market. Nevo (2001) studies breakfast cereals and so sources of common cost
shocks include changes in input prices; sources of common demand shocks (which are
ruled out) include national advertising campaigns.
In oligopoly markets prices typically sell at a markup over marginal cost. So if the
product’s own (˜xj , rj )’s are used as instruments, then so might the (˜x−j , r−j ) of other
products, giving us a lot of potential instruments. Moreover if price setting models
like the one in equation (13) are appropriate (and recall that they often have a lot
of explanatory power), the impact of the (x−j , r−j ) on pj will depend on whether the
product’s are owned by the same or by different firms. This type of reasoning dates
back at least to Bresnahan (1987), who notes the empirical importance of the idea
that markups will be lower in “crowded” parts of the product space and that they
will be higher when “nearby” products are owned by the same firm. BLP and Berry,
29
Levinsohn, and Pakes (1999) rely on this sort of argument to propose the use of functions
of rivals’ observed product characteristics, and of the ownership structure of products,
as instruments. Relatedly exogenous changes in competitive conditions across markets
are also candidate instruments (say due to the size of the market, or heterogeneity in
entry costs).
It is difficult to specify a priori how to make optimal use of the product characteristics to
predict markups. Both BLP and Berry, Levinsohn, and Pakes (1999) try approximations
to the “optimal instrument” formula suggested by Chamberlain (1986). This assumes
E[ξj |x˜j , x˜−j , rj , r−j ] = E[ωj |x˜j , x˜−j , rj , r−j ] = 0,
homoscedasticity, and ignores the within market dependence induced by the market
interactions. Chamberlain’s results then imply that the optimal instrument for our
problem is the derivative of these expectations with respect to the parameter vector.
In our context this will be a difficult to compute function of all the product characteristics. BLP tries to approximate this function “non-parametrically” using the exchangeable basis provided in Pakes (1993). Berry, Levinsohn, and Pakes (1999), try
an alternative approximation which is more direct, but also more computationally burdensome. They use a first-stage estimate of the parameter vector, θ, to recalculate
equilibrium prices with all values of ξ = ω = 0. They then compute the derivative of ξ
and ω with respect to θ at the first stage estimate of θ and the new equilibrium prices,
and use it as an instrument. I.e. instead of evaluating the mean of the derivative they
evaluate the derivative at the mean of the disturbance vector. Note that the instrument
is then a function only of exogenous variables, and so results in consistent estimators
(even though they are not quite efficient).
So far we have assumed mean independence of the unobservable characteristics, and, as
noted, there are plausible reason to believe that product characteristics themselves are
correlated with ξ. After all the product design team has at least some control over the
level of ξ, and the costs and benefits of producing different levels of the unobservable
characteristics might well vary with the observed characteristics of the product. One
possible solution would be to completely model the choice of product characteristics, as
in the dynamic models considered later in this chapter.
That said since p is typically not as hard to adjust as the other product characteristics,
the relationship between ξ and ˜x does not seem to be nearly as direct as that between
ξ and p (which is the reason it is often ignored; just as it was in traditional models of
demand and supply). So one might be willing to make some reduced form assumption
which allows us to proceed without all the detail of a dynamic game. In particular one
might try to use changes in demand over time, or across markets, for the same good to
30
control for the influence of unobserved product characteristics.
For example, suppose that we observe demand for the same product over time. It
might be reasonable to suppose that the product characteristics are correlated with the
unobservable in the year of product introduction. However one might also argue that
any changes in the level of unobserved characteristics over time are due to changes in
either perception of the product or in customer service that have little to do with the
initial x choices. So if t0 were the date of introduction of the good we might assume that
ξj,t = ξj,t0 + ηj,t+1. (16)
where ηj,t+1 is mean independent of the observed characteristics of all products. Alternatively we could assume that ξj,t followed a first order Markov process with only ξj,t0
,
and not the increments in the process, correlated with observed characteristics.
Relatedly if the data contains sales of the same product in many markets one could think
of restrictions on how the unobservable for a single product changes across markets. The
most straightforward example of this is to require ξ to be the same across markets. This
is quite a powerful restriction, and one might question it on the basis of differences in
the distribution of consumer preferences across markets that impact on their estimated
ξ
0
s. A weaker assumption would be that the difference between ξ’s for the same product
across markets is uncorrelated with the observed x. Similarly, some products within
a market may differ only by the the addition of some optional features and we could
restrict the way that ξ changes across products that vary only in their options.
1.5 Problems With the Framework.
We have motivated our discussion on demand estimation by noting how the recent literature dealt with the problems that arose in using representative agent models in product
space. There are many senses, however, in which the framework outlined above can be
too restrictive for particular problems. This section reviews some of the more obvious of
them. The impact of these problems depend upon the market one is analyzing and the
issues one is focusing on. Also, at least partial solutions to some of these problems are
available, and we will direct the reader to them where we can. In large part, however,
this section is an outline of agendas for future research on demand estimation for I.O.
problems.
We begin with multiple choice and/or dynamics, and then come back to the problem
in the static discrete choice model considered above. Most empirical studies simply
31
ignore issues related to multiple choices and/or dynamics. The hope is that the estimated demand system is still the best currently available approximation for analyzing
the question of interest. To us the surprising part of the results of those studies is that
the framework seems to provide a “reasonable” approximation to substitution patterns,
and even more surprisingly, a reasonable approximation to pricing patterns. This despite
the fact that we know that consumers’ demands and the market equilibrium outcomes
are products of much more complicated processes then those we model. Even so, as will
become clear presently, there are a number of issues of importance to I.O. which can not
be studied empirically without a more detailed understanding of multiple choice and/or
the dynamic aspects of demand.
Multiple Units of Demand.
There are many situations for which a model based on the choice of either one or zero
units of a good does not match reality11. Models for choosing a finite number of units
from a set of substitute goods require a specification for the utility from multiple units.
Then, at least in principle, we are back to a discrete choice for “tuples” of goods. However
to maintain tractability when the number of units can grow large the specification is likely
to require constraints which cut down the choice set by implying that some choices are
dominated by others (otherwise the size of the choice set grows as J
C, where J is the
number of products and C is the maximum number of purchases).
One example of the use of such constraints is Hendel’s (1999) two-stage multiple-unit
/ multiple good framework for the demand of a firm for computers. He simplifies the
problem by imagining that the firm faces a random, discrete number of tasks. For each
task, it chooses only one type (brand) of computer and, according to the random size of
the tasks, a number of computers to purchase. This explicitly accounts for decisions to
purchase multiple units of multiple kinds of goods.
Gentzkow (2004) considers a problem with a small number of goods, but where there are a
small number of choices. In that study of online and print newspapers, some of the goods
are potentially complements, and this requires a different set of modifications. Moreover,
as Gentzkow shows the determination of whether goods are in fact complements or
substitutes interacts with the issue of the form of consumer heterogeneity in subtle ways
reminiscent of the initial condition problems in panel data estimation (see Heckman
(1981).
11Dubin and McFadden (1984) provide an earlier example with one discrete choice and one continuous
choice.
32
A related problem involves continuous choice over multiple goods. If all goods are purchased in some positive amount by every consumer, then a traditional continuous demand
approach, equating marginal rates of substitution across all goods, is appropriate. But
many real-world consumer data problems involve a large number of goods with many
zero purchase decisions and many positive purchase decisions. Chan (2002) considers
the Kuhn-Tucker version of the traditional continuous choice problem to study soft drink
purchases.
Dynamic Demand.
Yet another set of problems arises when the demand for the good is inherently dynamic,
as occurs with either durable, storable, or experience goods. Models which are appropriate for dynamic demand estimation can become quite complex; they require forward
looking consumers whose behavior depends on the likely distribution of future (as well as
current) offerings. Moreover in a complete model these future offerings would, in turn,
depend on producer’s perceptions of consumer demand. A number of new studies make
simplifying assumptions which allow them to make some headway.
Both Hendel and Nevo (2002) and Erdem, Keane, and Imai (2003) consider a problem
of durable good demand in an explicitly dynamic framework. They consider shopping
decisions when consumer’s are allowed to store purchases, and use a reduced form assumption on the process generating prices. It has been clear to I.O. economists for some
time that we are going to have to model intertemporal substitution of this form in order
to understand “sales” in retail markets (see Sobel (1984)).
Two problems in this kind of study are that the rate of consumption (inventory reduction) at home is typically not observed and the dimension of the state space (which
involves both the current price vector, which predicts future prices, and also the vector
of household inventories of different brands.) In these models new purchases are added
to a single-index of home inventories, with different brands of product receiving different utility weights in the inventory stock. This single index of inventories reduces the
dimensionality of the state space. Another simplifying assumption is that unobserved
household consumption follows a simple rule.
Esteban and Shum (2002) consider a model of durable automobile purchases. They
assume a used-car market with zero transaction costs. The zero transaction costs imply
that the joint distribution of past choices and consumer characteristics are not a state
variable of the problem. Under these assumptions they are able to derive empirical
implications about the dynamic pricing problem of the durable goods manufacturer (in
determining current price the manufacturer has to worry about future aggregate supply
33
of the used goods). Many, if not most, manufacturing goods are durable.
Studies of demand for advertised experience goods include Erdem and Keane (1996),
Ackerberg (2003), and Crawford and Shum (2005). All of these papers feature Bayesian
consumers who learn both from experience and from adverting. This lead to a fairly
complex dynamic programming problems for the consumer. The studies largely ignore
the firm’s endogenous pricing and advertising decisions.
Problems With the Static Discrete Choice Specification.
There are also aspects of the static discrete choice specification of the model outlined
above whose flexibility, and/or implications, are not yet well understood. One such issue
is whether the second derivatives of the demand function are very flexibly estimated.
This will determine whether two goods are strategic substitutes or strategic complements,
and hence has implications for the analysis of the structure of strategic interaction, and
appears to be largely unexplored in the current literature. More generally there are a
host of questions on what we can learn non-parametrically about the structure of demand
from different kinds of data that we have not touched on here (for a discussion of some
of them, see Matzkin’s contribution to this volume).
A second such issue concerns the role of the i.i.d. “idiosyncratic match values”, the 
0
ijs, in
the models above. These are added to the model largely for computational convenience;
they do not seem to match any omitted causal demand determinant. Moreover the
presence of the ij has implications. They imply that each product is “born” with a
distribution of consumer tastes whose conditional distribution, conditional on the tastes
for other products, has support that ranges from minus to plus infinity. This implies that
every conceivable product, no matter its characteristics and price, will have a strictly
positive (though perhaps quite tiny) expected market share.
Given the standard ij ’s, each product will also have a positive cross-price effect with
every other product: competition is never completely local. Perhaps most problematic,
it also implies that if we define a consumer by a (z, ν) combination, every consumer’s
utility will grow without bound as we increase the number of products – regardless of
the characteristics or prices of the new products that are introduced. As a result there
is a worry about the ability of the model in (6) to provide an adequate approximation
to the benefits from introducing new goods12.
12We hasten to note that estimating the consumer surplus generated by new products is an extremely
difficult task in any framework. This because we typically do not have data on the demand for new
products at prices that are high enough to enable us to estimate the reservation prices of a large fraction
of consumers. The characteristic based demand model does use slightly more information in its estimation
34
To investigate these issues more fully, Berry and Pakes (2005) consider a “pure characteristic” model of demand. That model is exactly the model in equation (6) once we
omit the the ij terms. They consider the analytic properties of the model, then provide an estimation algorithm for it and explore its computational properties, and finally
provide Monte Carlo evidence on its performance. Song (2004) has used this model to
evaluate the gains from new semiconductor chips. The pure characteristics model is
somewhat more computationally burdensome then the model in equation (6), largely
because the equation for solving for δ for that model (the analogue to equation 9) is
not necessarily a contraction with modulus less than one. On the other hand its shares
are easier to simulate to sufficient accuracy. However the jury is still out on the major
question; the question of whether the pure characteristic model tends to provide a better
approximation to the consumer surplus gains from new goods then the model with the
ij .
Berry and Pakes (2005) and Bajari and Benkard (2005) discuss two different versions
of the “pure characteristics” model with “no ”s. Berry and Pakes (2005) consider a
discrete choice version of the model, with a utility function of
uij = xjβi − αipj + ξj , (17)
where βi and αi are random coefficients associated with consumer i’s tastes for characteristics and price of product j. Berry and Pakes suggest a BLP-style estimation
algorithm.
In contrast, Bajari and Benkard (2005) obtain an estimate of the unobservable demand
component, ξj , from the pricing side of the model rather than the demand side. The
argument is that in a “pure characteristics” model, prices must be strictly increasing
in ξ conditional on other x’s. Following on recent econometric literature, they show
that a monotonic transformation of the ξ can be obtained from data on prices and x’s.
This transformed ξ is then used in the demand-side analysis to control for unobserved
characteristics. Note, however, that consistency of this approach relies on asymptotics
in the number of products, and further requires the assumption that products enter
the market in such a way that eventually they “fill up” the product space (i.e., for
every product, it is assumed that eventually there will be other products whose observed
characteristics are arbitrarily close to those of the given product). In practice it is clear
that the approach requires data with many products per market, but there has not been
enough experience to date to know what “many” means in this context.
of consumer surplus gains than do demand models in product space, since it uses the price variance for
products with similar characteristics. However the results are still not terribly robust. Petrin (2002) for
example, reports large differences in consumer surplus gains from differences in specifications and data
sources.
35
1.6 Econometric Details.
This subsection summarizes results from Berry, Linton, and Pakes (2004) who provide
limit theorems for the parameter estimates from differentiated product models. The
actual form of the limit distributions depends on the type of data and type of model.
We will focus on the case where only one cross section of market level data is available.
Our purpose is to give the reader some indication of how the various estimation errors
that have been introduced are likely to effect the parameter estimates, and this is the
simplest environment in which to show that13.
Recall that the objective function minimized in the estimation algorithm or equation
(11) is a norm of
GJ (θ, sn, P ns) = 1
J
X
J
j=1
ξj (θ,s
n
, P ns)fj (w).
The ξj are defined implicitly as the solution to the system
s
n
j = σj (ξ, x; θ, P ns),
where σ(·) is defined in (8), the w satisfy E[ξ|w, θ0] = 0, s
n
is the observed vector of
market shares, and P
ns is notation for the vector of simulation draws used to compute
the market shares predicted by the model.
The objective function, kGJ (θ, sn, P ns)k, has a distribution determined by three independent sources of randomness: randomness generated from the draws on the product characteristics (both observed and unobserved, in the full model these are vectors
{ξ, x, r, ω ˜ }), randomness generated from the sampling distribution of s
n
, and that generated from the simulated distribution P
ns. Analogously there are three dimensions in
which our sample can grow: as n, as ns, and as J grow large.
The limit theorems allow different rates of growth for each dimension. Throughout we
take pathwise limits, i.e., we write n(J) and ns(J), let J → ∞, and note that our
assumptions imply n(J), ns(J) → ∞ at some specified rate. Note also that both s
n
and σ(ξ, θ, P) take values in RJ, where J is one of the dimensions that we let grow in
our limiting arguments. This is an unusual feature of the econometric model and causes
complications in the limiting arguments. As will become obvious sampling error (error in
13Cases in which there is data from many regional markets but the same goods are sold in each of
them will still have to deal with limits as the number of products grow large; it is just that then we
might also want to let the number of markets increase as we increase the number of products. Also in
cases with regional markets the computational problems we highlight will be even more severe, as then
we will have to compute ξ separately in each different market.
36
s
n
) plays an analogous role to simulation error (error in P
ns), so for notational simplicity
assume that n is sufficiently large that we do not need to worry about sampling error.
When there is no sampling (simulation) error we set n (ns) equal to zero.
We need to find an approximation for the objective function which allows us to separate
out the roles of the three sources of error. To this end write
ξ(θ, s0, P ns) = ξ(θ, s0, P0) + ξ(θ, s0, P ns) − ξ(θ, s0, P0)
	
. (18)
The function σ(ξ, θ, P) is differentiable in ξ, and its derivative has an inverse, say
H−1(ξ, θ, P) = 
∂σ(ξ, θ, P)
∂ξ0
−1
.
Abbreviate σo(θ, s, P) = σ(ξ(s, θ, P), θ, P) and Ho(θ, s, P) = H(ξ(s, θ, P), θ, P), and let
σ(ξ, P ns, θ) = σ(ξ, P0, θ) + ε
ns(θ).
Then from the fact that we obtain ξ from σ(·) = σ(ξ, P0, θ) + ε
ns(θ) it follows that
ξ(θ, s0, P ns) = ξ(θ, s0, P0) + H−1
o
(θ, s0, P0) {ε
ns(θ)} + r(θ, sn
, P ns)
where r(θ, sn, P ns) is a remainder term. Substituting into (18)
GJ (θ, sn, P ns) = GJ (θ, s0, P0) + 1
J
z
0H−1
o
(θ, s0, P0) {−ε
ns(θ)} +
1
J
z
0
r(θ, sn, P ns).
The limit theorems in Berry, Linton, and Pakes (2004) work from this representation
of GJ (θ, sn, P ns). To prove consistency they provide conditions which insure that: i)
the second and third terms in this equation converge to zero in probability uniformly
in θ, and ii) an estimator which minimized kGJ (θ, s0, P0)k over θ ∈ Θ would lead to a
consistent estimator of θ
0
.
Asymptotic normality requires, in addition, local regularity conditions of standard form,
and a limiting distribution for H−1
o
(θ, s0, P0) {−ε
ns(θ)}. The rate needed for this limit
distribution depends on how the elements of the J × J matrix H−1
o
(θ, s0, P0) grow, as J
gets large. It is easiest to illustrate the issues that can arise here by going back to the
simple logit model.
In that model; ui,j = δj + i,j , with the {i,j} distributed i.i.d. type II extreme value,
and δj = xjθ + ξj . Familiar arguments show that σj = exp[δj ]/(1 + P
q
exp[δq]), while
37
σ0 = 1/(1 + P
q
exp[δq]). In this case the solution to the contraction mapping in (9) is
analytic and
ξj (θ, so, Po) = (ln[s
o
j
] − ln[s
o
0
]) − xjβ.
Thus in this simple case
∂ξ
∂sj
|s
o =
1
s
o
j
.
Now consider how randomness effects the estimate of ξj (θ). In the simple logit model
the only source of randomness is in the sampling distribution of s
n
. That is we observe
the purchases of only a finite random sample of consumers. Letting their shares be s
n
we have, s
n − so = n
. The first order impact of this randomness on the value of our
objective function at any θ will be given by
H−1
o
(θ, s0) × n =
∂ξ
∂s|s=s
0
× 
n
.
This contains expressions like 
n
j
1
s
o
j
. In the logit model as J → ∞, s
o
j → 0. So as J grows
large the impact of any given sampling error grows without bound.
A similar argument holds for the estimator of BLP’s model, only in this more complicated
model there are two sources or randomness whose impacts increase as J grows large,
sampling error and simulation error. Consequently Berry, Linton, and Pakes show that
to obtain an asymptotically normal estimator of the parameter vector from this model
both n and ns must grow at rate J
2
. Note the similarity here to the reason that
simulation error is likely to make use of maximum likelihood techniques with survey
data computationally demanding; i.e. the impact of the simulation error on the objective
function increases as the actual shares get smaller. The computational implication here
is that for data sets with large J one will have to use many simulation draws, and large
samples of purchasers, before one can expect to obtain an accurate estimator whose
distribution is approximated well by a normal with finite variance.
Interestingly, this is not the case for the pure characteristic model discussed in the last
subsection. We will not provide the argument here but Berry, Linton, and Pakes (2004)
show that in that model both n and ns need only grow at rate J (and depending on the
pricing equilibrium, sometimes slower rates will do), for the normal limit distribution
to be appropriate. This gives the pure characteristic model a computational advantage
in calculating shares, though, as noted above, it is harder to compute the analogue
of the contraction mapping in (9) for the pure characteristics model, so it can still be
computationally demanding.
38
1.7 Concluding Remark.
The last decade has seen a rather dramatic change in the way I.O. researchers analyze
demand systems. There now is a reasonably substantial body of academic research using the new techniques, and it seems to indicate that, at least for many situations, they
allow us to get better approximations to substitution patterns and the likely demand for
new goods then had been possible previously. Perhaps not surprisingly then, the techniques have been picked up, to varying extents, by; the consulting community, various
government offices, and even by a part of the business community. On the other hand,
as we have tried to emphasize, there are empirically important issues and data sets that
the new techniques are not able to analyze – at least not without substantial further
developments. We welcome those developments. Moreover we hope that they will not
be judged by any absolute criteria but rather by the simple test of whether they allow
for improvements in our ability to empirically analyze one or more issue of substantive
interest.
2 Production Functions.
As noted in the introduction, the advent of new micro data sets on the inputs and
outputs from the production process has generated a renewed interest in the estimation
of production functions and their use in the analysis of productivity. We begin this
section by reviewing the basic simultaneity and selection issues that the recent literature
on production function estimation has faced. We then consider the traditional solutions
to these issues, pointing out why those solutions are not likely to be terribly helpful in
our context.
Next we introduce an approach based on explicit models of input choices and exit decisions that was first introduced in a paper by Olley and Pakes (1996). Our presentation
of the Olley-Pakes model will stress the assumptions they used which either we, or others
before us, see as questionable (at least in certain environments). These include assumptions on; the timing of input choices, the cost of changing the levels of different inputs
over time, the process by which productivity evolves over time, and the relationship of
investment to that process. The rest of the section focuses on ways of testing these
assumptions, and details recently proposed modifications to the estimation procedure
which might be used when they seem appropriate.
39
2.1 Basic Econometric Endogeneity Issues.
We can illustrate all issues that will concern us with simple Cobb Douglas production
technology
Yj = AjK
βk
j L
βl
j
with one output (Yj ) and two inputs; capital (Kj ) and labor (Lj ). Aj represents the
Hicksian neutral efficiency level of firm j, which is unobserved by the econometrician14.
Taking natural logs results in a linear equation
yj = β0 + βkkj + βllj + j (19)
where lowercase symbols represent natural logs of variables and ln(Aj ) = β0 + j . The
constant term β0 can be interpreted as the mean efficiency level across firms, while j
is the deviation from that mean for firm j. j might represent innate technology or
management differences between firms, measurement errors in output, or unobserved
sources of variance output caused by weather, machine breakdowns, labor problems, etc.
We have known since Marshak and Andrews (1944) that direct OLS estimation of (19)
is problematic. The problem is that the right hand side variables, capital and labor,
are generally chosen by the firm. If the firm has knowledge of its j (or some part of
j ) when making these input choices, the choices will likely be correlated with j . For
example, suppose that firms operate in perfectly competitive input and output markets
(wj , rj , and pj being the prices of labor, capital, and output respectively), that capital
is a fixed input, that firms perfectly observe j before choosing labor, and that firms’
current choices of labor only impact current profits and have no affect on future profits.
Then the firm’s optimal short-run choice of labor input is given by
Lj =

pj
wj
βle
β0+jK
βk
j
 1
1−βl
. (20)
14The methods discussed in this chapter are equally applicable to many other production functions.
As we shall see the major requirements will be that variable inputs have positive cross-partials with
productivity, and that the value of the firm is increasing in fixed inputs.
40
Since choice of Lj (and thus lj ) depends directly on j , OLS will generate biased coefficient estimates. In more general models, firms’ choices of Kj will also typically be
correlated with j
15
.
There is a second, less well documented, endogeneity problem often inherent in OLS
estimation of (19). Firm level datasets usually have a considerable level of attrition. For
example, over a wide range of manufacturing industries, Dunne, Roberts, and Samuelson
(1988) find exit rates higher than 30% between 5 year census pairs. In applied work,
one only has data on firms prior to exiting. If firms have some knowledge of j prior to
exiting, the firms that continue to produce will have j draws from a selected sample,
and the selection criteria will be partially determined by the other fixed inputs. Again
as a simple example, suppose that firms are monopolies that are exogenously endowed
with different fixed levels of capital. Firms then observe j , decide whether to exit or
not, and choose labor and produce if they have not exited. Also for simplicity suppose
that after production firms disappear, so that the firms have no dynamic considerations.
Firms in this situation will have an exit rule of the following form:
χ(j , Kj ; pj , wj , β) = 0 (or exit) iff Π(j , Kj ; pj , wj , β) < Ψ
where β is the set of parameters (β0, βl, βk) and Ψ is the non-negative selloff value of the
firm. Π is the argmax (over the variable input labor) of variable profits. This condition
states that firms exit if variable profits are not at least as high as the selloff value of the
firm16.
The key point is that this exit condition will generate correlation between j and Kj
conditional on being in the dataset (i.e. on not exiting). In the Cobb-Douglas case, both
j and Kj positively impact variable profits. As a result, selection will generate negative
correlation between j and Kj , since firms with higher Kj will be able to withstand lower
j without exiting. Thus, even if Kj is exogenous in the sense that it is uncorrelated
with j in the entire population of potentially active firms, selection can generate negative
correlation in one’s sample.
15Empirical results have lead practitioners to conclude that most often the bias imparted on the labor
coefficient α is larger than the bias imparted on the capital coefficient β. This is consistent with models
of input choice where labor is more easily adjustable than capital (i.e. labor is a “more variable” input
than capital). The intuition here is that because it is more quickly adjustable, labor is more highly
correlated with j .
16This is a very simple example of an exit rule. More realistic models of exit would be dynamic in
nature and distinguish between fixed and sunk costs; see the discussion below.
41
2.2 Traditional Solutions.
As is often the case, the two traditional solutions to these endogeneity problems are
instrumental variables and fixed effects. Before discussing these approaches, we make
two slight changes to our basic model. First, to explicitly consider the use of longitudinal
panel data, we index our variables by time t. Second, to be precise about where exactly
the endogeneity problems are coming from, we divide the unobservable jt into two
components, ωjt and ηjt, i.e.
yjt = β0 + βkkjt + βlljt + ωjt + ηjt. (21)
The ηjt here are intended to represent unobservables that are not observed (or predictable) by the firm before input and exit decisions at time t. As such, they will not be
correlated with these choices of inputs or exit behavior. On the other hand, we do allow
the possibility that ωjt is observed (or predictable) by firms when they choose inputs
and make exit decisions. Intuitively, ωjt might represent factors like managerial ability
at a firm, expected down-time due to machine breakdowns or strikes, or the expected
rainfall at a farm’s location. ηjt might represent deviations from expected breakdown
rates in a particular year or deviations from expected rainfall at a farm. Another valid
interpretation of ηjt is that it is classical measurement error in yjt that is uncorrelated
with inputs and exit decisions. The basic point here is that we have consolidated our
endogeneity problems into ωjt. ηjt is not a concern in that regard. We will often refer
to ωjt as the firms ”unobserved productivity”.
2.2.1 Instrumental Variables.
Instrumental variables approaches rely on finding appropriate instruments - variables
that are correlated with the endogenous explanatory variables but do not enter the
production function and are uncorrelated with the production function residuals. Fortunately, the economics of production suggests some natural instruments. Examining
input demand functions (such as (20)) suggests that input prices (rjt and wjt) directly
influence choices of inputs. In addition, these prices do not directly enter the production
function. The last necessary condition is that the input prices need to be uncorrelated
with ωjt. Whether this is the case depends on the competitive nature of the input markets that the firm is operating in. If input markets are perfectly competitive, then input
prices should be uncorrelated with ωjt since the firm has no impact on market prices.
This is the primary assumption necessary to validate input price instruments. Note why
42
things break down when firms have market power in input markets. If this is the case,
input prices will be a function of the quantity purchased inputs, which will generally
depend on ωjt.
17
While using input prices as instruments may make sense theoretically, the IV approach
has not been uniformly successful in practice. We believe there are at least four reasons
for this. First input prices are often not reported by firms, and when firms do report the
labor cost variable, i.e. wjt, is often reported in a way that makes it difficult to use. Labor
costs are typically reported as average wage per worker (or average wage per hour of
labor). Optimally, we would want this variable to measure differences in exogenous labor
market conditions faced by firms. Unfortunately, it may also pick up some component
of unmeasured worker quality. Suppose we as econometricians do not observe worker
quality, and that some firms employ higher quality workers than others. Presumably,
the firms with higher quality workers must pay higher average wages. The problem
here is that unobserved worker quality will enter the production function through the
unobservable ωjt. As a result, ωjt will likely be positively correlated with observed wages
wjt, invalidating use of wjt as an instrument.
Second, to use prices such as rjt and wjt as instruments requires econometrically helpful
variation in these variables. While input prices clearly change over time, such time
variation is not helpful when one wants to allow flexible effects of time in the production
function (e.g. allowing β0 to be a flexible function of t). One generally needs significant
variation in rjt and wjt across firms to identify production function coefficients. This
can be a problem as we often tend to think of input markets as being fairly national in
scope. One might not expect, for example, the price of capital or labor market conditions
to vary that much between states. Summarizing, to use the IV approach one 1) has to
observe significant variation in input prices across firms in the data, and 2) believe that
this variation is due primarily to differences in exogenous input market conditions, not
due to differences in unobserved input quality.
A third problem with IV is that it relies fairly strongly on an assumption that ωjt evolves
exogenously over time, i.e. firms do not choose an input that effects the evolution of
ωjt. Allowing ωjt to be affected by chosen inputs that we do not control for is very
problematic econometrically for the IV approach, for then it would be hard to imagine
finding valid instruments for observed input choices. One would need to find variables
that affect one input choice but that do not affect other input choices. In general this
17Another possible instrument is output prices, as long as the firm operates in competitive output
markets. These instruments have been used less frequently, presumably because input markets are
thought to be more likely to be competitive. Other related instruments are variables that shift either
the demand for output or the supply of inputs. While these types of instruments are typically harder
to come by, one can argue that they are valid regardless of the competitive nature of input or output
markets.
43
will be hard to do, since individual input choices typically depend on all input prices.
Finally, the IV approach only addresses endogeneity of input choice, not endogenous exit.
Endogenous exit will tend to invalidate the direct use of input prices as instruments. The
reason for this is that it is probable that the exit decision will be based in part on input
prices. For example, we might expect that firms who face higher input prices to be
more likely to exit (i.e. would exit at a higher ωjt). This is likely to generate positive
correlation between the instruments and the residuals in the the production function.
While direct application of IV in this situation is problematic, it is possible that one
could combine the population orthogonality assumptions with a selection model (e.g.
Gronau (1974), Heckman (1974, 1976, 1979)) to generate a consistent estimator of the
production function parameters.
2.2.2 Fixed Effects.
A second traditional approach to dealing with production function endogeneity issues is
fixed effects estimation. In fact, fixed effects estimators were introduced to economics
in the production function context (Hoch (1962), Mundlak (1961)). Fixed effects approaches make explicit use of firm panel data. The basic assumption behind fixed effects
estimation is that unobserved productivity ωjt is constant over time, i.e.
yjt = β0 + βkkjt + βlljt + ωj + ηjt. (22)
This allows one to consistently estimate production function parameters using either
mean differencing, first differencing, or least squares dummy variables estimation techniques. First differencing, for example, leads to
yjt − yjt−1 = βk(kjt − kjt−1) + βl(ljt − ljt−1) + (ηjt − ηjt−1). (23)
Given the assumption that the ηjt’s are uncorrelated with input choices ∀t,
18 this equa18The assumption that ηjt’s are uncorrelated with input choices (and possibly entry/exit decisions) at
all time periods t is often described as a ”strict” exogeneity assumption. One can often estimate these
fixed effects models under weaker, ”sequential” exogeneity assumptions, i.e. that ηjt’s are uncorrelated
with input choices at all time periods ≤ t. See Wooldridge (2002) for a discussion of these issues.
44
tion can be consistently estimated by OLS.19 Note that this approach simultaneously
solves the selection problem of endogenous exit, at least if exit decisions are determined
by the time invariant ωj (and not by the ηjt’s). While fixed effects approaches are fairly
straightforward and have certainly been used in practice, they have not been judged to
be all that successful at solving endogeneity problems in production functions either.
Again, there are a number of reasons why this may be the case.
First, it is clearly a strong assumption that ωj is constant over time. This is especially
true given the longer time frames for which panel data is now becoming available. In
addition, researchers are often interested in studying periods of data containing major
economic environmental changes (e.g. deregulation, privatization, trade policy changes,
. . . ). Typically these changes affect different firms’ productivities differently, and those
firms that the change impacts positively will be more likely to increase their inputs and
less likely to exit20.
A second potential problem with fixed effects estimators is that when there is measurement error in inputs, fixed effects can actually generate worse estimates than standard
level (OLS) estimators. Griliches and Hausman (1986) note that when inputs are more
serially correlated over time than is input measurement error, differencing can lower the
signal to noise ratio in the explanatory variables.21 This can generate higher biases in
fixed effects estimators than in OLS estimators, even if ωj is constant over time and
correlated with the explanatory variables22.
Lastly, fixed effects estimators simply have not performed well in practice. One often
gets unreasonably low estimates of capital coefficients23. Even one of the seminal papers,
Hoch (1962), for example, finds estimates of returns to scale around 0.6 - almost certainly
an unrealistically low number. Another empirical finding that appears to contradict the
fixed effect assumption concerns the comparison of fixed effects estimates on balanced
19 Note that generic OLS standard errors are wrong because the residuals will be correlated across
observations.
20 The restriction that ωj is constant over time is one that has been relaxed (in parametric ways) in
the dynamic panel data literature, e.g. Chamberlain (1984), Arellano and Bond (1991), Arellano and
Bover (1995), and Blundell and Bond (1999). For example, these methods can allow ωjt to be composed
of a fixed effect plus an AR(1) process.
21 By signal to noise ratio, Griliches and Hausman mean the variance in an observed explanatory
variable due to true variance in the variable, vs. variance in the observed explanatory variable due to
measurement error. This signal to noise ratio is inversely related to the bias induced by measurement
error.
22 Note that in this case (i.e. when there is measurement error in inputs), both fixed effects and OLS
estimators are biased. Also, note that the more structural approaches discussed later in this chapter are
likely also prone to this critique.
23 ”Unreasonable” is clearly not a completely precise statement here. We are referring to cases where
the estimated capital coefficient is considerably below capital’s cost share or where returns to scale are
extremely low.
45
panels (containing only observations for firms appearing throughout the sample) to those
on the full panel. As mentioned above, if ωj is constant over time, fixed effects estimation
completely addresses selection and input endogeneity problems. As a result, one should
obtain similar fixed effects estimates whether one uses the balanced sample or the full
sample. Olley and Pakes (1996), for example, finds very large differences in these two
estimates, suggesting that the fixed effects assumption is invalid. That said, whether
or not one takes fixed effects estimates as serious estimates of structural production
function parameters, the fixed effect decomposition of variation into within and between
components often provides a useful reduced form look at a dataset.
2.3 The Olley and Pakes (1996) Approach.
A recent paper by Olley and Pakes (1996) (henceforth OP) takes a different approach
to solving both the simultaneity and selection problems inherent in production function
estimation. Their empirical context is that of telecommunications equipment producers
(using data from the U.S. Census Bureau’s longitudinal research database). The basic
empirical goal is to measure the impact of deregulation and the breakup of AT&T on
measures of plant level productivity. Our focus is on the OP methodology for addressing
the endogeneity problems rather than the actual empirical results.
As we work through the OP approach, it is useful to keep in mind three types of assumptions that will be important in the approach. First there are assumptions on timing
and the dynamic nature of inputs. Timing refers to the point in time when inputs are
chosen by the firm relative to when they are utilized in production. “Dynamic nature”
refers to whether the input choices of the current period effect the cost of input use in
future periods; if it does not the input is labelled non-dynamic and if it does the input is
labelled as dynamic (and its current value becomes a “state variable” in the problem).
Second, there will be a scalar unobservable assumption. This assumption limits the dimensionality of the econometric unobservables that impact firm behavior. Third, there
will be a strict monotonicity assumption on the investment demand function - basically
that investment level is strictly monotonic in the scalar unobservable (at least for firms
whose investment level is strictly positive). We will see that this last assumption can
be generated by more basic assumptions on economic primitives. While some of these
assumptions can be relaxed in various ways, we delay that discussion until the next
subsection.
Lastly, note that we focus on how to use the OP methodology in practice. We do not
address the higher level technical aspects of the methodology, e.g. semi-parametric consistency proofs and alternative standard error derivations for their two-step estimators.
For discussion of these issues, e.g. see Pakes and Olley (1996) and the literature they cite.
46
One might also look at Wooldridge (2004), who presents a concise, one-step, formulation
of the OP approach for which standard error derivations are more straightforward24.
This one-step approach may also be more efficient than the standard OP methodology.
The rest of this section discusses in detail the workings of the OP methodology. We start
by describing a simple, bare bones, version of the model and methodology that ignores
potential selection problems. We then move on to the full OP model, which does address
selection. Lastly, we discuss caveats and extensions of the OP procedure.
2.3.1 The Model.
The OP approach considers firms operating through discrete time, making production
choices to maximize the present discounted value (PDV) of current and future profits.
The environment is as follows. First, the assumed production function is similar to (21),
with an additional input ajt
yjt = β0 + βkkjt + βaajt + βlljt + ωjt + ηjt (24)
the natural log of the age (in years) of a plant. The interest in the age coefficient stems
from a desire to separate out cohort from selection effects in determining the impact of
age of plant on productivity.
Second, unobserved productivity ωjt is assumed to follow an exogenous first order Markov
process. Formally,
p

ωjt+1| {ωjτ }
t
τ=0 , Ijt
= p (ωjt+1|ωjt) (25)
where Ijt is the firms entire information set at time t. This is simultaneously an econometric assumption on unobservables and an economic assumption on how firms form
their perceptions on (i.e. learn about) the evolution of their productivity over time.
Specifically, a firm in period t, having just observed ωjt, infers that the distribution of
ωjt+1 is given by p (ωjt+1|ωjt). Firms thus operate through time, realizing the value of
24Though Woolridge deals with input endogeneity, he does not explicitly consider the selection issue.
However similar ideas can be used when one needs to incorporate selection corrections.
47
ωjt at period t and forming expectations of future ωj ’s using p (ωjt+1|ωjt). Note that
this first-order Markov assumption encompasses the fixed effects assumption where ωjt
is fixed over time (i.e. ωjt = ωj ). OP also assume that p (ωjt+1|ωjt) is stochastically increasing in ωjt. Intuitively, this means that a firm with a higher ωjt today has a ”better”
distribution of ωjt+1 tomorrow (and in the more distant future). Lastly, note that the
ωjt process is assumed to be a time-homogeneous Markov process, i.e. p is not indexed
by t
25
.
Third, capital is assumed to be accumulated by firms through a deterministic dynamic
investment process, specifically
kjt = (1 − δ)kjt−1 + ijt−1
Here we will assume that ijt−1 is chosen by the firm at period t − 1. That is we are
assuming that the capital that the firm uses in period t was actually decided upon at
period t − 1; so it takes a full production period for new capital to be ordered, received,
and installed by firms26. Note that capital is a fixed (rather than variable) input.
Lastly, OP specify single period profits as
π(kjt, ajt, ωjt, ∆t) − c(ijt, ∆t).
Note that labor ljt is not explicitly in this profit function - the reason is that labor is
assumed to be a variable and non-dynamic input. It is variable in that (unlike capital),
ljt is chosen at period t, the period it actually gets used (and thus it can be a function
of ωjt ). It is non-dynamic in the sense that (again, unlike capital) current choice of
labor has no impact on the future (i.e. it is not a state variable). This non-dynamic
assumption rules out, for example, fixed hiring or firing costs of labor. We discuss
relaxing this assumption in section 2.4. For now π(kjt, ajt, ωjt, ∆t) can be interpreted as
a ”conditional” profit function - conditional on the optimal static choice of labor input.
25This assumption is not as strong as it might seem, as, e.g. one can easily allow average productivity
to vary across time by indexing β0 by t, i.e. β0t. The assumption can also be relaxed in some cases, i.e.
allowing pt (ωjt+1|ωjt) to be indexed by t.
26We note that there is a long literature on trying to determine the distributed lag which translates
investment expenditures into a productive capital stock (see for e.g. Pakes and Griliches (1984) and
the literature cited there), and one could incorporate different assumptions on this distributed lag into
the OP framework (for some examples see Ackerberg et. al (2005)). OP themselves also tried allowing
current investment to determine current capital, but settled on the specification used here.
48
Note also that both π(·) and c(·) depend on ∆t
, which represents the economic environment that firms face at a particular point in time. ∆t could capture input prices,
characteristics of the output market, or industry characteristics like the current distribution of the states of firms operating in the industry. The OP formulation allows all
these factors to change over time, although they are assumed constant across firms in a
given time period. Including market structure in the state space allows some of the competitive richness of the Markov-perfect dynamic oligopoly models of Ericson and Pakes
(1995)27.
Given this economic environment, a firm’s maximization problem can be described by
the following Bellman equation:
V (kjt, ajt, ωjt, ∆t) =
max 
Φ(kjt, ajt, ωjt, ∆t),
maxijt≥0 {π(kjt, ajt, ωjt, ∆t) − c(ijt, ∆t) + βE [V (kjt+1, ajt+1, ωjt+1, ∆t+1)|kjt, ajt, ωjt, ∆t, ijt]}

kjt, ajt and ωjt are sufficient to describe the firm specific component of the state space
because labor is not a dynamic variable and because (kjt, ajt, ωjt) (and the control iit )
are sufficient to describe firms perceived distributions over future (kjt+1, ajt+1, ωjt+1).
The Bellman equation explicitly considers two decisions of firms. First is the exit decision
- note that Φ(kjt, ajt, ωjt, ∆t) represents the sell off value of the firm. Second is the
investment decision ijt, which solves the inner maximization problem. Under appropriate
assumptions28, we can write the optimal exit decision rule as
χjt =

1 (continue)
0 (exit)
if ωjt ≥ ω(kjt, ajt, ∆t) = ωt(kjt, ajt)
otherwise,
(26)
27See Gowrisankaran (1995), Doraszelski and Satterthwaite (2004), and the third section of this chapter
for more discussion of such equilibria.
28 Other than assuming that an equilibria exists, the main assumption here is that the difference in
profits between continuing and exiting is increasing in ωjt. Given that ωjt positively affects current
profits and that the distribution p(ωjt+1|ωjt) is stochastically increasing in ωjt, the value of continuing
is clearly increasing in ωjt. Thus as long as Φ(kjt, ωjt, ∆t) either doesn’t depend on ωjt, decreases in
ωjt, or doesn’t increase too fast in ωjt, this will be satisfied. Note that to get the specific selection bias
discussed in Section 1.1 above (i.e. kjt negatively correlated with ωjt), we also need the difference in
returns between continuing and exiting to be increasing in kjt.
49
and the investment demand function as
ijt = i(kjt, ajt, ωjt, ∆t) = it(kjt, ajt, ωjt). (27)
Note the slight change in notation - we are now representing the dependence on ∆t
through the subscript t. See Pakes (1994) for a discussion of conditions under which
this investment demand function is strictly increasing in ωjt in the region where ijt >
0. That is, conditional on kjt and ajt, firms with higher ωjt optimally invest more.
This is an intuitive result - because p(ωjt+1|ωjt) is assumed stochastically increasing in
ωjt , ωjt positively impacts the distribution of all future ωjτ ’s. Since ωjτ ’s positively
impact the marginal product of capital in future periods τ , current investment demand
should increase. The importance of this strict monotonicity condition will be apparent
momentarily.
2.3.2 Controlling for Endogeneity of Input Choice.
Given the setup of the model, we can now proceed with the OP estimation strategy. We
first focus on dealing only with the endogeneity of input choice, i.e. we assume there are
no selection problems due to exit. We will also assume for now that investment levels
are always positive, i.e. ijt > 0, ∀(j, t). Later we will relax both these assumptions.
Given that (27) is strictly monotonic in ωjt, it can be inverted to generate
ωjt = ht(kjt, ajt, ijt) (28)
Intuitively, this says that conditional on a firm’s levels of kjt and ajt, its choice of
investment ijt ”tells” us what its ωjt must be. Note that the ability to ”invert” out ωjt
depends not only on the strict monotonicity in ωjt, but also the fact that ωjt is the only
unobservable in the investment equation.
This is the scalar unobservable assumption mentioned earlier. This, for example, means
that there can be no unobserved differences in investment prices across firms29, no other
29Recall that changes in the price of investment over time are permitted as they are picked up by the
function h through its dependence on t.
50
state variables that the econometrician does not observe, and no unobserved separate
factors that affect investment but not production. It also prohibits ωjt from following
higher than a first order Markov process.30 We discuss both tests for this assumption
and the possibilities for relaxing it in section 2.4.
Substituting (28) into the production function (24) gives
yjt = β0 + βkkjt + βaajt + βlljt + ht(kjt, ajt, ijt) + ηjt. (29)
The first stage of OP involves estimating (29) using semi-parametric methods that treat
the inverse investment function ht(kjt, ajt, ijt) non-parametrically. Note the advantages
of treating ht(kjt, ajt, ijt) non-parametrically. it(·) (and thus its inverse ht(·)) are complicated functions that depend on all the primitives of the model (e.g. demand functions,
the specification of sunk costs, the form of conduct in the industry, etc.). These functions are also solutions to a potentially very complicated dynamic game. The OP nonparametric approach therefore avoids both the necessity of specifying these primitives,
and the computational burden that would be necessary to formally compute ht(·).
Given the non-parametric treatment of ht(kjt, ajt, ijt), it is clear that β0, βk and βa
cannot be identified using (29). If, for example, ht(kjt, ajt, ijt) is treated as a polynomial
in kjt, ajt and ijt, the polynomial will be colinear with the constant, kjt, and ajt terms.
Thus, we combine these terms into φt(kjt, ajt, ijt), i.e.
yjt = βlljt + φt(kjt, ajt, ijt) + ηjt. (30)
Representing φt with a high order polynomial in kjt, ajt and ijt (an alternative would be
to use kernel methods, e.g. Robinson (1988)) and allowing a different φt for each time
period, OP estimate this equation to recover an estimate of the labor coefficient βb
l
. To
summarize this first stage, the scalar unobservable and monotonicity assumptions essentially allow us to ”observe” the unobserved ωjt - this eliminates the input endogeneity
problem in estimating the labor coefficient. Note that it is important here that labor is
assumed to be a non-dynamic input - if labor had dynamic implications, it would enter
the state space, and thus the investment function and φt. As a result, βl would not be
30If, for example, ωjt followed a second order process, both ωjt and ωjt−1 would enter the state space
and the investment decision. With two unobservables in the investment function, it would not be possible
to invert out ωjt in the current model.
51
identified in this first stage. Again, this is an assumption that can potentially be relaxed
- see section 2.4.
The second stage of OP identifies the capital and age coefficients βk and βa. First, note
that the first stage provides an estimate, φbjt, of the term
φt(kjt, ajt, ijt) = β0 + βkkjt + βaajt + ωjt.
If one uses a polynomial approximation to φt(kjt, ajt, ijt), φbjt is just the estimated sum
of the polynomial terms for a particular (kjt, ajt, ijt) pair. This means that given a
particular set of parameters (β0, βk, βa), we have an estimate of ωjt for all j and t
ωbjt(β0, βk, βa) = φbjt − β0 − βkkjt − βaajt. (31)
Next decompose ωjt into its conditional expectation given the information known by the
firm at t − 1 (denote this by Ijt−1) and a residual, i.e.
ωjt = E [ωjt|Ijt−1] + ξjt (32)
= E [ωjt|ωjt−1] + ξjt
= g(ωjt−1) + ξjt
for some function g. The second line follows from the assumption that ωjt follows an
exogenous first order Markov process. By construction, ξjt is uncorrelated with Ijt−1
One can think of ξjt as the innovation in the ω process between t − 1 and t that is
unexpected to firms. The important thing is that given the information structure of the
model, this innovation ξjt is by definition uncorrelated with kjt and ajt. The reason is
that kjt and ajt are functions of only the information set at time t−1 . Intuitively, since
kjt was actually decided on at time t − 1 (from the investment decision ijt−1), it cannot
be correlated with unexpected innovations in the ω process that occurred after t − 1.
Lastly, note that since the stochastic process generating ωjt has been assumed constant
over time, the g function need not be indexed by t.
31
31Were we to allow p (ωjt+1|ωjt) to vary across time, we would simply index g by t.
52
Next, consider rewriting the production function as:
yjt − βlljt = β0 + βkkjt + βaajt + ωjt + ηjt. (33)
Substituting in both (32) and (31) results in:
yjt − βlljt = β0 + βkkjt + βaajt + g(ωjt−1) + ξjt + ηjt (34)
= β0 + βkkjt + βaajt + g(φjt−1 − β0 − βkkjt−1 − βaajt−1) + ξjt + ηjt
= βkkjt + βaajt + ˜g(φjt−1 − βkkjt−1 − βaajt−1) + ξjt + ηjt (35)
where ˜g encompasses both occurrences of β0 in the previous line. The key point in (34)
is that, as argued above, the residual ξjt +ηjt is uncorrelated with all the right-hand side
variables.
We do not observe βl or φjt−1, but we do have estimates of them from the first stage.
Substituting βˆ
l and φˆ
jt−1 for their values in the equation above, and treating ˜g nonparametrically we obtain √
n consistent estimates of βk and βa. If one uses polynomials
to approximate ˜g, NLLS can be used for estimation32.
Alternatively one can adapt the suggestion in Wooldridge (2004) to combine both stages
into a single set of moments and estimate in one step. This should be more efficient than
the OP approach (as it uses the information in the covariances of the disturbances, and
any cross equation restrictions). The moment condition in this case is
E

ηjt ⊗
(ξjt + ηjt) ⊗
f1(kjt, ajt, ijt, ljt)
f2(kjt, ajt, kjt−1, ajt−1, ijt−1)

= 0
32An alternative way to construct a moment condition to estimate (34) is as follows (see Ackerberg,
Caves, and Frazer (2004)). Given βk and βa, construct ωbjt = φbjt − βkkjt − βaajt ∀t. Non-parametrically
regress ωbjt on ωbjt−1 to construct estimated residuals ξbjt (note that if using polynomial approximation,
this can be done using linear methods (since βk and βa are given)). Construct a moment condition
interacting ξbjt with kjt and ajt. Estimation then involves searching over (βk, βa) space to make this
moment close to zero.
53
where f1 and f2 are vector valued instrument functions, and ⊗ is the Kronecker product
operator. Appropriate choices for f1 and f2 lead to moments similar to those used by
OP. Note that there is a different set of conditioning variables for the moment in ηjt
than that in the moment for ξjt + ηjt ( since ljt can be correlated with ξjt)
33
.
2.3.3 Controlling for Endogenous Selection.
Next we relax the assumption that there is no endogenous exit. Firms now exit according
to the exit rule given in (26). A first important observation is that the first stage of the
OP procedure is not affected by selection. The reason is that by construction, ηjt, the
residual in the first stage equation (30), represents unobservables that are not observed
(or predictable) by the firm before input and exit decisions. Thus there is no selection
problem in estimating (30). Intuitively, the fact that in the first stage we are able to
completely proxy ωjt means that we can control for both endogenous input choice and
endogenous exit.
In contrast, the second stage estimation procedure is affected by endogenous exit. Examining (34), note that the residual contains not only ηjt, but ξjt. Since the firms exit
decision in period t depends directly on ωjt (see (26)), the exit decision will be correlated
with ξjt, a component of ωjt
34
.
We now correct for the selection. Starting from (33), take the expectation of both sides
conditional on both the information at t − 1 and on χjt = 1 (i.e. being in the dataset at
t). This results in
E [yjt − βlljt|Ijt−1, χjt = 1] = E [β0 + βkkjt + βaajt + ωjt + ηjt|Ijt−1, χjt = 1]
= β0 + βkkjt + βaajt + E [ωjt|Ijt−1, χjt = 1] (36)
33As Wooldridge notes, one can add further lags of variables to these instrument functions, increasing
the number of moments; though more lags will not be able to be used on the observations for the initial
years.
34This correlation relies on OP allowing firms to know the realization of ξjt before making the exit
decision. Otherwise exit would not cause a selection problem. The longer the time period between
observations the more serious the selection problem is likely to be. This point comes out clearly in OP’s
comparison of results based on their “balanced” panel (a data set constructed only from the observations
of plants that were active throughout the sample period), to results from their full panel (a panel which
keeps the observations on exiting firms until the year they exit and uses observations on new startups
from the year they enter). Selection seemed a far larger problem in the balanced than in the full panel.
54
The second line follows because 1) kjt and ajt are known at t − 1, and 2) ηjt is by
definition uncorrelated with either Ijt−1 or exit at t. Focusing on the last term, we have
E [ωjt|Ijt−1, χit = 1] = E [ωjt|Ijt−1, ωjt ≥ ωt(kjt, ajt)] (37)
=
Z ∞
ωt(kjt,ajt)
ωjt
p(ωjt|ωjt−1)
R ∞
ωt(kjt,ajt)
p(ωjt|ωjt−1)dωjt
dωjt (38)
= g(ωjt−1, ωt(kjt, ajt))
The first equality follows from the exit rule. The second and third equalities follows from
the exogenous first order Markov process assumption on the ωjt process.
While we do know ωjt−1 conditional on the parameters (from (31)), we do not directly
observe ωt(kjt, ajt). Modelling ωt(kjt, ajt) as a non-parametric function of kjt and ajt
might be a possibility, but this would hinder identification of βk and βa due to collinearity
problems. What we can do is try to control for ωt(kjt, ajt) using data on observed exit.
Recall that our exit rule is given by
χjt =

1 (continue)
0 (exit) according as ωjt
≥
< ωt(kjt, ajt). (39)
This means that the probability of being in the data (at period t) conditional on the
information known at t − 1 is:
Pr(χjt = 1|Ijt−1) = Pr(ωjt ≥ ωt(kjt, ajt)|Ijt−1) (40)
= Pr(χjt = 1|ωjt−1, ωt(kjt, ajt)) = eϕet(ωjt−1, ωt(kjt, ajt))
= ϕet(ωjt−1, kjt, ajt) = ϕt(ijt−1, kjt−1, ajt−1) = Pjt
The second to last equality holds because of (28), and the fact that kjt and ajt are
deterministic functions of ijt−1, kjt−1, and ajt−1.
Equation (40) can be estimated non-parametrically, i.e. modelling the probability of
surviving in t as a non-parametric function of ijt−1, kjt−1, and ajt−1. OP do this
55
in two alternative ways - first using a probit model with a 4th order polynomial in
(ijt−1, kjt−1, ajt−1) as the latent index, second using kernel methods. For a plant characterized by (ijt−1, kjt−1, ajt−1), these estimates allow us to generate a consistent estimate
of the probability of the plant surviving to period t (Pbjt).
Next, note that as long as the density of ωjt given ωjt−1 is positive in an area around
ωt(kjt, ajt), (40) can be inverted to write ωt(kjt, ajt) as a function of ωjt−1 and Pjt
35, i.e.
ωt(kjt, ajt) = f(ωjt−1, Pjt). (41)
Substituting (41) into (37) and (36), and using (31) gives us
E [yjt − βlljt|Ijt−1, χjt = 1] = β0 + βkkjt + βaajt + g(ωjt−1, f(ωjt−1, Pjt) (42)
= β0 + βkkjt + βaajt + g
0
(ωjt−1, Pjt)
= β0 + βkkjt + βaajt + g
0
(φjt−1 − β0 − βkkjt−1 − βaajt−1, Pjt)
This is similar to (36), only differing in the additional Pjt term in the non-parametric
g
0
function. Pjt controls for the impact of selection on the expectation of ωjt - i.e. firms
with lower survival probabilities who do in fact survive to t likely have higher ωjt’s than
those with higher survival probabilities.
Equation (42) implies that we can write
yjt − βlljt = β0 + βkkjt + βaajt + g
0
(φjt−1 − β0 − βkkjt−1 − βaajt−1, Pjt) + ζjt
= βkkjt + βaajt + ge(φjt−1 − βkkjt−1 − βaajt−1, Pjt) + ζjt + ηjt (43)
where, as in (34), the two β0 terms have been encompassed into the non-parametric function ge. By construction the residual in this equation satisfies E [ζjt + ηjt|Ijt−1, χjt = 1] =
35Formally, (40) implies that Pjt = eϕet
(ωjt−1, ωjt). With positive density of ωjt around ωjt,
eϕet
is
strictly monotonic in ωjt, so this can be inverted to generate (41).
56
0. Substituting Pˆ
jt, φˆjt and βˆ
l
for Pjt, φjt and βl, (43) can also be estimated with NLLS,
approximating ge with either a polynomial or a kernel 36.
In this estimation procedure information on βk and βa is obtained by comparing labor
productivities of firms with the same ωjt−1 and Pjt but different kjt and ajt. In addition,
since the functions ϕt(·) and Pt(·) vary across t with changes in industry conditions
(while g(·) is assumed constant over time), it also uses information from variation in
inputs across firms in different time periods that have the same ωjt−1 and Pjt.
In the selection literature, Pbjt is referred to as the propensity score - for discussion of
these techniques, see, e.g. Heckman (1974, 1978, 1979), Rosenbaum and Rubin (1983),
Heckman and Robb (1985), and Ahn and Powell (1993). An important difference between
OP and this selection literature is that controlling for the propensity score is not sufficient
for OP’s model; they require a control for both ωjt−1 and for ωjt−1.
A number of recent papers have applied the OP procedure successfully. As an example
consider Table 3, which displays the results from the food processing industry in Pavcnik
(2002) - this is the first out of the eight industries in her table 2. Comparing the OLS to
the OP estimates, we see the changes that we should expect. Returns to scale decrease
(consistent with positive correlation between unobserved productivity and input use),
with the coefficients on the more variable inputs accounting for all of the fall (consistent
with this correlation being more pronounced for the variable inputs). Consistent with
selection, the capital coefficient rises moving from OLS to OP. The fixed effects estimates
are the most difficult to understand, as they generate a coefficient for capital near zero,
and an estimate of economies of scale below .9. These results are indicative of those for
the other industries in Pavcnik’s table 2. The average of the returns to scale estimate
across industries when estimated by OLS is 1.13, when estimated by OP it is 1.09, and
when estimated by Fixed Effects it is .87. The average of the capital coefficients across
industries from OLS is .066, from OP .085, and from Fixed Effects only .021 (with two
industries generating negative capital coefficients).
Olley and Pakes (1995) themselves compare their estimates to estimates obtained using
OLS and fixed effect on both a balanced panel (a panel constructed only from firms that
were operating during the entire fifteen year sample period) and from the full sample
(constructed by keeping firms that eventually exit until the year prior to their exit and
introducing new entrants as they appear). The difference between the balanced sample
estimators and OP estimators on the full sample are truly dramatic, and those between
the OLS and fixed effect estimators on the full sample and the OP estimators are similar
to those reported above (though somewhat larger in absolute value). In both papers, the
OP estimator generates standard errors for the labor coefficient that are not too different
36OP try both the kernel and a polynomial with only minor differences in results.
57
Table 3: Production Function Estimates from Pavcnik (2002)∗
OLS Fixed Effects Olley-Pakes
Unskilled Labor 0.178 0.210 0.153
(0.006) (0.010) (0.007)
Skilled Labor 0.131 0.029 0.098
(0.006) (0.007) (0.009)
Materials 0.763 0.646 0.735
(0.004) (0.007) (0.008)
Capital 0.052 0.014 0.079
(0.003) (0.006) (0.034)
∗ From Pavcnik, N. (2002) ”Trade Liberalization, Exit, and Productivity Improvements:
Evidence from Chilean Plants,” The Review of Economic Studies 69, January, pp. 245-76
then those generated by OLS, but, as might be expected, standard errors for the capital
coefficient do increase (though much less so in the OP results than in Pavcnik’s).
2.3.4 Zero Investment Levels.
For simplicity, we assumed above that investment levels for all observations were nonzero. This allowed us to assume that the investment equation was strictly monotonic
in ωjt everywhere (and hence could be inverted to recover ωit for every observation).
Observations with zero investment call into question the strict monotonicity assumption.
However, the OP procedure actually only requires investment to be strictly monotonic in
ωjt for a known subset of the data. OP themselves take that subset to be all observations
with it > 0, i.e. they simply do not use the observations where investment equals 0.
Even with this selected sample, first stage estimation of (29) is consistent. Since ωjt
is being completely proxied for, the only unobservable is ηjt, which is by assumption
uncorrelated with labor input and with the selection condition iit > 0. Second stage
estimation of (43) is also consistent when OP discard the data where ijt−1 = 0 (φbjt−1 −
β0 − βkkjt−1 − βaajt−1 is not computable when ijt−1 = 0). The reason is that the
error term in (43) is by construction uncorrelated with the information set Ijt−1, which
contains the investment level ijt−1. In other words, conditioning on ijt−1 = 0 doesn’t
say anything about the unobservable ζjt.
While the OP procedure can accommodate zero investment levels, this accommodation is
58
not without costs. In particular, there is likely to be an efficiency loss from discarding the
subset of data where ijt > 0. Levinsohn and Petrin (2003) (henceforth LP) suggest an
alternative estimation routine whose primary motivation is to eliminate this efficiency
loss. They start by noting that in many datasets, particularly those from developing
countries, the set of observations with zero investment can be quite large. For example, in
LP’s dataset on Chilean plants more than 50% of the observations have zero investment
(note that in OP’s U.S. plant data, this proportion is much less, ≈ 8%). To avoid a
potentially large efficiency loss, LP suggest using variables other than investment to
proxy for the unobserved ωjt. In particular LP focus on firms’ choices of intermediate
inputs (e.g. electricity, fuels, and/or materials) - these are rarely zero.37
Consider the production function
yjt = β0 + βkkjt + βlljt + βmmjt + ωjt + ηjt (44)
with additional input mjt (e.g. materials). LP assume that like labor, mjt is a variable (i.e. chosen at t), non-dynamic input, and consider the following material demand
equation
mjt = mt(kjt, ωjt). (45)
As with the OP investment equation, the demand equation is indexed by t to allow,
e.g. input prices, market structure, and demand conditions to vary across time38. LP
state conditions under which this demand equation is monotonic in ωjt. Given this
monotonicity, estimation proceeds analogously to OP. First, (45) is inverted to give:
ωjt = ht(kjt, mjt). (46)
37 An alternative to LP might be to augment the original OP procedure with a more complete model
of investment and/or distributional assumptions on ω, allowing one to utilize the zero investment observations.
38 Given that materials are a static choice (in contrast to dynamic investment), one might be more
willing to make parametric assumptions on this input demand function (since it depends on fewer primitives, e.g. it does not depend on expectations about the future). However, there are caveats of such an
approach, see section 2.4.1
59
Next, (46) is substituted into (44) to give:
yjt = β0 + βkkjt + βlljt + βmmjt + ht(kjt, mjt) + ηjt (47)
Treating the ht function non-parametrically results in the following estimating equation
yjt = βlljt + φt(kjt, mjt) + ηjt (48)
where βk and βm are not separately identified from the non-parametric term. As in OP,
the first stage of LP involves estimating (48) to obtain βb
l and φbjt. The second stage of
LP again proceeds following OP, the main difference being that the parameter on the
intermediate input, βm, still needs to be estimated. Moving the labor term to the left
hand side and using (32) gives:39
yejt = βkkjt + βmmjt + ge(φjt−1 − βkkjt−1 − βmmjt−1) + ξjt + ζjt, (49)
and nonparametric estimates of φjt and of ˜g(·) are used in estimation.
Note that since kjt is assumed decided at t − 1, it is orthogonal to the residual, ξjt + ηjt.
However, since mjt is a variable input, it is clearly not orthogonal to ζjt, the innovation
component of ωjt. LP address this by using mjt−1 as an instrument for mjt in estimation
of (49). In their application LP find biases that are generally consistent with those
predicted by OP, but some differences in actual magnitudes of coefficients.
2.4 Extensions and Discussion of OP.
The OP model was designed to produce estimates of production function coefficients
which are not subject to biases due to simultaneity and selection problems generated
by the endogeneity of input demands and exit decisions. We begin this section with a
39While the LP procedure does not formally address selection, they note that their procedure could
be extended to control for it in the same way as OP.
60
test of whether the coefficient estimates obtained using OP’s assumptions are robust to
different sources of misspecification.
There are a variety of reasons why this test could fail and the rest of this subsection considers some of the more likely candidates. Each time a source of possible misspecification
in OP’s assumption is introduced, we consider modifications to their estimation techniques which produce consistent estimates of production function coefficients under that
misspecification. This is in keeping with our belief that different modifications are likely
to be appropriate for different industries and data sets. Though the extended models
may well be of more general interest, as they typically will produce richer dynamics with
more detailed policy implications, we limit ourselves to considering their implications for
estimating production function coefficients.
In this context we first investigate relaxing assumptions on the dynamic implications of
inputs (e.g. that labor choices today have no dynamic implications) and on the timing
of input choices. We then investigate the potential for relaxing the scalar unobservable
assumptions of OP. Most of the discussion regarding the timing and dynamic implications of inputs is based on Ackerberg, Caves, and Fraser (2004) (ACF) (also see Buettner
(2004a) for some related ideas), while much of the discussion on non-scalar unobservables is taken from Ackerberg and Pakes (2005). We also briefly discuss two recent
contributions by Buettner (2004b) and Greenstreet (2005).
2.4.1 A Test of Olley and Pakes’ Assumptions.
This subsection combines results from section 4.1 in OP with results from ACF. Broadly
speaking, there are two questionable implications of the assumptions used in OP that are
central to their estimation strategy. First there is the implication that, conditional on
capital and age, there is a one to one mapping between investment and productivity (we
give reasons for doubting this implication below). Second there is the direct assumption
that the choice of labor has no dynamic implications; i.e. that labor is not a state variable
in the dynamic problem.
Focusing on the second assumption first, assume instead that there are significant hiring
or firing costs for labor, or that labor contracts are long term (as in, for example, unionized industries). In these cases, current labor input choices have dynamic implications,
labor becomes a state variable in the dynamic problem, and equation (28) becomes
ωjt = ht(kjt, ljt, ajt, ijt). (50)
Now the labor coefficient will not be identified in the first stage; i.e. from equation (34)
61
- the first stage can not separate out the impact of labor on production, or βl
, from its
impact on the h(·) function.
ACF point out that under these assumptions βl can still be identified from the second
stage. To see this note that the second stage is now
yjt = βlljt + βkkjt + βaajt + ge(φjt−1 − βlljt−1 − βkkjt−1 − βaajt−1) + ξjt + ηjt. (51)
After substituting φbjt−1 for φjt−1, we can estimate the production function parameters
using a semiparametric GMM procedure related to the above. Note, however, that
if we maintain the rest of OP’s assumptions, then ljt differs from kjt in that labor
can adjust to within period variation in productivity. This implies that unlike kjt, ljt
can be correlated with ξjt. As a result we need to use an “instrument” for ljt when
estimating equation (51). A fairly obvious instrument is ljt−1. Since ljt−1 was decided
on at t − 1, it is uncorrelated with ξjt, and ljt and ljt−1 are typically highly correlated.
With this modification, estimation can proceed as before using, say, a polynomial or
kernel approximation to ge.
Note that even though the first stage does not directly identify any of the parameters of
the model in this procedure, we still need the first stage to generate estimates of φbjt−1.
Indeed we still need (an extended version) of the assumptions that generates the first
stage equation. Before we needed the assumption that conditional on values for (kjt, ajt)
there was a one to one map between productivity and investment. Now we need the
assumption that conditional on values of (kjt, ajt, ljt) there is a one to one map between
productivity and investment.
In fact equation (51) is closely related to the test for the inversion proposed in Olley and
Pakes (1995). Recall that they assume that labor is not a dynamic input. In that case
when they subtract their first stage estimate βˆ
l times l from both sides of their second
stage equation they obtain
yjt −βˆ
l
ljt = (βl −βˆ
l)ljt +βkkjt +βaajt +ge(φˆ
jt−1 −βkkjt−1 −βaajt−1) +ξjt +ηjt, (52)
which is an equation with over identifying restrictions40.
In particular the term (βl−βˆ
l)l in equation (52) should be zero if the inversion which lead
to the estimate of the labor coefficient was a good approximation to reality. Further the
inversion implies that what we must subtract from our estimate of φjt−1 to obtain lagged
productivity is determined by the contribution of (kjt−1, ajt−1) to production of yjt−1,
40We have omitted a term that results from substituting φˆjt−1 for the true φjt−1 in this equation.
The additional term’s impact on the parameter estimates is op(1/
√
J), and so does not effect their limit
distributions.
62
i.e. by (βk, βa). These coefficients also determine the contribution of (kjt, ajt) to yjt given
ωjt−1. If our inversion were seriously in error we would expect that φjt−1 − βkkjt−1 −
βaajt−1 would not be perfectly correlated with ωjt−1 and as a result there would be a
residual component of productivity we are not controlling for. Provided there was an
endogeneity problem in the first place, this residual should be correlated with (kjt, ajt).
Thus OP allow the coefficients of (kjt, ajt) to differ from those of (kjt, ajt) in equation
(52), use (lj,t−1, kjt, ajt) and powers and lags of these variables as instruments, and then
test whether; βˆ
l − βl = 0, and whether the coefficients on the current lagged values of k
and a are equal41.
As noted when the current labor choice has dynamic implications the first stage estimate of βl obtained by OP is inconsistent (regardless of whether the inversion is correct).
However even if labor is dynamic equation (51) does still generate over identifying restrictions; the existence of the inversion implies that the current and lagged values of
(l, k, a) should enter in this equation with the same factors of proportionality. I.e. if the
inversion is correct then what we must subtract from our estimate of φjt−1 to obtain
lagged productivity is determined by the contribution of (ljt−1, kjt−1, ajt−1) to production of yjt−1, i.e. by (βl
.βk, βa). These coefficients also determine the contribution of
(ljt, kjt, ajt) to yjt given ωjt−1. That is if we were to estimate
yjt = β
∗
l
ljt + β
∗
kkjt + β
∗
aajt + ge(φjt−1 − βl
ljt−1 − βkkjt−1 − βaajt−1) + ξjt + ηjt, (53)
and then test whether (β
∗
l
, β∗
k
, β∗
a
) = (βk, βl, βa), with a large enough data set we should
reject the null of equality if assumptions which underlie the inversion are wrong. Given
the additional parameters one will need additional instruments to estimate this specification. Natural instruments would be those used in OP, i.e. (lj,t−1, kjt, ajt) and powers
and lags of these variables.
Two other points about the test. First, the OP test conditions on the fact that labor
is variable (i.e. it is not a state variable) and endogenous (current labor is correlated
with ξjt), and then tests whether the inversion is a good approximation. We could have
proceeded by conditioning on the inversion and then test one or both of the assumptions
that; labor is dynamic and/or labor choices are fixed prior to the realization of ξt. We
would do this by estimating the two equations in ACF simultaneously and then imposing
constraints. The constraint to be tested in asking whether labor can be treated as a
non-dynamic (or variable) input is whether φ(ljt, ijt, kjt, ajt) = βlljt + φ(ijt, kjt, ajt). To
test whether labor is endogenous in the sense that it can react to ξjt we estimate the
system once using ljt−1 as an instrument for ljt in equation (51) and once using ljt as an
instrument on itself. Exactly what it makes sense to condition on (and what to test for)
is likely to depend on the characteristics of the industry being studied. Alternatively we
41Note that we can not use both current and lagged values of ajt as instruments for the two are
collinear. We could, however, use different functions of ajt as additional instruments.
63
could improve the power of the omnibus test in equation (51) by estimating the first stage
in ACF simultaneously with this equation and then asking (β
∗
l
, β∗
k
, β∗
a
) = (βl, βk, βa). If
that is accepted we could then try the additional (nested) constraints implied by an
assumption that labor is not endogeous42.
Finally a word of caution on the usefulness of these tests. First we have made no attempt
to look at the power of these tests. Though OP find very precise estimates of differences
in coefficients from (53), their data seems to deliver more precise estimates than many
other data sets (see, for e.g. ACF). Second it is important to realize that the test that
(β
∗
l
, β∗
k
, β∗
a
) = (βk, βl, βa) is designed to ask the limited question of whether making
our approximations greatly hinders our ability to obtain reasonable production function
coefficient. As a result we are using the difference in these coefficients, normalized by
the variance-covariance of those differences, as our metric for “reasonableness”. There
are other metrics possible, one of which would be to have some prior knowledge of the
characteristics of the industry the researcher is working on (and we expect these results
to vary by industry). Further there may well be independent reasons for interest in the
timing of input decisions or in our invertibility assumption (see the discussion below),
and a test result that our approximations do not do terrible harm to production function
estimates does not imply that they would do little harm in the analysis of other issues
(for e.g. in the analysis of the response of labor hiring to a change in demand, or in the
response of investment to an infrastructure change which increases productivity).
2.4.2 Relaxing Assumptions on Inputs.
This subsection assumes that there is an inversion from productivity to investment conditional on the state variables of the problem, and investigates questions regarding the
nature of the input demands given this inversion. ACF note that there are two dimensions along which we can classify inputs in this context, and the two dimensions have
different implications for the properties of alternative estimators. First inputs can either
be variable (correlated with ξjt) or fixed (uncorrelated with ξjt). Second the inputs can
either be dynamic, i.e. be state variables in the dynamic problem and hence conditioned
on in the relationship between productivity and investment, or static. So if we generalize,
42Note that both these ideas; that one can allow labor to have dynamic effects, and that some of the
assumptions behind these procedures are testable - are related to the the dynamic panel literature cited
above (e.g. Arellano and Bond (1991), Arellano and Bover (1995), and Blundell and Bond (1999)) in
that further lags of inputs are typically used as instruments. If one were willing to assume that the ηj,t
are independently distributed across time then the residuals should be uncorrelated with past values of
output also. However if ηjt represented serially correlated measurement error in the observations on yt
then the ηjt may be serially correlated, and we could not expect a zero correlation between past output
and the disturbance from (53). ACF flesh out the distinction between their methods and the dynamic
panel literature further.
64
and allow for inputs of each of the four implicit types we have
yjt = β
vsXvs
jt + β
vdXvd
jt + β
fsX
fs
jt + β
f dX
f d
jt + ωjt + ηjt (54)
where the input acronyms correspond to these dimensions, e.g. Xvs
jt represent variable,
non-dynamic inputs, while X
f d
jt represent fixed, dynamic inputs, and so on.
The various coefficients can be identified in different ways. β
vs, like labor in the original
OP framework, can be identified either in the first stage, or in the second stage using
Xvs
jt−1
as an instrument (because Xvs
jt is variable and thus potentially correlated with ξjt,
it cannot be used as an instrument in the second stage). β
f d
, like capital in the original
OP framework, cannot be identified in the first stage, but it can be identified in the
second stage using either X
f d
jt or X
f d
jt−1
(or both) as instruments. β
vd, the coefficients
on the inputs that are variable and dynamic, also cannot be identified in the first stage,
but can be identified in the second stage using Xvd
jt−1
as an instrument. Lastly, β
fs can
be identified either in the first stage or in the second stage using either X
fs
jt or X
fs
jt−1
(or
both) as instruments.
Note also that if we have any static or fixed inputs we have over identifying restrictions.43
This over identification can potentially be useful in testing some of the timing assumptions. For example, suppose one starts by treating capital as a fixed, dynamic input.
One could then estimate the second stage using both kit−1 and kit as instruments, an
over identified model. In the GMM context, one could test this over identification with
a J-test (Hansen(1982)). Since kit is a valid instrument only when capital is truly fixed
(yet kit−1 is a valid instrument regardless) rejection of the specification might be interpreted as evidence that capital is not a completely fixed input. Consistent estimation
could then proceed using only kit−1 as an instrument. Again, the Wooldridge (2004)
framework makes combining these multiple sources of identification and/or testing very
convenient.
ACF also look deeper into the various assumptions on inputs. They note that, under the
assumption that ljt is a variable input, for it to have the independent variance needed
to estimate our first stage equation (30), there must be a variable, say zjt, that impacts
firms’ choices of ljt but that does not impact choices of investment at t. This variable
zjt must also have some variance that is independent of ωjt and kjt. If this were not the
43In all these cases, further lags (prior to t − 1) of the inputs can be used as instruments and thus as
over identifying restrictions, although it is not clear how much extra information is in these additional
moment conditions, and one will not be able to use these additional lags in the initial time periods.
65
case, e.g. if ljt = ft(ωjt, kjt), then one can show that ljt is perfectly collinear with the
nonparametric function in equation (30), implying that one cannot estimate βlfrom that
equation. Note that the variable zjt does not need to be observed by the econometrician.
Thus, to proceed as OP do we need the demand function for labor to be
ljt = ft(ωjt, kjt, zjt)
where zjt are additional factors that affect demand for labor (or more generally, demand
for the variable inputs) with non-zero conditional variance (conditional on ωjt, kjt). Note
that the zjt cannot be serially correlated. If this were the case, then zjt would become
part of the state space, influence ijt, and one would not be able to do the inversion44
Even with this restriction, there are at least two possible zjt’s here; i.i.d. firm specific
input price shocks, and i.i.d. random draws to the environment that cause differences
in the variance of ηjt over time (since the profit function is a convex function of η
the variance in this variable will affect labor demand). The latter could be associated
with upcoming union negotiations, the likelihood of machine break downs due to age of
equipment, or the approach of maintenance periods. One problem with the i.i.d. input
price shock story is that it is somewhat at odds with the assumptions that all other
components of prices are constant across firms and that the other unobservables (ωjt) in
the model are serially correlated over time.
ACF provide two additional ways of overcoming this problem. First they note that if
one weakens OP’s timing assumptions slightly, one can still identify ljt in the first stage.
Their observation also reopens an avenue of research on the timing of input decisions
which dates back at least to Nadiri and Rosen (1974). Suppose that ljt is actually not
a perfectly variable input, and is chosen at some point in time between periods t − 1
and the completion of period t. Denote this point in time as t − b, where 0 < b < 1.
Suppose that ω evolves between the subperiods t − 1, t − b, and t according to a first
order Markov process, i.e.
p (ωjt|Ijt−b) = p (ωjt|ωjt−b), and p (ωjt−b|Ijt−1) = p (ωjt−b|ωjt−1).
In this case, labor input is not a function of ωjt, but of ωjt−b, i.e.
44Note also that observing zjt would not help in this serially correlated case. While one would now be
able to do the inversion, zjt would enter the non-parametric function, again generating perfect collinearity.
66
ljt = ft(ωjt−b, kjt).
Since ωjt−b cannot generally be written as a function of kjt, ajt, and ijt, ljt will not
generally be collinear with the non-parametric term in (30), allowing the equation to
be identified. The movement of ω between t − b and t is what breaks the collinearity
problem between ljt and the non-parametric function. The second alternative suggested
by ACF avoids this collinearity problem by abandoning the first stage identification of
the labor coefficient. Instead, they suggest identifying the labor coefficient in the second
stage using ljt−1 as an instrument.
Importantly, ACF argue that this collinearity problem is more severe when using the
LP procedure. They contend that it is considerably harder to tell a believable story
in which the assumptions of LP hold and where ljt varies independently of the nonparametric function in (48). The reason for this is that it is hard to think of a variable
zjt that would affect a firms’ labor choices but not their material input choices (either
directly or indirectly through the labor choice).45 ACF suggest a couple of procedures
as alternatives to LP.46 The first, based on the discussion above, again involves simply
identifying the labor coefficient in the second stage. This can be done using either ljt
or ljt−1 to form an orthogonality condition, depending on what one wants to assume
about the timing of the labor choice. Moreover, it can also be done in a manner that
is also consistent with labor having dynamic effects. The second procedure is more
complicated and involves sequentially inverting the value of ωjt at each point in time at
which inputs are chosen. While this procedure depends on independence (rather than
mean independence) assumptions on innovations in ωjt, it has the added advantage of
allowing one to infer something about the point in time that labor is chosen. Bond
and Soderbom (2005) make a somewhat related point regarding collinearity. They argue
that in a Cobb-Douglas context where input prices are constant across firms, it is hard
if not impossible to identify coefficients on inputs that are perfectly variable and have
no dynamic effects. This is important for thinking about identification of coefficients on
Xvs
jt in the above formulation.
45ACF note that one probably will not observe this perfect collinearity problem in practice (in the
sense that the first stage procedure will actually produce an “estimate”). However, they point out that
unless one is willing to make what they argue are extremely strong and unintuitive assumptions, the lack
of perfect collinearity in practice must come entirely from misspecification in the LP model.
46An alternative approach to dealing with these collinearity problems might be to model the input
demand functions (investment or materials) parametrically. If g() is parametric, one doesn’t necessarily
have this collinearity problem. However, at least in the LP situation this does not guarantee identification.
ACF show that in the Cobb-Douglas case, substituting in the implied parametric version of the material
input function leads to an equation that cannot identify the labor coefficient.
67
2.4.3 Relaxing the Scalar Unobservable Assumption.
The assumption of a scalar unobserved state variable is another aspect of the OP approach that might be a source of concern. We begin with three reasons for worrying
about this assumption and then provide a way of modifying the model to account for
each of them. In each case we bring information on additional observables to bear on
the problem. As a result, one way of looking at this section is as a set of robustness tests
conducted by asking whether the additional observables affect the results.
Our three concerns in order of increasing difficulty are as follows. First productivity
itself is a complex functions of many factors, and it may not be appropriate to assume
that one can represent it as a first order Markov process. Second investment might well
respond to demand factors that are independent of the firm’s productivity. Then there is
no longer a one to one mapping between investment and productivity given capital and
age. Consequently we can not do the inversion in equation (28) underlying the first stage
of the OP procedure. Finally, at least in some industries we often think of two sources
of increments in productivity, one that results from the firm’s own research investments,
and one whose increments do not depend on the firm’s behavior. A process formed from
the sum of two different first order Markov processes is not generally a first order Markov
process, and if one of those processes is “controlled” it may well be difficult to account
for it in the same way as we can control for exogenous Markov processes.
First assume that productivity follows a second-order (rather than first-order) Markov
process. This changes the investment demand equation to
ijt = it(kjt, ajt, ωjt, ωjt−1). (55)
Since there are the two unobservables (ωjt, ωjt−1) the investment equation cannot be
inverted to obtain ωjt as a function of observables, and the argument underlying the
first stage of the OP process is no longer valid.
One possible solution to the estimation problem is through a second observed control of
the firm. Suppose, for example, one observes firms’ expenditures on another investment
(advertising, expenditure on a distributor or repair network), say sit
47. Then we have
the bivariate policy function

ijt
sjt 
= Υt(kjt, ajt, ωjt, ωjt−1).
If the bivariate function Υt ≡ (Υ1,t, Υ2,t) is a bijection in (ωjt, ωjt−1) (i.e. it is onto),
47One can modify this argument to allow also for a second state variable, the stock of advertising or
the size of the repair network, provided that stock is known up to a parameter to be estimated.
68
then it can be inverted in ωjt to obtain:
ωjt = Υ−1
t
(kjt, ajt, ijt, sjt).
Given this assumption the first stage proceeds as in OP, except with a higher dimensional
non-parametric function to account for current productivity (it is a function of sjt as
well as (kjt, ajt, ijt)).
OP’s second stage is modified to be
yejt = βkkjt +βaajt +ge(φbjt−1 −βkkjt−1 −βaajt−1, φbjt−2 −βkkjt−2 −βaajt−2)+ξjt +ηjt,
where yejt = yjt − βl ˆ
jt and the φbjt variables are obtained from the first stage estimates
at t − 1 and t − 2. Note that since the conditional expectation of ωjt given Ijt−1 now
depends on ωjt−2 as well as ωjt−1, we need to use estimates of φ from two prior periods.
The extension to control for selection as well is straightforward. Moreover, provided the
number of observed control variables is at least equal to the order of the Markov process,
higher order Markov processes can be handled in the same way.
We now move on to allow investment to depend on an unobservable demand shock that
varies across firms, in addition to the (now first-order) ωjt process. Suppose that the
demand shock, µjt, also follows a first-order Markov process that is independent of the
ωjt process. Then the investment function will be a function of both unobservables, or
ijt = it(kjt, ajt, ωjt, µjt). Again we will assume the existence of a second control and use
it to allow us to substitute for ωjt in the first stage of OP’s procedure.
More precisely, assume we also observe the firms’ pricing decisions, pjt. At the risk of
some notational confusion, again let the bivariate policy function determining (ijt, pjt)
be labelled Υ(·), and assume it is a bijection in (ωjt, µjt) conditional on (kjt, ajt). Then
it can be inverted to form
ωjt = Υ−1
t
(kjt, ajt, ijt, pjt), (56)
and one can proceed with the first stage of estimation as above.
For the second stage observe that since the µjt process is independent of the ωjt process
the firm’s conditional expectation of ωjt given Ijt−1 only depends on ωjt−1. Thus, the
second stage is
yejt = βkkjt + βaajt + ge(φbjt−1 − βkkjt−1 − βaajt−1) + ξjt + ηjt. (57)
Note that the demand shock, if an important determinant of ijt, may help with the
precision of our estimates, as it generates independent variance in φb.
69
The estimation problem becomes more complicated if, for some reason, the two Markov
processes are dependent. The problem is that in this case, the firm’s conditional expectation of ωjt given Ijt−1 depends on both ωjt−1 and µjt−1. Then equation (57) will have
to be amended to allow ge(·) to also depend on µjt−1. If we let
µjt−1 = Υ−1
2,t−1
(kjt−1, ajt−1, ijt−1, pjt−1), (58)
our second stage can then be written as
yejt = βkkjt + βaajt + ge(ωjt−1, µjt−1) + ξjt + ηjt (59)
= βkkjt +βaajt +ge(φjt−1 −βkkjt−1 −βaajt−1, Υ
−1
2,t−1
(kjt−1, ajt−1, ijt−1, pjt−1)) +ξjt +ηjt.
Unfortunately, this equation cannot identify βk and βa since it requires us to condition on
a nonparametric function of (kjt−1, ijt−1, ajt−1). More formally, since kjt = (1−δ)kjt−1+
ijt−1 (and ajt = ajt−1 + 1), there is no remaining independent variance in (kjt, ajt) to be
used to identify βk and βa.
To avoid this problem, we need an explicit ability to solve for or estimate µjt−1. This
would generally require demand side data. For example the Berry, Levinsohn, and
Pakes (1995) demand estimation procedure produces estimates of a set of ”unobserved
product characteristics” which might be used as the µjt’s. Of course once one brings
in the demand side, there is other information that can often be brought to bear on
the problem. For example, the production function estimates should imply estimates of
marginal cost which, together with the demand system, would actually determine prices
in a “static” Nash pricing equilibrium (see the first section of this chapter). We do not
pursue this further here.
Finally we move to the case where there are two sources of productivity growth, one
evolving as a controlled Markov process, and one as an exogenous Markov process. In
this case the production function is written as
yjt = β0 + βkkjt + βaajt + βlljt + ω
1
jt + ω
2
jt + ηjt,
where ω
1
jt is the controlled, and ω
2
jt is the exogenous, first-order Markov process.
Assume now that we have data on both R&D expenditures, say rt, which is the input
of the controlled process, and a “technology indicator” or Tt (like patents, or licensing
fees) which is an output of the controlled process. As above, assume the policy functions
for physical and R&D investment are a bijection, so we can write
ω
1
jt = Υ−1
1t
(kjt, ajt, ijt, rjt)
ω
2
jt = Υ−1
2t
(kjt, ajt, ijt, rjt). (60)
70
Now the first stage consists of using the technology indicator to isolate ω
1
jt. In other
words, we assume
Tjt = ω
1
jtθ + η2jt, (61)
where η2,t is mean independent of all the controls. We then substitute a nonparameteric
function of (kjt, ajt, ijt, rjt) for ω
1
jt in equation (61). This provides us with an estimate
of ω
1
jtθ, say Υˆ −1
1tj .
Our second stage mimics the first stage of OP except we treat Υˆ −1
1tj as an input. That
is, we estimate
yjt = βlljt + θ
−1Υˆ −1
1tj + φ(kjt, ajt, ijt, rjt) + η1jt, (62)
where
φ(kjt, ajt, ijt, rjt) ≡ βkkjt + βaajt + ω
2
jt.
Then, without a selection correction, the third stage becomes
y˜jt = βaajt + βkkjt + ˜g(φjt−1 − βkkjt−1 − βaajt−1, Υˆ −1
1tj ) + ξjt + η1jt,
Once again, we can modify this to allow for selection by using the propensity score as
an additional determinant of ˜g(·).
Buettner (2004b) explores a related extension to OP. While he only allows one unobserved
state variable, he does allow the distribution of ωjt to evolve endogenously over time, i.e.
firms invest in R&D and these investments affect the distribution of ωjt (conditional on
ωjt−1).48 Unlike the above, Buettner does not assume that a ”technology indicator” is
observed. He develops a dynamic model with investments in R&D and physical capital
that generates invertible policy functions such that the first stage of OP can be directly
applied (and the labor coefficient can be estimated). However, second stage estimation is
problematic, since the conditional expectation of ωjt now depends on the full state vector
through the choice of R&D. Furthermore, with the endogenous productivity process, he
cannot rely on exogenous variation (such as changes in the economic environment over
time) for identification. It remains to be seen whether this problem can be solved.
Greenstreet (2005) proposes and utilizes an alternative model/methodology that, while
related to the above procedures, does not require the first stage inversion. This is a very
48Recall that ”endogenous” evolution of ωjt is problematic for IV approaches.
71
nice attribute since as a result, the procedure does not rely at all on the key scalar unobservable and monotonicity assumptions of the OP/LP/ACF procedures. Greenstreet
achieves this by making a different assumption on firms’ information sets. Specifically,
instead of observing ωjt and ηjt individually (after production at t), firms only ever
observe the sum ωjt + ηjt. Because of this alternative informational assumption, the
econometrician does not need the first-stage inversion to recreate the information set of
the firms. While this does avoid the scalar unobservable and monotonicity assumptions,
Greenstreet’s approach still relies on similar timing assumptions, involves a slightly more
complicated learning process than the above procedures (requiring Kalman filtering), and
also generates some new initial conditions problems that require additional assumptions
to solve.
2.5 Concluding Remark.
The increase in the availability of plant and/or firm level panels together with a desire to
understand the efficiency implications of major environmental and policy changes has led
to a renewed interest in productivity analysis. Most of this analysis is based on production function estimates, and the literature has found at least two empirical regularities.
First, there are indeed large efficiency differences among firms and those differences are
highly serially correlated. Second, at least in many environments, to obtain realistic production function estimates the researcher must account for the possibility of simultaneity
and selection biases.
Put differently, to study either the changes in the allocative efficiency of production
among firms of differing productivities, or the correlates of productivity growth within
individual establishments, we first have to isolate the productivity variable itself. Since
firms’ responses to the changes in the environment being studied typically depend on
how those changes impacted their productivity, movements in productivity can not be
isolated from changes in input and exit choices without an explicit model of how those
choices are made.
The appropriateness of different models of how these decisions are made will undoubtedly
depend on the environment being studied. We have presented a number of alternatives,
and discussed their properties. However this is an empirically driven subfield of estimation, and there are undoubtedly institutional settings where alternative frameworks
might be better to use. It is not the precise framework that is important, but rather the
fact that productivity studies must take explicit account of the fact that changes in productivity (or, if one prefers, sales for a given amount of inputs) in large part determine
how firms respond to the changes being studied, and these must be taken into account
in the estimation procedure.
72
3 Dynamic Estimation.
This chapter considers structural estimation of dynamic games. Despite a blossoming
empirical literature on structural estimation of static equilibrium models, there has been
relatively little empirical work to date on estimation of dynamic oligopoly problems.
Four exceptions are Gowrisankaran and Town (1997), Benkard (2004), Jofre-Benet and
Pesendorfer (2003), and Ryan (2005). The literature’s focus on static settings came about
not because dynamics were thought to be unimportant to market outcomes, but rather
because empirical analysis of dynamic games was seen as too difficult. In particular, while
some of the parameters needed to analyze dynamic games could be estimated without
imposing the dynamic equilibrium conditions, some could not and, until very recently,
the only available methods for estimating these remaining parameters were extremely
burdensome, in terms of both computation time and researcher time.
This computational complexity resulted from the need to compute the continuation values to the dynamic game in order to estimate the model. The direct way of obtaining
continuation values was to compute them as the fixed point to a functional equation, a
high order computational problem. Parameter values were inferred from observed behavior by computing the fixed point that determines continuation values at different trial
parameter values, and then searching for the parameter value that makes the behavior
implied by the continuation values “as close as possible” to the observed behavior. This
“nested fixed point” algorithm is extremely computationally burdensome because the
continuation values need to be computed many times.
However, a recent literature in industrial organization (Aguirregabiria and Mira (2006),
Bajari, Benkard, and Levin (2006), Jofre-Benet and Pesendorfer (2003), Pakes, Ostrovsky, and Berry (2006), and Pesendorfer and Schmidt-Dengler (2003)) has developed
techniques that substantially reduce the computational and programming burdens of
estimating dynamic games. This literature extends a basic idea that first appeared in
the context of single agent problems in Hotz and Miller (1993). Hotz and Miller (1993)
provided a set of assumptions under which one could obtain a nonparametric estimate
of continuation values without ever computing the fixed point49. Rust (1994) suggests
the extension of these ideas to the context of dynamic games. The recent literature in
industrial organization has shown that, at least under a certain set of assumptions, these
approaches can be extended to estimate continuation values in a wide variety of dynamic
games, even in the presence of multiple equilibria.
49In related work Olley and Pakes (1996) use nonparametrics to get around the problem of computing
the fixed point needed to obtain an agent’s decision rule in a multiple agent framework; but they use
the nonparametric estimates to control for unobservables and do not recover the implied estimates of
continuation values.
73
This chapter summarizes the currently available techniques for estimating dynamic
games, concentrating on this recent literature. The chapter proceeds as follows. We
first outline the goals of the estimation procedure and consider what might be gained
by modelling dynamics in an oligopoly situation. Then we present a general framework
for dynamic oligopoly problems, with three simple examples from the recent literature.
Next we overview existing estimation methods, providing details for the three examples.
We conclude with a brief discussion of techniques available to ameliorate one (of many)
outstanding problems; that of serially correlated unobserved state variables.
We note that there are at least two issues that appear in the literature and are not
considered here. First we do not consider identification issues (at least not directly).
Our feeling is that many of the parameters determining behavior in dynamic games can
be estimated without ever computing an equilibrium, and those parameters that remain
depend on the nature of the problem and data availability. Second, we do not consider
“timing” games, such as those in Einav (2003) and in Schmidt-Dengler (2004). Our only
excuse here is our focus on the evolution of market structure in oligopolies.
3.1 Why Are We Interested?
One contribution of the recent literature is that it provides a means of obtaining information about certain parameters that could not be obtained via other methods. For
example, the sunk costs of entry and the sell-off values (or costs) associated with exit are
key determinants in the dynamics of market adjustments to policy and environmental
changes. Knowledge of the level of sunk costs is critical, for example, in a regulatory
authority’s decision of whether to approve a merger, or in the analysis of the likely impacts of changes in pension policy on shut down decisions. However, actual data on sunk
costs are extremely rare. Besides being proprietary, and thus hard to access, sunk costs
can also be very difficult to measure. Thus, in many cases the only option for learning
the extent of sunk costs may be to infer them from equilibrium behavior using other
variables that we can observe. Since sunk costs are only paid once upon entry, while
firms may continue to operate for many periods, inferring the level of sunk costs from
equilibrium behavior requires a dynamic framework. Similar arguments can be made
regarding the parameters determining, among other diverse phenomena, the transaction
costs of investments (including installment, delivery, and ordering costs), the costs of
adjusting output rates or production mix, and the extent of learning-by-doing.
There are a number of other uses for techniques that enable us to empirically analyze
dynamic games. For example, there are many industries in which an understanding of the
nature of competition in prices (or quantities) requires a dynamic framework. In such
cases, the empirical literature in industrial organization has often used static models
74
to approximate behavior that the authors are well aware is inherently dynamic. For
example, there has been much work on identifying and estimating the form of competition
in markets (e.g., Bresnahan (1982, 1987), Lau (1982)). This literature typically compares
a static Nash equilibrium with particular static “collusive” pricing schemes. In reality,
the set of collusive pricing schemes that could be supported in equilibrium depends on the
nature of the dynamic interactions (e.g., Abreu, Pearce and Stacchetti (1986), Green and
Porter (1984), Rotemberg and Saloner (1985), Fershtman and Pakes (2000)). A related
point is that static price or quantity setting models are known to be inappropriate when
future costs depend directly on the quantity sold today, as in models with learning by
doing or adjustment costs, and/or when future demand conditions depend on current
quantities sold, as in models with durable goods, experience goods, the ability to hold
inventory, and network externalities.
Similarly, most of the existing empirical literature on entry relies on two-period static
models. While these models have proven very useful in organizing empirical facts, the two
period game framework used makes little sense unless sunk costs are absent. Therefore,
the results are not likely to be useful for the analysis of policy or environmental changes
in a given market over time. This leaves us with an inability to analyze the dynamic
implications of a host of policy issues, and there are many situations where dynamics
may substantially alter the desirability of different policies. For example, Fershtman
and Pakes (2000) show that because collusive behavior can help promote entry and
investment, it can enhance consumer welfare. Similarly, a static analysis would typically
suggest that mergers lower consumer welfare by increasing concentration, whereas a
dynamic analysis might show that allowing mergers promotes entry, counterbalancing
the static effects.
3.2 Framework.
This section outlines a framework for dynamic competition between oligopolistic competitors that encompasses many (but not all) applications in industrial organization.
Examples that fit into the general framework include entry and exit decisions, dynamic
pricing (network effects, learning-by-doing, or durable goods), dynamic auction games,
collusion, and investments in capital stock, advertising, or research and development.
The defining feature of the framework is that actions taken in a given period affect
future payoffs, and future strategic interaction, by influencing only a set of commonly
observed state variables. In particular we will assume that all agents have the same
information to use in making their decisions, up to a set of disturbances that have only
transitory effects on payoffs.
We use a discrete time infinite horizon model, so time is indexed by t = 1, 2, ...,∞. At
75
time t, prevailing conditions are summarized by a state, st ∈ S ⊂ R
G, that reflects
aspects of the world relevant to the payoffs of the agents. Relevant state variables might
include firms’ production capacities, the characteristics of the products they produce,
their technological progress up to time t, the current market shares, stocks of consumer
loyalty, or simply the set of firms that are incumbent in the market. We assume that
these state variables are commonly observed by the firms. Note that we have not yet
specified which state variables are observed by the econometrician. This distinction will
be made in the applications below.
Given the state st at date t, the firms simultaneously choose actions. Depending on the
application, the firms’ actions could include decisions about whether to enter or exit
the market, investment or advertising levels, or choices about prices and quantities. Let
ait ∈ Ai denote firm i’s action at date t, and at = (a1t, ..., aN t) the vector of time t
actions, where Ntis the number of incumbents in period t (entry and exit, and hence
Nt, are endogenous in these models).
We also assume that before choosing its action each firm, i, observes a private shock νit ∈
R, drawn independently (both over time and across agents) from a distribution G(·|st).50
Private information might derive from variability in marginal costs of production that
result, say, from machine breakdowns, or from the need for plant maintenance, or from
variability in sunk costs of entry or exit. We let the vector of private shocks be νt =
(ν1t, ..., νN t).
In each period, each firm earns profits equal to πi(at, st, νit). Profits might include
variable profits as well as any fixed or sunk costs, including the sunk cost of entry and
the selloff value of the firm. Conditional on the current state, s0, and the current value of
the firm’s private shock, νi0, each firm is interested in maximizing its expected discounted
sum of profits,
E
"X∞
t=0
β
tπi(at
, st, νit)

s0, νi0
#
, (63)
where the expectation is taken over rival firms’ actions in the current period as well as
the future values of all state variables, the future values of the private shock, and all
rivals’ future actions. We assume firms have a common discount factor β.
The final aspect of the model is to specify the transitions between states. We assume
that the state at date t + 1, denoted st+1, is drawn from a probability distribution
P(st+1|st, at). The dependence of P(·|st, at) on the current period actions at reflects
50Here we assume that firm i’s private shock is a single scalar variable. However, as will be seen in the
examples below, there is no conceptual difficulty in allowing the shock to be multi-dimensional.
76
the fact that some time t decisions may affect future payoffs, as is clearly the case if
the relevant decision being modelled is an entry/exit decision or a long-term investment.
Of course, not all the state variables necessarily depend on past actions; for example,
one component of the state could be a transitory iid shock that affects only the current
payoffs, such as an iid shock to market demand.
Note that we have assumed that firms’ private information does not influence state
transitions directly (i.e., it only influences transitions through its impact on ait). For
example, incumbent firms care only about whether or not a potential entrant enters
the market, and not what the entrant’s sunk cost of entry was. On the other hand this
assumption does rule out applications where firms’ investment outcomes are their private
information (e.g., Fershtman and Pakes (2005)).
We are interested in equilibrium behavior. Because the firms interact repeatedly and the
horizon is infinite, there are likely to be many Nash, and even subgame perfect equilibria,
possibly involving complex behavioral rules. For this reason, we focus on pure strategy
Markov perfect equilibria (MPE).
In our context a Markov strategy for firm i describes the firm’s behavior at time t as
a function of the commonly observed state variables and firm i’s private information at
time t. Formally, it is a map, σi: S ×R → Ai. A profile of Markov strategies is a vector,
σ = (σ1, ..., σn), where σ : S × R
n → A. A Markov strategy profile, σ, is a MPE if there
is no firm, i, and alternative Markov strategy, σ
0
i
, such that firm i prefers the strategy σ
0
i
to the strategy σi given its opponents use the strategy profile σ−i. That is, σ is a MPE
if for all firms, i, all states, s, and all Markov strategies, σ
0
i
,
Vi(s, νi|σi, σ−i) ≥ Vi(s, νi|σ
0
i
, σ−i). (64)
If behavior is given by a Markov profile σ, firm i’s present discounted profits can be
written in recursive form:
Vi(s, νi|σ) = Eν−i

πi(σ(s, ν), s, νi) + β
Z
Vi(s
0
, ν0
i
|σ)dG(ν
0
i
|s
0
)dP(s
0
|σ(s, ν), s)

. (65)
3.2.1 Some Preliminaries.
The framework above is a generalization of the Ericson and Pakes (1995) model. The
existence proofs for that model that are available have incorporated additional assumptions to those listed above (see Gowrisankaran (1995), and Doraszelski and Satterthwaite
(2004)). Typically, however, the algorithms available for computing an equilibrium do
77
find an equilibrium even when the available sets of sufficient conditions for existence are
not satisfied (i.e. the algorithm outputs policies and values that satisfy the fixed point
conditions that define the equilibrium up to a precision determined by the researcher).
There may, however, be more than one set of equilibrium policies (for an explicit example
see Doraszelski and Satterwaite (2004)).
If the regularity conditions given in Ericson and Pakes (1995) are satisfied, each equilibrium generates a finite state Markov chain for the st process. That is, the vector of
state variables can only take on a finite set of values, a set we will designate by S, and
the distribution of the future {sτ }∞
τ=t
conditional on all past history depends only on
the current value of st. Every sequence from this finite state Markov chain will, in finite
time, wander into a subset of the states called a recurrent class or an R ⊂ S, and once
in R will stay there forever. Every s ∈ R will be visited infinitely often51.
Throughout we assume that agents’ perceptions of the likely future states of their competitors depend only on st (i.e., we assume that st
is a complete description of the state
variables observed by the firms). As detailed by Pakes, Ostrovsky, and Berry (2006),
this implies that there is only one equilibrium policy for each agent that is consistent
with the data generating process; at least for all st ∈ R. To see this it suffices to note
that since we visit each point in R infinitely often, we will be able to consistently estimate the distribution of future states of each firm’s competitors given any st ∈ R.
Given that distribution, each agent’s best response problem is a single agent problem.
Put differently, since reaction functions are generically unique, once the agent knows the
distribution of its competitors’ actions, its optimal policy is well defined. Thus, given
the data generating process, policies are well defined functions of the parameters and
the state variables. Consequently, standard estimation algorithms can be used to recover
them.52
Finally, in all of the examples below we will assume that the discount factor, β, is one
of the parameters that is known to the econometrician. It is a straightforward extension
to estimate the discount parameter. However, our focus here is on obtaining estimates
of parameters that we have little other information on.
51Formally, the dynamics of the model are described by a Markov matrix. Each row of the matrix
provides the probability of transiting from a given s to each possible value of s ∈ S. Ericson and Pakes
(1995) also provide conditions that imply that the Markov matrix is ergodic, that is there is only one
possible R.
52Note that if our data consists of a panel of markets, this implicitly assumes that, conditional on st,
the policy rule (our σ) in one market is the same as in the other.
78
3.2.2 Examples.
The framework above is general enough to cover a wide variety of economic models. We
provide three examples below. In general the objects that need to be recovered in the
estimation are the period profit function, π(·), the transition probabilities, P(st+1|st, at),
and the distribution of the private shocks, G(·|s).
Example 1: A Simple Model of Entry/Exit.
This example is based on Pakes, Ostrovsky, and Berry (2006). Let the state variables
of the model be given by a pair, st = (nt, zt), where nt denotes the number of firms
active at the beginning of each period, and ztis a vector of profit shifters that evolve
exogenously as a finite state. In the model, operating profits are determined solely by
these variables. In any period, t, in which a firm is active it earns profits equal to
π˜(nt, zt; θ).
The model focuses on entry and exit. In each period, each incumbent firm receives a
random draw, denoted φit, determining the selloff value of the firm. The selloff values
are assumed to be private information. However, their distribution is commonly known
to the agents. The firm chooses to exit if the selloff value of the firm is greater than the
expected discounted value of continuing in the market. Otherwise, the firm continues in
the market.
Entry is described similarly. For ease of exposition, we assume that there are E potential
entrants each period, where E is known to the agents.53 Each period, each potential
entrant firm receives a random draw, denoted κit, determining its sunk cost of entry.
As above, the entry cost is private information, but its distribution is commonly known.
The firm enters the market if the expected discounted value of entering is greater than
the entry cost. Otherwise, the entrant stays out of the market and earns nothing.
To see how this model fits into the general framework, let χit = 1 for any firm i that is
active in the market in period t, and χit = 0 otherwise. Note that we assume that when
an incumbent firm exits χit = 0 thereafter. In that case the period profit function is,
πi(at, st, νit) = {χit = 1}π˜(nt, zt; θ) + (χit − χi,t−1)
−φit − (χit − χi,t−1)+κit,
53The extension to a random number of entrants is straightforward. See Pakes, Ostrovsky, and Berry
(2006) for details.
79
where the notation {χit = 1} denotes an indicator function that is one if the firm is
active and zero otherwise, the notation f
+ ≡ {f > 0}f, for any function f, and similarly
f
− ≡ {f < 0}|f|. On the right hand side, χ represents firms’ actions, a; n and z
represent the states, s; and φ and κ represent the private shocks, ν.
Note that while this model does not allow for observed heterogeneity among incumbent
firms, this can be achieved by allowing for multiple entry locations. We consider this
extension below. Note further that this model is a special case of the Ericson and Pakes
(1995) model in which investment is not modelled. We add investment back to the model
in the next example.
Example 2: An Investment Game with Entry and Exit.
This example is a straightforward extension of the Ericson and Pakes (1995) model due
to Bajari, Benkard, and Levin (2006). Similarly to the above example, there are a set
of incumbent firms competing in a market. Firms are heterogeneous, with differences
across firms described by their state variables, sit, which are commonly known. For
ease of exposition, we will omit any other exogenous profit shifters from the set of state
variables.
Each period, firms choose investment levels, Iit ≥ 0, so as to improve their state the
next period. Investment outcomes are random, and each firm’s investment affects only
its own state so that there are no investment spillovers. Therefore, each firm’s state
variable, sit, evolves according to a process P r(si,t+1|sit, Iit).
Here are some examples of models that are consistent with this framework.
(i) Firms’ state variables could represent (one or more dimensions of) product quality,
where investment stochastically improves product quality.
(ii) Firms’ state variables could represent the fraction of consumers who are aware
of the firm’s product, where investment is a form of advertising that increases
awareness (e.g. Doraszelski and Markovich, 2004).
(iii) Firms’ state variables could represent capital stock, where investment increases a
firm’s capital stock.
Firms earn profits by competing in a spot market. Because quantity and price are
assumed not to influence the evolution of the state variables, they are determined in
80
static equilibrium conditional on the current state. In any period, t, in which a firm is
active in the market it earns profits equal to
qit(st, pt; θ1) (pit − mc(sit, qit; θ2)) − C(Iit, νit; θ3), (66)
where qit is quantity produced by firm i in period t, ptis the vector of prices, mc is
the marginal cost of production, νit represents a private shock to the cost of investment,
θ = (θ1, θ2, θ3) is a parameter vector to be estimated, and we have assumed that the
spot market equilibrium is Nash in prices.
The model also allows for entry and exit. Each period, each incumbent firm has the
option of exiting the market and receiving a scrap value, Φ, which is the same for all firms
(this differs from the prior example in which there is a distribution of exit costs). There
is also one potential entrant each period with a random entry cost, κit.
54 The entrant
enters if the expected discounted value of entering exceeds the entry cost. As above, the
entry cost is assumed to be private information, but its distribution is commonly known.
Relative to the general framework above, current period returns are given by
πi(at, st, νit) ={χit = 1}

qit(st, pt; θ1) (pit − mc(sit, qit; θ2)) − C(Iit, νit; θ3)

+
(χit − χi,t−1)
−Φ − (χit − χi,t−1)+κit.
On the right hand side, prices (p), investment (I), and entry/exit (χ) are the actions
(a), while the private shocks are the shock to investment (νit) and the entry cost (κit).
Example 3: A Repeated Auction Game with Capacity Constraints.
This example is based on Jofre-Benet and Pesendorfer (2003). In this example, a set of
incumbent contracting firms compete in monthly procurement auctions. The auctions
are heterogeneous because the contracts that come available each month are of differing
size and scope. The firms bidding on the contracts are also heterogeneous as each has
a different cost of completing each contract. In a given month, each firm also has a
different backlog of contracts, which might affect its ability to take on new contracts.
Let zt be the characteristics of the contract to be auctioned in month t, including both the
contract size (in dollars), and the number of months required to complete the contract.
We assume that zt evolves exogenously as a finite state. Let ωi,t be the backlog of work
for firm i in period t and ωt = (ω1,t, ..., ωN,t) be the vector of backlogs. A firm’s backlog
54It is straightforward to generalize the model to have a random number of potential entrants each
period.
81
of work represents the remaining size in dollars, and the remaining number of days
left until completion of each contract previously won by the firm. It therefore evolves
deterministically depending on the current auction outcome according to the map,
ωt+1 = Γ(ωt, zt, j),
where j is the winner of the time t auction and the map Γ is known. The state variables
of the model are st = (ωt, zt). All states are assumed to be common knowledge.
Each firm also has a different cost, cit, for each contract that is private information to
the firm. Bidders’ costs are drawn independently from a distribution G(cit|ωit, ω−it, zt)
that is commonly known.
In each period, each firm views its cost for the contract being offered and then chooses
a bid, bit. Each firm earns current profits equal to
πi(at, st, νit) = (bit − cit){bit ≤ min
j
(bjt)} (67)
where the indicator function takes the value one if firm i submits the lowest bid and
hence wins the auction (assume there are no ties). On the right hand side the bids (bjt)
are the action variables (at) and the costs cit are the private shocks (νit).
Note that the state variables do not directly enter current profits in this model. However,
the state variables influence all firms’ costs and hence a firm’s optimal bid depends on
the current state both through its own costs directly and through the firm’s beliefs about
the distribution of rivals’ bids. For the same reason, expected profits are also a function
of the current state.
Note also that an important distinction between the investment model above and this
example is that, in this example, each firm’s choice variable (in this case, its bid) affects
the evolution of all firms’ states. In the investment model above, a firm’s investment
affects only the evolution of its own state. This distinction is important because many
I.O. models share this feature. For example, models of dynamic pricing (learning by
doing, network effects, or durable goods) would have this feature when firms compete
in prices (though not if firms compete in quantities). Such models can be placed in the
general EP framework we have been using, but to do so we need to adjust that framework
to allow the control that affects the distribution of current profits (bids, quantities, or
prices) to also have an impact on distribution of future states; see the discussion in Pakes
(1998). We note that to our knowledge Jofre-Benet and Pesendorfer (2003) were the first
to show that a two-step estimation approach was feasible in a dynamic game.
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3.3 Alternative Estimation Approaches.
In order to conduct policy analysis in any of the economic models above, it is typically
necessary to know all of the parameters of the model, including the profit function, the
transition probabilities, and the distribution of the exogenous shocks. Often many of the
parameters can be estimated “off line”, that is, without needing to compute equilibria of
the dynamic game. At one extreme here is Benkard’s (2004) analysis of the commercial
aircraft industry. He was able to obtain a large amount of cost data on sunk as well as
marginal costs which, together with generally available information on demand, enabled
him to estimate all the parameters he needed off line. Given these parameters he could
focus on computing the dynamic implications of alternative policies.
However, such an approach is rarely possible. More typically, at least cost data are unavailable, either because they are proprietary and hence difficult for researchers to access,
or because they are hard to measure. In static settings we often solve the problem of a
lack of cost data by inferring marginal costs from their implications in an equilibrium
pricing equation. A similar approach can be taken in this dynamic setting. However,
characterizing the relationship between the data generating process and equilibrium play
in the models above is complicated by the fact that the model involves repeated interactions.
Observed behavior in the model represents the solution to a maximization problem that
involves both the profit function, which typically has a known parametric form, and the
value function, which results from equilibrium play and therefore has unknown form. For
example, the value of entering a market depends both on current profits, and expected
future profits, which in turn depend on future entry and exit behavior. In order to describe the data generating process, then, we need the ability to compute the equilibrium
continuation values.
Thus, conceptually, estimation of dynamic models can be separated into two main parts.
The first part involves obtaining the continuation values for a given parameter value, θ.
The second part is to use the continuation values obtained in the first part to maximize
an objective function in the parameters, θ. Note that the continuation values must be
obtained for many different values of θ in order to perform this maximization, and thus
the first part is the source of most of the computational burden of the estimation. The
key differences in estimation approaches lie in the details of how each of these two parts
is performed.
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3.3.1 The Nested Fixed Point Approach.
The nested fixed point approach is a logical extension of the method of Rust (1987) to
games. The general idea is as follows:
1. Given a parameter vector, θ, compute an equilibrium to the game, V (s; θ), numerically, using the computer.
2. Use the computed values, V (s; θ), to evaluate an objective function based on the
sample data.
3. Nest steps one and two in a search routine that finds the value of θ that maximizes
the objective function.
A framework capable of computing equilibria to models like those above has existed
for some time (Pakes and McGuire (1994)), and recent papers by Pakes and McGuire
(2001), Doraszelski and Judd (2004), and Weintraub, Benkard, and Van Roy (2005)
enable significant improvements in computational times, at least in some problems (for
a discussion of these, and other alternatives, see Doraszelski and Pakes (2006)). All of
these algorithms rely on similarities between the dynamic framework above and dynamic
programming problems. The general idea of these algorithms is to start with an initial
guess at the value function, V
0
(s; θ), and substitute that into the right hand side of the
Bellman equation (equation (65)). Then, at each state point and for each firm, solve
the maximization equation on the right hand side of (65) yielding a new estimate of
the value function, V
1
(s; θ). This procedure is iterated until convergence is achieved,
so that the new and old value functions are the same. Unlike single agent problems,
in the context of a game, convergence of the algorithm is not guaranteed (the mapping
is not a contraction) and, indeed, initial iterations will often seem to move away from
equilibrium. However, in practice the algorithms typically converge and, once they do,
the value functions obtained must represent an equilibrium.
An important feature of the nested fixed point algorithm is that the first step is performed
without using any data. As a result, the value functions are obtained precisely; that is,
they contain no sampling error. This lack of sampling error makes the second part of
the algorithm, in which the parameters are estimated, straightforward.
On the other hand the algorithm is computationally burdensome. For models rich enough
to use in empirical work, it is often difficult to compute an equilibrium even once, and in
the nested fixed point algorithm it is necessary to compute an equilibrium once for each
iteration of the maximization routine; implying that up to hundreds, if not thousands,
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of fixed points must be calculated. Moreover, setting up an efficient algorithm often
requires a large amount of complex computer programming, creating a relatively large
burden on researcher time. As a result there are very few examples in the literature
where the nested fixed point algorithm has been applied to estimate parameters.
One exception is Gowrisankaran and Town (1997), who use a nested fixed point approach
to apply a model similar to the investment model above to data for the hospital industry.
In each iteration of the estimation they compute an equilibrium using the algorithm of
Pakes and McGuire (1994). They then estimate the model using a GMM objective
function that matches cross-sectional moments such as average revenue per hospital,
average expenditures per hospital, average investment per hospital, and average number
of hospitals of each type (nonprofit and for-profit) per market. The nested fixed point
approach was feasible in their application because their model was parsimonious and
there were never more than three hospitals in any market in the data.
Another difficulty with the nested fixed point algorithm arises from the fact that dynamic
oligopoly models can admit more than one equilibria. While the assumptions given above
in principle allow the researcher to use the data to pick out the correct equilibrium,
actually achieving this selection using the nested fixed point algorithm is likely to be
difficult. Moreover, equilibrium selection has to take place for every candidate value of
the parameters to be estimated. Alternative sets of assumptions could be used to select
different equilibria, but unless we were willing to assume “a priori” that equilibrium
was unique, somehow we must investigate the issue of the relationship between the
equilibrium computed in the algorithm, and that observed in the data.
3.3.2 Two-Step Approaches.
The biggest obstacle to implementing the nested fixed point algorithm in practice is
the heavy computational burden that results from the need to compute equilibria for
each trial parameter value. Fortunately, the recent literature (Aguirregabiria and Mira
(2006), Bajari, Benkard, and Levin (2006), Jofre-Benet and Pesendorfer (2003), Pakes,
Ostrovsky, and Berry (2006), and Pesendorfer and Schmidt-Dengler (2003)) has derived methods for estimating dynamic oligopoly models that impose the conditions of
a dynamic equilibrium without requiring the ability to compute an equilibrium. The
new literature sidesteps the equilibrium computation step by substituting nonparametric functions of the data for the continuation values in the game. These nonparametric
estimates are in general much easier to compute than the fixed point calculations in
the nested fixed point algorithm. As a result, these methods have substantially lower
computational burden.
85
Below we outline five different two-step methods of estimating dynamic games. The
overall approach is similar throughout, but each method does both the first and second
steps of the estimation differently. To our knowledge, Hotz and Miller (1993) were
the first to show that it was possible to estimate the continuation values in a dynamic
programming problem nonparametrically instead of computing them. In a single agent
dynamic discrete choice problem, Hotz and Miller showed that the agent’s dynamic choice
problem mimics a static discrete choice problem with the value functions replacing the
mean utilities. Thus, the agent’s continuation values can be obtained nonparametrically
by first estimating the agent’s choice probabilities at each state, and then inverting
the choice problem to obtain the corresponding continuation values. This inversion is
identical to the one commonly used in discrete choice demand estimation to obtain the
mean utilities.
We begin our discussion of estimation by showing that if the game has only discrete
actions, and there is one unobserved shock per action for each agent in the game, then
under the information structure given in the general framework above, estimators very
similar to those of Hotz and Miller (1993) can still be used (see also Aguirregabiria
and Mira (2006)). Sticking with the single agent framework, Hotz et. al (1994) use
estimated probabilities to simulate sample paths. They then calculate the discounted
value of utility along these paths, average those values for the paths emanating from a
given state, and use these averages as the continuation values at that state. The Bajari,
Benkard, and Levin (2006) paper discussed below shows that related ideas can be used
to incorporate continuous controls into estimation strategies for dynamic games.
Pakes, Ostrovsky, and Berry (2006) also consider dynamic discrete games but, instead of
inverting the choice problem, they estimate the continuation values directly by computing
(approximately) the average of the discounted values of future net cash flows that agents
starting at a particular state actually earned in the data (at least up to the parameter
vector of interest). Econometrically, they use a nonparametric estimate of the Markov
transition probabilities that determine the evolution of the state of the system to form
an analytic estimate of the probability weighted average of the discounted returns earned
from different states. Given equilibrium play, these averages will converge to the true
expected discounted value of future net cash flow, that is of the continuation values we
are after.
Bajari, Benkard, and Levin (2006) instead begin by projecting the observed actions on
the state variables to compute nonparametric estimates of the policy functions of each
agent at each state. Then they use the estimated policies to simulate out the discounted
values of future net cash flows. This procedure is computationally light even in models
with large state spaces and is easily applied to models with continuous controls, such
as investment, quantity, or price (including models with both discrete and continuous
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controls like the investment game above). Given equilibrium play, the continuation values
obtained in this fashion will be consistent estimates of the continuation values actually
perceived by the agents.
Berry and Pakes (2002) provide an alternative approach for estimating models with
continuous controls that is likely to be useful when the dynamic environment is complex,
but sales and investment data are available. They assume that current period net returns
are observable up to a parameter vector to be estimated, but do not require that the
state variables of the model be observed, or even specified (so it would not be possible
to estimate policy functions conditional on those state variables as in Bajari, Benkard
and Levin). They derive an estimating equation from the first order condition for the
continuous control (investment in our example) by substituting observed profit streams
for the expected profit streams, and noting that the difference must be orthogonal to
information known at the time investment decisions are made.
Jofre-Benet and Pesendorfer (2003) provide an estimator for the dynamic auction model.
They show that it is possible to derive an expression for the equilibrium continuation
values in the auction game that involves only the bid distributions. Since bids are observed, the bid distributions can be recovered nonparametrically from the data and then
substituted into these expressions. Provided that agents are bidding close to optimally,
the continuation values obtained from this procedure will be consistent estimates of the
continuation values perceived by the agents.
In many of the cases we consider several of the methods could be used in estimation.
In these cases it is not currently known how the methods compare to one another on
such dimensions as computational burden and econometric efficiency. Hybrid methods
are also possible in which features of two or more of the approaches could be combined.
We expect these issues to be sorted out in the future.
Finally, there are also some costs associated with the two-step approaches. First, because
the continuation values are estimated rather than computed, they contain sampling error.
This sampling error may be significant because these models often have state spaces that
are large relative to the available data. As we will see below, this influences the properties
of the second step estimators in important ways. To summarize, the choice of second
stage estimation method will be influenced as much or more by a need to minimize small
sample bias caused by error in the continuation value estimates as it is by the desire to
obtain asymptotic efficiency.
Second, for the two step approaches to produce estimators with desirable properties the
data must visit a subset of the points repeatedly. Formally the requirement for the limit
properties of the estimators is that all states in some recurrent class R ⊆ S be visited
87
infinitely often. Moreover equilibrium strategies must be the same every time each point
in R is visited. Whether or not this assumption is reasonable for the problem at hand
depends on the nature of the available data and the institutional setting which generated
it. If the data consists of a time series on one market then we would require stationarity
of the process over time. There are different ways to fulfill this requirement in panels
(i.e. when we follow a cross section of markets over time); one possibility is that the
initial state in each market is a random draw from a long run ergodic distribution. Note
that the nested fixed point approach has a weaker data requirement.
These costs must be weighed against the benefit that the two-step estimators eliminate
most of the computational burden of the nested fixed point approach. Indeed, the entire
two-step algorithm might well have less computational burden than one iteration of the
nested fixed point algorithm.
3.4 A Starting Point: Hotz and Miller.
Because of the similarity of this section to the previous literature on single agent problems, we will keep this section short, concentrating mainly on extending Hotz and Miller
to games. For more detail on the approach in single agent problems see Hotz and Miller
(1993), Hotz et al. (1994), Magnac and Thesmar (2002), and Rust (1994). See also Aguirregabiria and Mira (2006) and Pesendorfer and Schmidt-Dengler (2003) for a discussion
in the context of entry games.
The idea behind Hotz and Miller’s estimation method for single agent problems is to set
up a dynamic discrete choice problem such that it resembles a standard static discrete
choice problem, with value functions taking the place of standard utility functions. This
allows a two step approach in which a discrete choice model is used as a first step for
recovering the value functions, and the parameters of the profit function are recovered
in a second step once the value functions are known.
We make two simplifying assumptions that will assist in the exposition. First, we suppose
that agents’ current profits do not depend on rivals’ actions (though they do depend on
rival’s states whose evolution depends on those actions). Second, we assume that the
unobserved shocks are additive to profits. In that case, current profits are given by,
πi(at, st, νit) = ˜π(ait, st) + νit(ait),
where νit is agent i’s vector of profitability shocks and νit(ait) is the shock associated
with agent i’s action ait.
The first assumption simplifies the agents’ choice problem because, if agents’ current
88
profits depend on rivals’ actions then, since rivals’ actions depend on their own current
shocks, in its own maximization problem each agent would have to integrate current
profits over all rivals’ current actions. This would not change the overall approach but
would complicate the computations below (we would need to integrate over distributions
of competitors outcomes to compute the expected profits; see Aguirregabiria and Mira,
2006, for a model in which profits do depend on rivals’ actions). The second simplification, additive separability in the private shocks, is also not strictly required. If the error
terms entered profits nonlinearly then we could rewrite the problem in terms of expected
profits and an additively separable projection error and work with that framework instead. However, such an approach does have the unattractive property that it changes
the interpretation of the profit function. Thus, it is typically the case that in practice
people assume that the profit function has additive structural error terms.
With these assumptions the Bellman equation can be simplified to (suppressing the
subscripts)
V (s, ν) = max
a

π˜(a, s) + ν(a) + β
Z
V (s
0
, ν0)dG(ν
0
|s
0
)dP(s
0
|s, a)

. (68)
Equation (68) represents a discrete choice problem in which the mean utilities are given
by,
va(s) = ˜π(a, s) + β
Z
V (s
0
, ν0)dG(ν
0
|s
0
)dP(s
0
|s, a). (69)
Thus, since the private shocks are independent across time and across agents, the choice
probabilities for a given agent can be generated in the usual manner of a static discrete
choice problem
P r(a|s) = P r(va(s) + ν(a) ≥ va
0(s) + ν(a
0
), ∀a
0
). (70)
Assuming that the data consists of a large sample of observations on states and actions,
the probability of each action at each state, P r(a|s), can be recovered from the data.
In that case, the left hand side of (70) is known, at least asymptotically. Let P(s) be
the vector of choice probabilities for all feasible actions. Hotz and Miller show that
for any distribution of the private shocks there is always a transformation of the choice
probabilities such that
va(s) − v1(s) = Qa(s, P(s)). (71)
That is, the differences in the choice specific value functions can be written as a function
of the current state and the vector of choice probabilities. The transformation on the
right hand side is the same as the inversion used in the discrete choice demand estimation
89
literature. Berry (1994) proves that the solution is unique. Berry, Levinsohn, and
Pakes (1995) provide a transformation from the data to the mean utilities which is a
contraction, and hence enables the researcher to actually compute the mean utilities (for
more discussion see the first part of this chapter).
In general, this transformation can be used to recover the normalized choice specific value
functions, va − v1, at each state, using the estimated choice probabilities. If the distribution of the private shocks is known, the mapping does not depend on any unknown
parameters. For example, in the case of the logit,
Qa(s, P(s)) = ln(P r(a|s)) − ln(P r(a1|s)). (72)
However, in general the mapping may be a function of unknown parameters of the
distribution of the private shocks.
Note that, as in static discrete choice models, only the value differences can be recovered
nonparametrically. Thus, some further information is required to obtain the values
themselves. This difficulty is not just a feature of this particular estimation approach,
but comes from the underlying structure of the discrete choice framework, in which only
utility differences are identified from the observed choices. One consequence of this is
that, even if the discount factor and the distribution of private shocks are completely
known, the profit function can not be recovered nonparametrically (see Magnac and
Thesmar (2002) for a detailed proof and analysis for single agent dynamic discrete choice
problems, and Pesendorfer and Schmidt-Dengler (2003) for results extended to dynamic
discrete games). This feature is inherent to the dynamic discrete choice setup and carries
through to the context of a dynamic discrete game. As noted earlier our feeling is that
the appropriate resolution of identification issues, such as this one, is context specific
and will not be discussed here.
To obtain the continuation values from the choice specific values we can use the fact
that,
V (s, ν) = max
a
{va(s) + ν(a)} . (73)
Because the continuation values are obtained by inverting from the observed choice
probabilities, the structure of the profit function has not yet been imposed on them, and
they are not yet functions of the profit function parameters. In order to estimate the
profit function parameters, Hotz and Miller iterate the Bellman equation once, inserting
the estimated continuation values on the right hand side,
Vb(s; θ) = Z
max
a

π˜(a, s; θ) + ν(a) + β
Z
Vb(s
0
, ν0)dG(ν
0
|s
0
)dP(s
0
|s, a)

dG(ν|s), (74)
90
Note that for some distributions such as those of type GEV the integral on the right
hand side has an analytic form. In other cases it can be simulated.
These new estimates of the continuation values contain the profit function parameters (θ)
and can be used in an estimation algorithm to estimate θ. The way this is typically done
is to compute new predicted choice probabilities, (70), based on the new continuation
value estimates, Vb(s; θ). Then, these choice probabilities can be used to construct either
a pseudo-likelihood or some other GMM objective function that matches the model’s
predictions to the observed choices.
As noted above, the nonparametric estimates of the continuation values and transition
probabilities on the right hand side of (74) introduce estimation error into the second
stage objective function nonlinearly. Hotz and Miller show that if this estimation error
disappears quickly enough then the estimator obtained is consistent and asymptotically
normal. However, there are other methods that may be preferable in this context to a
pseudo likelihood. Because of the nonlinearity of the pseudo likelihood in the continuation values, estimation error in the continuation values causes increased small sample
bias in the parameter estimates obtained using this method. We discuss methods that
at least partially address this problem in the next section.
3.5 Dynamic Discrete Games: Entry and Exit.
In this section we consider estimation of the entry/exit game in example one using the
methods of Pakes, Ostrovsky, and Berry (2006) (henceforth, POB). We maintain the
assumption that all of the state variables, (nt, zt), are observed and that the number of
entrants (et) and exitors (xt) are also observed. Entry and exit costs are assumed not to
be observed and are the objects of interest in the estimation. We discuss the possibilities
for estimation when there are one or more unobserved state variables in section 3.8.1.
Consider first exit behavior. Redefining the value function from the start of a period,
prior to the point at which the private scrap value is observed, the Bellman equation for
incumbent firms is given by (t subscript suppressed),
V (n, z; θ) = ˜π(n, z; θ) + βEφ [max{φi, V C(n, z; θ)}] (75)
where V C denotes the continuation value of the firm, which equals
V C(n, z; θ) ≡
X
z
0
,e,x
V (n + e − x, z0; θ)P(e, x|n, z, χ = 1)P(z
0
|z). (76)
In the above equation, e and x denote the number of entering and exiting firms, and
P(e, x|n, z, χ = 1) denotes the incumbent’s beliefs about the likely number of entrants
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and exitors starting from state (n, z) conditional on the incumbent itself continuing
(χ = 1).
If the equilibrium continuation values, V C(n, z; θ), were known, then it would be straightforward to construct a likelihood function since the probability of exit is given by
P r(i exits|n, z; θ) = P r(φi > V C(n, z; θ)), (77)
and is independent across firms. Thus, we need to find a simple way to construct the
equilibrium continuation values using observed play.
The continuation values represent the expected discounted value of future profits conditional on the incumbent continuing. They are a function of the profit function, ˜π(n, z; θ),
which determines future profits at each state (n, z), and the processes determining the
evolution of the state variables, n and z. The profit function is known up to the parameters, θ. Therefore, in order to construct the continuation values as a function of
the parameters, we need only estimate the evolution of the number of firms, which is
determined by entry and exit, and the evolution of the profit shifters, P(z
0
|z). The easiest way to do this is to use their empirical counterparts. Starting from a certain state,
to estimate the evolution of the number of firms we can use the actual evolution of the
number of firms each time that state was observed in the data. Similarly, we can use
the observed evolution of the profit shifters to estimate the process P(z
0
|z). That way
the estimated continuation values reflect, approximately, the actual profits of firms that
were observed in the data. The next subsection outlines this process in detail.
3.5.1 Step 1: Estimating Continuation Values.
To facilitate estimation of the continuation values, it helps to rewrite the Bellman equation in terms of the continuation values, V C,
V C(n, z; θ) = X
n0,z0

π˜(n
0
, z0; θ)+βEφ

max{φi, V C(n
0
, z0; θ)}


P(n
0
|n, z, χ = 1)P(z
0
|z),
(78)
where to shorten the notation we let n
0 ≡ n + e − x.
Next, rewrite (78) in vector form. Let V C(θ) be the #S × 1 vector representing
V C(n, z; θ) for every (n, z) pair, and define ˜π(θ) similarly. Also let Mi be the #S × #S
matrix whose (i, j) element is given by P(nj |ni, zi, χ = 1)P(zj |zi). This is the matrix
whose rows give us the equilibrium transition probabilities from a particular (n, z) to
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each other possible (n, z). Note that if we were not conditioning on χ = 1 an unbiased
estimate of the rows of this matrix could be obtained by simply counting up the fraction
of transits from (n, z) that were to each other state. Since the continuation value the
agent cares about is the continuation value should the agent continue, these estimates
have to be modified for conditioning on χ = 1, see the discussion below.
With this notation, (78) becomes,
V C(θ) = Miπ˜(θ) + βMiEφ [max{φi, V C(θ)}] . (79)
In this last equation, ˜π(θ) is a known vector (up to θ). In a structural model the
distribution of φ would also typically be known up to a parameter vector. Therefore,
the only unknowns in the equation are Mi and V C(θ). If Mi were known, V C(θ) could
be calculated as the solution to the set of equations, (79). We discuss the estimation of
Mi below and turn first to the solution for V C(θ).
One of the insights of POB is that the expectations term on the right hand side of
(79) can sometimes be simplified, making computation of V C(θ) simple. Expanding the
expectations term at a single state (n, z) gives,
Eφ [max{φi, V C(n, z; θ)}] =P r(φi < V C(n, z; θ)) ∗ V C(n, z; θ)+
P r(φi > V C(n, z; θ)) ∗ Eφ [φi|φi > V C(n, z; θ)]
=(1 − px(n, z)) ∗ V C(n, z; θ) + px(n, z) ∗ Eφ [φi|φi > V C(n, z; θ)]
where px(n, z) is the probability of exit at state (n, z). Provided that the distribution of
scrap values is log-concave, the above equation is a contraction mapping (see Heckman
and Honore (1990)). In that case, given estimates of M and p
x
, the equation can be
solved for V C(·) in a straightforward manner. Moreover, when the distribution of scrap
values is exponential, a distribution often thought to be reasonable on a priori grounds,
Eφ [φi|φi > V C(n, z; θ)] = σ + V C(n, z; θ),
where σ is the parameter of the exponential, and
Eφ [max{φi, V C(n, z; θ)}] = (1 − px(n, z)) ∗ V C(n, z; θ) + px(n, z) ∗ [V C(n, z; θ) + σ]
= V C(n, z; θ) + σpx(n, z).
Substituting this expression into (79) and iterating gives,
V C(θ) = Mi[˜π(θ) + βσpx] + (Mi)
2
[˜π(θ) + βσpx] + (Mi)
3V C(θ) + . . .
=
X∞
τ=1
(Mi)
τ
[˜π(θ) + βσpx]
= (I − βMi)
−1Mi
[˜π(θ) + βσpx].
(80)
93
The only thing that remains is to estimate Mi and px using the data. Both can be
estimated in a variety of different ways, but the simplest approach, and the one supported
by POB’s Monte Carlo results, is to use their empirical counterparts. Let
T(n, z) = {t : (nt, zt) = (n, z)}
be the set of periods in the data with the same state (n, z). Then, the empirical counterpart to px is
pbx(n, z) = 1
#T(n, z)
X
t∈T(n,z)
xt
n
.
Due to the Markov property, pbx(n, z) is a sum of independent draws on the exit probability, and therefore it converges to px(n, z) provided #T(n, z) → ∞.
Similarly, the matrix Mican be estimated element-by-element using
Mci
i,j =
P
t∈T(ni,zi)
(ni − xt)1{(nt+1, zt+1) = (nj , zj )}
P
t∈T(ni,zi)
(ni − xt)
.
This expression weights the actual observed transitions from (ni, zi) in different periods
by the number of incumbents who actually continue in those periods. This weighting
corrects the estimated transition probabilities for the fact that incumbents compute
continuation values under the assumption that they will continue in the market.
Note that because this procedure uses empirical transition probabilities it never requires
continuation values or transition probabilities from points not observed in the data. As a
result there is no need to impute transition probabilities or continuation values for states
not visited.55 Since typical data sets will only contain a small fraction of the points in
S, this reduces computational burden significantly.
Substituting the estimated transition and exit probabilities into (80) provides an expression for the estimated continuation values,
V Cd(θ, σ) = (I − βMci)
−1Mci
[˜π(θ) + βσpbx] (81)
Note first that the estimates of continuation values using the expression in (81) are, approximately, the averages of the discounted values of the incumbents who did continue.56
This makes the relationship between the data and the model transparent. Provided only
55Strictly speaking this is only true if the last period’s state in the data was visited before. If it were
not we would have to impute transition probabilities for it.
56This is only approximately true because the transitions for all firms that reached a state (n, z)
are used to compute transitions for each firm, so information is pooled across firms in computing the
continuation values.
94
that the specification of the profit function is correct, the actual average of realized continuation values should be close to the expected continuation values used by the agents
in making their decisions.
Second, note how easy it is to compute the estimated continuation values. If the discount
factor is known, then,
V Cd(θ, σ) = A˜π˜(θ) + ˜aσ (82)
where A˜ = (I − βMci)
−1Mci and ˜a = β(I − βMci
)
−1pbx. Both A˜ and ˜a are independent
of the parameter vector and can therefore be computed once and then held fixed in the
second step of the estimation.
Finally, note that the parameters of the entry distribution do not enter into the calculation of the continuation values. The reason for this is that sunk costs of entry are paid
only once at the time of entry. After that, the sunk costs distribution only affects profits
indirectly through rival firms’ entry decisions. Thus, all that matters for computing continuation values is the probability of entry, not the associated level of sunk costs. As a
result the computational burden of the model does not depend in any major way on the
form of the entry cost distribution, a fact which is particularly useful when we consider
models with multiple entry locations below.
Entry behavior can be described similarly. A potential entrant enters the market if the
expected discounted value of entering is greater than the entry cost, i.e., if χ
e
is the
indicator function which is one if the potential entrant enters and zero elsewhere
βV E(n, z; θ) ≥ κ,
where
V E(n, z; θ) ≡
X
z
0
,e,x
V (n + e − x, z0; θ)P(e, x|n, z, χe = 1)P(z
0
|z),
similarly to V C before. The main difference here is that the entrant is not active in the
current period and therefore forms beliefs slightly differently from the incumbent.
The incumbent forms beliefs conditional on it remaining active. The entrant forms beliefs
based on it becoming active. In vector form, the expression for the entrants’ continuation
values is
V E(θ, σ) = Me(˜π + βV C(θ) + βpxσ),
where the elements of Merepresent a potential entrant’s beliefs about the distribution
over tomorrow’s states conditional on that entrant becoming active. An estimator for
Methat is analogous to the one above is given by
Mce
i,j =
P
t∈T(ni,zi)
et1{(nt+1, zt+1) = (nj , zj )}
P
t∈T(ni,zi)
et
.
95
Accordingly, a consistent estimator of V Eb (θ, σ) is given by,
V Ed(θ, σ) = B˜π˜(θ) + ˜bσ (83)
where B˜ = Mce(I + βA˜), and ˜b = βMce(˜a + pbx).
3.5.2 Step 2: Estimating the Structural Parameters.
If the continuation values (V E and V C) were known, any of a number of method of moments algorithms would provide consistent estimators of (θ, σ) and maximum likelihood
would provide the efficient estimator. Since V Ed and V Cd are consistent estimators of
the unknown continuation values, an obvious way to obtain a consistent estimator is to
substitute them for V C and V E in any of these algorithms and proceed from there. For
example, the implied “pseudo” maximum likelihood estimator would maximize
l(xt, et|θ, σ) = (nt − xt) log F
φ
h
V Cd(nt, zt; θ, σ)
i
+ xtlog h1 − F
φ
(V Cd(nt, zt; θ, σ))i
+ etlog F
κ
h
V Ed(nt, zt; θ, σ)
i
+ (E − et) log h1 − F
κ
(V Ed(nt, zt; θ, σ))i,
where F
φ
is the distribution of scrap values and F
κ
is the distribution of entry costs.
POB stress the importance of remembering that V Ed and V Cd contain sampling error.
Though this sampling error does converge to zero with sample size, the fact that we have
to estimate separate continuation values for each sample point means that, for standard
sample sizes, the sampling error should not be ignored. This has implications both for
the choice of estimators, and for how we compute standard errors for any given choice.
In this context there are two problems with the pseudo maximum likelihood estimator.
First since it does not “recognize” that there is sampling error in the probabilities it uses,
events can occur that the likelihood assigns zero probability to, no matter the value of
(θ, σ) (even though the true probabilities of these events are nonzero; POB shows that
this tends to occur in their two location model). If this happens even once in the data,
the pseudo maximum likelihood estimator is not defined. Second, even if the pseudo
likelihood is well defined, it’s first order condition involves a function that is both highly
nonlinear in, and highly sensitive to, the sampling error. The nonlinearity implies that
the impact of the sampling error on the first order conditions will not average out over
sample points. The sensitivity is seen by noting that the first order effect of the sampling
error on the log likelihood will be determined by one over the probabilities of entry and
exit, and these probabilities are typically quite small.
96
POB consider two alternatives to the likelihood approach. The first is a pseudo minimum
χ
2
estimation algorithm that minimizes the sum of squares in the difference between the
observed and predicted state specific entry and exit rates (i.e., the entry and exit rates
for each observed (n, z) pair), where the predicted state specific entry and exit rates are
given by
E[xt|nt, zt] = nt ∗ P r(φi > V C(nt, zt; θ, σ)), and
E[et|nt, zt] = E ∗ P r(κ < V E(nt, zt; θ, σ)).
Their second estimator matches the overall entry and exit rates (across all observed
state pairs) to those predicted by the model, or more generally takes a sum of squares in
the differences between the predicted and actual entry and exit rates at different states
multiplied by a known function of the state variables at those states.
They show that in finite samples the pseudo minimum χ
2
estimator has an extra bias term
that reflects the sampling covariance between the estimated probability and its derivative
with respect to the parameter vector, and their Monte Carlo evidence indicates that the
extra bias term can have large effects. Thus they prefer the simplest method of moments
algorithm and show that with moderately sized samples this estimator is both easy to
calculate and performs quite well.
The second general point is that the variance of the second stage estimates, (θb,σb), depends on the variance of the first stage estimates.57 It is possible to use standard
semi-parametric formulae to obtain the asymptotic variance of the parameter estimates
analytically. However these formula are somewhat complex and can be difficult to evaluate. Moreover there is little reason to do the calculation. Since we have a complete
model and the computational burden of obtaining estimates is minimal it is relatively
easy to obtain estimates of standard errors from a parametric bootstrap.
For an empirical example which uses these techniques see Dunne, Klimek, Roberts and
Xu (2005). They estimate the parameters of a dynamic entry game from data on entry
and exit of dentists and chiropractors in small towns. They first estimate the variable
profit function (which depends on the number of active competitors) from observed data
on revenues and costs. They then employ POB’s method to provide estimates of the
sunk costs of entry and of exit values. Their parameters could be used, for example, to
predict the effect of a government subsidy intended to increase the number of medical
service professionals in small towns.
57This follows from the fact that the derivative of the objective function with respect to the estimates
of V C and V E are not conditional mean zero.
97
3.5.3 Multiple Entry Locations.
We now show how to generalize the model to allow for observed heterogeneity among
incumbents. We do this by allowing entrants to choose from multiple entry locations.
For ease of exposition, we will consider only two locations. However, expanding this to
a larger number is straightforward.
Entrants have entry costs (κ1, κ2) in the first and second locations respectively, where
entry costs are drawn from a distribution, F
κ
(κ1, κ2|θ), that is independent over time
and across agents. Note that we place no restrictions on F
κ
so that entry costs of the
same potential entrant at the different locations may be freely correlated. Once in a
particular location, the entrant cannot switch locations, but can exit to receive an exit
fee. Exit fees are an iid draw from the distribution F
φ
1
(·|θ) if the incumbent is in location
one, and an iid draw from F
φ
2
(·|θ) if the incumbent is in location two.
The Bellman equation for an incumbent in the two location model is,
V1(n1, n2, z; θ) = ˜π1(n1, n2, z; θ) + βEφ [max{φi, V C1(n1, n2, z; θ)}] ,
where the subscript “1” indicates the value function for a firm in location one and the
continuation values are,
V C1(n1, n2, z; θ) ≡
X
z
0
,e1,e2,x1,x2
V1(n1+e1−x1, n2+e2−x2, z0; θ)P(e1, e2, x1, x2|n1, n2, z, χ = 1)P(z
0
|z).
Behavior of incumbent firms is identical to before, with the probability of exit given by
(77) except using the new continuation values. However, because they have potentially
different continuation values and different scrap values, firms in location one will in
general behave differently than firms in location two.
Behavior of potential entrant firms is different from before because potential entrant
firms now have three options. They can enter location one, enter location two, or not
enter at all. A potential entrant will enter into location 1 if and only if it is a better
alternative than both not entering anywhere, and entering into location 2, i.e. if,
βV E1(n1, n2, z; θ) ≥ κ1, and βV E1(n1, n2, z; θ) − κ1 ≥ βV E2(n1, n2, z; θ) − κ2.
The entry process therefore generates a multinomial distribution with probabilities derived from the inequalities above.
Firms’ beliefs are now comprised of the probability of exit for an incumbent in location
one, the probability of exit for an incumbent in location two, the three entry probabilities
98
(location one, location two, or not at all), and the distribution determining the evolution
of the profit shifters. In computing the continuation values we now have to consider all
of these together.
Consistent estimates of a location-one incumbent’s perceived transition probabilities from
state (ni1, ni2, zi) to (nj1, nj2, zj ) are obtained analogously to before using
Mci,1
i,j =
P
t∈T(ni1,ni2,zi)
(ni1 − xt1)1{(nt+1,1, nt+1,2, zt+1) = (nj1, nj2, zj )}
P
t∈T(ni1,ni2,zi)
(ni1 − xt1)
.
Similarly, estimates of a potential location-one entrant’s perceived transition probabilities
can be obtained using
Mce,1
i,j =
1
#T(ni1, ni2, zi)
P
t∈T(ni1,ni2,zi)
et11{(nt+1,1, nt+1,2, zt+1) = (nj1, nj2, zj )}
P
t∈T(ni1,ni2,zi)
et1
.
As before these estimates are not exactly equal to the empirical frequency of state transitions but are a weighted average based on the fact that, when computing continuation
values, an incumbent assumes it will continue, and a potential entrant assumes that it
will enter.
As in the single location model, given the matrix inversion formula for continuation
values, the computational burden of obtaining estimates for the parameters of the model
is minimal. Indeed in their Monte Carlo results POB show that in two location models
with relatively large data sets (on the order of 7500 observations) one finds estimates in
under fifteen minutes on a five year old desktop computer. Most of that computation
time is devoted to computing the Markov transition matrix and its inverse. The time
required to compute the inverse can grow polynomially in the number of distinct states
and, at least given market size, this typically increases with the number of locations.
Whether it does or not depends on the structure of the matrix being inverted, and the
way one computes the inverse. Models which only allow transitions to “near by” states,
which are likely to dominate in I.O. applications, should not be as problematic in this
respect.
Second, though the estimators remain consistent when the number of entry states is
increased, their small sample properties may change. In particular, the estimates of
the continuation values will become noisier in small samples and this is likely to cause
increased small sample bias and variance in the second stage estimates. POB show that
the use of smoothing techniques, such as those discussed in the next section, can be
helpful in this context.
99
3.6 Models with Discrete and Continuous Controls: Investment Games.
In this section we consider Bajari, Benkard and Levin’s (2006) (henceforth, BBL) estimation approach in the context of the investment model in example two. The main
conceptual difference in BBL’s general approach that separates it from the above methods is that, instead of estimating continuation values directly, BBL first estimate policy
functions. Then, the estimated policy functions are used to simulate the continuation
values. As noted earlier this is similar to the single agent approach of Hotz et. al. (1994),
but BBL show that there are assumptions and techniques that allow the researcher to
use this approach in a wide class of models, including models with both discrete and
continuous controls such as investment models, some models of dynamic pricing, and
dynamic auction problems. The assumptions used do carry with them some restrictions,
and we will try to be clear on those restrictions below.
The presence of both discrete and continuous controls in the investment model affects
both the first and second stage of the estimation. In particular, the second stage is
augmented in order to incorporate information from firms’ investment, as well as its
entry/exit, choices. Additionally, when the stress is on investment we generally consider
models with larger state spaces, and, as noted above, both computation of the estimates
of continuation values, and the precision of those estimates, can become problematic.
BBL introduce simulation techniques that, depending on the structure of the model,
can cause a significant reduction in the computational burden of obtaining estimates of
the continuation values. They also use techniques that smooth estimated continuation
values across states to lower the mean square error of those estimates.
In the investment model from example two there are three policies (entry, exit, and
investment) that are set in dynamic equilibrium, and one policy (price) that is set in
static equilibrium. Since the pricing equilibrium is consistent with a large past literature
on demand estimation, we will not consider estimation of the demand and marginal cost
functions (θ1 and θ2) here as they would typically be estimated using existing methods.
Instead, we will treat those parameters as if they were known and focus on estimation
of the investment cost function (θ3) and the entry and exit costs parameters.
We assume that all of the state variables, s, are observed, as well as entry, exit, quantity,
price, and investment levels. Entry and exit costs, the cost of investment function,
and marginal costs are not observed. Note that it would be possible for some of the
state variables to be unobserved as long as they could be recovered beforehand during
estimation of the demand and cost systems. We discuss the issue of unobserved states
further in section 3.8.1.
Let ˜πi(s) represent the profits earned by firm i in the spot market equilibrium at state
100
s. Since the demand and marginal cost functions are assumed to be known, the function ˜πi(·) is also known, as the spot market equilibrium can be computed from these
primitives.
Firms maximize the expected discounted value of profits. From the beginning of a period
(prior to realization of the private shock), and for incumbent firms, this is
E
X∞
t=0
β
t

{χit = 1} (˜πi(st) − C(Iit, νit; θ3)) + (χit − χi,t−1)
−Φ

s0

, (84)
where χit = 1 indicates that the incumbent continues in the market at period t and
χit = 0 indicates that the incumbent exits, and it is understood that each exiting firm
receives the same exit value and never operates thereafter. Note that unlike in the
entry/exit example above, in this model we assume that the incumbent’s choice of its
discrete control (whether or not to exit) is not subject to a random cost shock.
For expositional (and computational) simplicity we will assume the following quadratic
cost of investment function,
C(I, ν; θ3) = {I ≥ 0}(θ3,0 + θ3,1I + θ3,2I
2 + θ3,3νI). (85)
The indicator function for I ≥ 0 above allows for an adjustment cost that is incurred
only if investment is nonzero. Zero investment is a phenomenon that is often observed
and can easily result from either flatness of the value function reflecting low returns
to investment (see Ericson and Pakes (1995)), or non convex investment costs (e.g.,
Caballero and Engle (1999)).
Potential entrant firms’ expected discounted values are similar to (84), except that in
the initial period they must pay a random entry cost, κit, in order to enter. We assume
that entrants take one period to setup the firm and therefore do not earn profits in the
spot market and do not invest until the subsequent period. For ease of exposition, we
also assume that entrants always enter at the same initial state.58
3.6.1 Step 1: Estimating Continuation Values.
The goal of the first step of the estimation procedure is to compute the continuation
values given by the expected discounted values in (84), under equilibrium strategies.
These expected discounted values are functions of the profits earned at each state and
the probability distributions determining future states and actions conditional on the
starting state, s0.
58It is straightforward to allow entrants’ initial state to be randomly determined.
101
BBL compute estimates of the continuation values by first estimating policies for each
state, then using the estimated policies to simulate sample paths of industry states and
actions, and then evaluating discounted profits on each sample path. In order to do
this we need both the ability to simulate sample paths of states and actions, and the
ability to evaluate profits along those paths given the states and actions in each period.
Evaluating profits requires a (possibly flexible) description of the profit function and
knowledge of the distribution of the private shocks, at least up to a parameter vector to
be estimated. We treat these two as “primitives” of the dynamic model.
The evolution of the states depends on firms’ entry, exit, and investment policies. BBL
recover these policy functions using the observed data. In our investment model, private
information is known to the firms before any actions are taken, so in MPE, strategies for
investment, exit and entry are functions of both the states and this private information,
I(st, νit), χ(st, νit), and χ
e
(st, κit),
where χ is the exit policy function for incumbent firms, χ
e
is the entry policy function
and χ
e = 1 indicates that the entrant enters the market. Since potential entrants cannot
invest in the first period, entry strategies depend only on the random entry cost. Both
the investment and exit strategies depend only on the shock to the marginal cost of
investment.
Consider first exit and entry. The optimal exit policy has the form of a stopping rule
χi,t = 1 iff νit ≤ ν¯(st).
All we require is a nonparametric estimate of the probability that χ = 1 conditional
on st. Similarly, there a critical entry level of κ conditional on st that determines
entry, and the entry policy is obtained as a nonparametric estimate of the probability
of entry conditional on st. In both cases we also have the restriction that the policies
are exchangeable in rivals’ states. In models with large state spaces, such that there are
some states in the data with few or zero observations, it would typically be optimal to
employ some smoothing in these estimates. In their Monte Carlo studies, BBL found
that local linear methods worked well for this.
As far as investment is concerned, one can show that, conditional on a firm continuing
in the market, investment is a (weakly) monotone function of νit, I(st, νit). Thus, if we
knew the distribution of investment at each state, F(Iit|st), we could map the quantiles
of ν into investment levels at each state. More precisely, the investment policy function
is given by
F
−1
I|s
(G(ν|s)).
The function G is a primitive of the model known up to a parameter vector, and the
function F can be recovered nonparametrically.
102
There is an additional complication and that is that investment is not observed for firms
that exit the market, which happens if νit > ν¯(st). However, since both the exit and
investment rules are monotonic in the shock, this is handled easily. Conditional on a
firm continuing in the market, we observe the distribution of investment conditional on
s that corresponds to νit ≤ ν¯(st). Therefore, if we first estimate the probability of exit
at each state, and then recover the distribution of investment at each state conditional
on staying in the market, then we have a complete description of the optimal exit and
investment policy functions.59 Note also that in the simulations below it is important
that we maintain this link between exit and investment since one draw on the private
shock to investment, νit, determines both policies.
If there was a second unobservable, say a random exit fee φit, then the exit decision would
depend on both (νit, φit). The probability of exit could still be obtained as above, but
the distribution of investment conditional on not exiting would depend on both νit and
φit. Then, without further restrictions it would not be possible to invert the observed
distribution of investment to obtain the policy decision as a function of ν conditional on
s and not exiting.60
There also remains the question of how best to estimate the investment function, and
this depends to some extent on its likely properties. Here it is important to keep in mind
that investment is a complicated function of the primitives. Indeed the only restriction
we have on its form is that it is exchangeable in the states of the competitors (which is
already embodied in the definition of s). Standard nonparametric approaches assume a
certain amount of smoothness that is not necessarily guaranteed by the primitives of the
model. The theoretical properties of the investment function in the EP model depend
on the underlying properties of the family {P(sit+1|Iit, sit)}. If conditional on sit the
points of support of this family do not depend on Iit
61, then by appropriate choice of
primitives one can ensure that the investment function is smooth; see EP, the Monte
Carlo evidence in BBL, and the generalizations of this in Dorazelski and Satterwaithe
(2004). In their Monte Carlo studies BBL also found that local linear regression worked
well for estimating the investment function.
Assume now that, for each state, we have consistent estimators of the entry probability,
the exit probability, the investment distribution, and the distribution of future states.
59We cannot use the data to learn what an exiting firm would have invested had it stayed in the
market, but it is not necessary to know this.
60Note that the problem here is that there is more than one error influencing the choice of investment.
Therefore, a feasible alternative would be to allow a random exit cost but no shock to the marginal cost
of investment.
61This assumption allows for stochastic outcomes to investment processes which is an assumption
often made in Industrial Organization. However it does rule out the deterministic accumulation models
traditionally used in growth theory.
103
This is all one needs to compute consistent estimates of the continuation values in (84)
as a function of the parameters. To do so analytically, however, would involve high
dimensional integration, so what BBL do is show how to extend the “forward simulation”
idea in Hotz, Miller, Saunders and Smith (1994) to simplify the analysis of the more
complex problems they deal with.
Starting from a given state, s0, one draw is taken on the shock to the marginal cost of
investment for the firm of interest, νi. This draw determines the firm’s investment and
exit policies through the estimated policy functions above (i.e., the same draw determines
the correct quantile for both investment and exit, as discussed above). These policies,
along with the state and the value of the private shock, determine current profits as
a function of the parameters. Draws are then taken for the investment shocks for the
remaining incumbent firms, and one draw on the entry distribution is taken for the
potential entrant. These draws, along with draws determining the outcomes of each
firm’s investment process, determine s1. The process is repeated to obtain one simulated
path of states and the associated discounted stream of profits. Many such paths can
be simulated to obtain an estimate of Vi(s0). Consistency of the estimation algorithm
requires that the number of simulated paths goes to infinity.
This forward simulation procedure is not too computationally burdensome, though one
does have to hold one set of simulation draws in memory and use these same draws
to evaluate the continuation values at the different values of θ tried in the estimation
algorithm. Moreover, much of what computational burden remains disappears when we
deal with models that are linear in the parameters. For example, suppose we consider the
investment model above where the private shock to investment has a normal distribution,
ν ∼ N(0, 1). (The investment shock is normalized to be standard normal without loss of
generality because its mean and variance parameters are absorbed into the parameters
θ3,0 and θ3,3.) Since all of the parameters enter the continuation values linearly, they
can be factored out as follows,
Vi(s0; σi, σ−i) =E
X∞
t=0
β
t
{χit = 1}π˜i(st) − θ3,0E
X∞
t=0
β
t
{χit = 1}{Iit ≥ 0}−
θ3,1E
X∞
t=0
β
t
{χit = 1}{Iit ≥ 0}Iit − θ3,2E
X∞
t=0
β
t
{χit = 1}{Iit ≥ 0}I
2
it−
θ3,3E
X∞
t=0
β
t
{χit = 1}{Iit ≥ 0}Iitνit + ΨE
X∞
t=0
β
t
(χit − χi,t−1)
−
≡Wi(s0; σi, σ0
i
)
0


1
θ3
Ψ

 ,
(86)
104
where Wi(s0, σi
, σ−i) represents the expected discounted value terms above when i follows
policy σi and rival firms follow policy σ−i. The estimated continuation values are then
computed by plugging in the estimated policies and simulating the expectations terms,
Vbi(s0; σbi, σb
0
i
) = Wci(s0; σbi, σb
0
i
)
0


1
θ3
Ψ

 .
The key observation here is that if the model is linear in the parameters, then the
parameters factor out of the continuation value calculations. In that case the W terms
need only be computed once, and the continuation values at different values of the
parameter vector can be obtained by multiplying two small dimensional vectors.
This simplification is an extension of the one used in the entry/exit example above,
except here we exploit linearity in the investment cost parameters as well as the linearity
in the period profits. Since the continuation values need to be calculated many times
in the second step of the estimation, and since computing continuation values is the
primary source of computational burden, such simplifications can lead to a substantial
reduction in the overall computational burden of the estimator.
3.6.2 Step 2: Estimating the Structural Parameters.
As with the entry model above, once the continuation values have been estimated there
are potentially many ways of estimating the structural parameters. The main difference
is that now there is one continuous control variable (investment) in addition to the two
discrete controls (entry/exit), and we want to use the information in the continuous
control to help estimate the parameters. Accordingly all the issues that arose in the
discussion of estimation of the entry/exit model are also relevant here. In particular
there is error in the estimated continuation values that can contaminate the second
stage estimates, so it is desirable to find a second step estimator that is close to linear
in the estimated continuation values.
There are at least three possible estimators; (i) an inequality estimator that finds a value
of the parameter vector that insures that the observed policies generate higher simulated
continuation values than alternative policies (see below), (ii) a method of moments estimator that fits the mean of the policies implied by the simulated continuation values (i.e.,
at each state in the data you substitute the simulated continuation values into the right
hand side of the Bellman equation (65) and solve for the optimal policy) to nonparametric estimates of the policies at each state, and (iii) a method of moments estimator
that fits the nonparametric estimates of the distribution of the polices to the distribution
of policies predicted by the simulated continuation values at each state. BBL provide
105
Monte Carlo evidence on the first two of these. Here we review the inequality estimator,
that is the estimator found by satisfying the optimality inequalities (64) that define the
MPE for the simulated values.
At the true values of the parameters, for all states, s0, all firms, i, and all alternatives,
σ
0
i
, it must be that
Wi(s0; σi, σ−i)
0


1
θ3
Ψ

 ≥ Wi(s0; σ
0
i
, σ−i)
0


1
θ3
Ψ

 .
Let x refer to a particular (i, s0, σ0) combination, such that x indexes inequalities, and
let
g(x; θ3, Ψ) = (Wi(s0; σi, σ−i) − Wi(s0; σ
0
i
, σ−i))0


1
θ3
Ψ

 .
Then it must be the case that g(x; θ3, Ψ) ≥ 0 at the true values of the parameters for
every x.
A natural thing to do in estimation would be to compute g at the estimated policies
from the first stage and then find the values of the parameters that best satisfy the
entire set of inequalities. However, when there are continuous controls this is difficult
because there are too many possible alternative policies. Instead, BBL use simulation to
choose a small subset of the inequalities to impose in estimation. The inequalities can be
chosen according to any random rule that selects all of them asymptotically. However,
it is important to remember that the exact rule used will influence efficiency. In their
Monte Carlo studies, for investment alternatives BBL use policies of the form
I
0
(st, νit) = Ib(st, νit) + 
0
where 
0
is drawn from a normal distribution with mean zero and standard deviation
chosen by the researcher. Alternative entry and exit policies were chosen similarly by
shifting the cutoff rule by an amount 
0 drawn from a normal distribution.
Suppose niinequalities are sampled, and let gbns(x; θ3, Ψ) be a simulator for g(x; θ3, Ψ)
evaluated at the estimated policy functions, where ns is the number of simulation draws
used to simulate each Wi term. Then the inequality estimator minimizes the objective
function
1
nI
XnI
k=1
1{gbns(xk; θ3, Ψ) < 0}gbns
(xk; θ3, Ψ)2.
Because the estimator is computationally light, it is easy to choose (nI , ns) to be large
enough that the simulation contributes nothing to the variance of the estimator. All
106
of the variance comes from error in the estimation of the continuation values. BBL
work out the asymptotic distribution of the estimator. However, the expression is difficult to evaluate and in practice the simplest way to compute standard errors is to use
subsampling or a bootstrap.
The inequality estimator has several advantages. One is that it is very easy to implement
even in complex models. It is conceptually simple and requires a minimum of computer
programming, the main programming burden being the forward simulation routine. Additionally, the method can be used with very little alteration even if the model is only
set-identified. In that case, all that is required is to use an alternative method for computing standard errors (see BBL, as well as Chernozhukov, Hong, and Tamer (2004) for
details).
However, one potential disadvantage of the approach is that, similarly to the pseudolikelihood methods shown in the examples above, the estimator is nonlinear in the first
stage estimates, and therefore the estimates obtained are likely to contain small sample
bias. For that reason, BBL also tested a natural alternative estimator based on a set
of moment conditions that match the observed choice data. The general idea of this
estimator is to substitute the estimated continuation values into the right hand side
of the Bellman equation and then solve for an optimal policy rule conditional on those
continuation values. This estimator is linear in the estimated continuation values, though
those values are still nonlinear functions of the estimated policies. The expected value
of the optimal policy is then matched against the average policy observed at each state
in the data. In their Monte Carlo studies BBL found that this second estimator did help
reduce small sample bias in the second stage estimates.
For an empirical example that uses these techniques see Ryan (2005). He estimates
the parameters of a dynamic oligopoly model of US cement producers. In the first
stage he estimates the static profits demand and cost parameters using demand data
and a static equilibrium assumption. He also estimates the entry, exit, and investment
policy functions using data on the set of firms operating in a panel of markets and their
capacities. In the second stage he uses BBL’s inequality estimator to estimate the sunk
costs of entry and exit, as well as the adjustment costs of investment. He finds that the
1990 amendments to the Clean Air Act significantly raised the sunk costs of entry in
the cement industry, and that a static analysis would have missed an associated welfare
penalty to consumers.
107
3.6.3 An Alternative Approach.
Berry and Pakes (2002) provide an alternative approach for estimating models with
continuous choice variables that uses quite different assumptions from POB or BBL.
They assume that profits are observable up to a parameter vector to be estimated, but
do not require that the state variables that determine current and expected future profits
are observed, and do not even require the researcher to specify what those state variables
are. In applications where the environment is complex, but sales and investment data
are quite good, this alternative set of assumptions can be quite attractive.
Let the random variable τi refer to the period in which firm i exits the market. Then,
firm i’s continuation value in the investment game starting at state s0 is
Vi(st) = E
"Xτi
r=t
β
r−t
(˜πi(sr) − C(σ(sr); θ3)) + β
τi−tΦ

st
#
(87)
where σ is the equilibrium policy function. Note that we have assumed there is no private
shock to investment; an assumption that is needed for the consistency of this estimator.
Berry and Pakes note that, if firms have rational expectations, then the actual discounted
stream of profits earned by a given firm is an unbiased estimate of its expected discounted
profits. Suppose that profits (˜πit), investment (Iit), and exit (χit) are observed. Then
the actual discounted sum of profits earned by the firm (corresponding to (87)) is
Vbi(st; θ3, Φ) ≡
Xτi
r=t
β
r−t
(˜πir − C(Iir; θ3)) + β
τi−tΦ, (88)
where, in a slight abuse of notation, τi now refers to the actual period in which the firm
exited. By rational expectations we have that, at the true values of the parameters,
Vbi(st; θ3, Φ) = Vi(st) + it
where E[it|st] = 0.
A unique feature of the Berry and Pakes approach is that the estimated continuation
values here are unbiased. However, in contrast to POB and BBL, Berry and Pakes
(2002) do not have a first stage that provides consistent estimates of continuation values.
Since the state variables are assumed not to be observed, there is no longer any way of
identifying a set of data points that correspond to the same state vector. Thus, there
is no way to average out across observations so as to obtain consistent estimates of the
continuation values, as in POB and BBL.
108
Berry and Pakes get around the problem of having only unbiased, and not consistent
estimates of continuation values, by using an estimating equation that has the error in the
estimated continuation value entering linearly. More precisely, their estimating equation
is derived from the first order condition for the firm’s continuous control. Conditional
on investment being strictly positive (a condition that is determined by the information
available when the investment decision is made, and hence that is independent of the
realization of it), that first order condition is obtained by setting the derivative of (87)
equal to zero. Using the cost of investment function after eliminating the i.i.d. shock to
investment this gives us
0 = −θ3,1 − 2 ∗ θ3,2 ∗ Iit + β
X
st+1
Vi(st+1)
∂
∂I P(st+1|Iit, st
, χit = 1)
= −θ3,1 − 2 ∗ θ3,2 ∗ Iit + β
X
st+1
Vi(st+1)
∂
∂I P(st+1|Iit, st
, χit = 1)
P(st+1|Iit, st, χit = 1) P(st+1|Iit, st
, χit = 1)
= −θ3,1 − 2 ∗ θ3,2 ∗ Iit + βE Vi(st+1)
∂ ln P(st+1|Iit, st, χit = 1)
∂I

st, Iit, χit = 1
(89)
Adding and subtracting a term in Vb(st+1; θ3, Ψ) gives
0 = −θ3,1 − 2 ∗ θ3,2 ∗ Iit + Vbi(st+1; θ3, Φ)∂ ln P(st+1|Iit, st
, χit = 1)
∂I + ηit(θ3, Φ) (90)
where we have defined
ηit(θ3, Φ) ≡ βE Vi(st+1)
∂ ln P(st+1|Iit, st, χit = 1)
∂I

st, Iit, χit = 1
− Vbi(st+1; θ3, Φ)∂ ln P(st+1|Iit, st
, χit = 1)
∂I , (91)
and consequently
E[ηit(θ3, Φ)|st] = 0, (92)
at the true values of the parameters vector. Condition (92) follows from the twin facts
that Vi(st+1) = Vbi(st+1; θ3, Φ) − i,t+1 and
E

∂ ln P(st+1|Iit, st, χit = 1)
∂I 
i,t+1 = 0,
as the derivative is a function of information known at t. It follows that (90) provides a
set of conditional moment restrictions that can be used as the basis for estimation.
109
There are a number of disadvantages of this approach. One that can potentially be corrected is that as presented in Berry and Pakes (2002) the algorithm does not incorporate
the additional information in the data that comes from the choice of discrete controls
(e.g. entry and exit), or from controls chosen to be at a corner of the choice set (e.g.,
Ii,t = 0). One could add a set of inequality constraints to the Berry-Pakes model to
account for entry and exit and the Ii,t = 0 case. Also, as mentioned above, it is difficult
to incorporate a shock to the cost of investment into this model.
However the major difference between this model and the other models discussed above
is that Berry and Pakes do not need to specify and control for all the state variables in
the dynamic system. This is an obvious advantage for complex problems. Of course,
if we can not identify and control for all the state variables of the system, we can not
make use of averaging techniques that enable us to use information on similar states
to construct estimates of the policies and returns at a given state. In problems where
the state variables are easy to identify and control for, averaging techniques can be very
helpful in reducing variance. It remains to be seen if hybrids can be developed that make
effective use of all of these techniques.
3.7 A Dynamic Auction Game.
In this section we consider estimation of the auction model in example three. This
section closely follows Jofre-Benet and Pesendorfer (2003) (henceforth, JP). We assume
that all bids, contract characteristics, and bidders’ state variables are observed. A unique
feature of the auction model is that the period payoff function is not a function of any
unknown parameters. The goal of estimation, then, is to recover the distribution of
bidders’ privately known costs at each state.
Since the outcome of the auction affects not only current profits but also the firm’s
backlog, firms choose their bids so as to maximize the expected discounted value of future
profits. Recall that zt provides the characteristics of the contracts to be auctioned in
month t and evolves as a Markov process, ωi,t provides the backlog of work of firm i
in period t, and if ωt = (ω1,t, . . . , ωN,t), then ωt+1 = Γ(ωt, zt, j) where j is the winning
bidder.
It is convenient to write the maximization problem from the beginning of a period, prior
to realization of the private shock and prior to realization of the contract characteristics.
Then firms choose their bidding strategy so as to maximize the expected discounted sum
110
E
"X∞
t=0
β
t
(bit − cit){bit ≤ min
j
(bjt)}

ω0, z−1
#
, (93)
where z−1 refers to last period’s contract and where the expectation is defined over rival’s
bids in all periods as well as over the firm’s own costs in all periods. Due to the Markov
structure, this maximization problem can be written recursively,
Vi(ωt, zt−1) = Z Z max
bit 
(bit − cit)P r(i wins|bit, ωt, zt)+
β
X
N
j=1
P r(j wins|bit, ωt, zt)Vi(Γ(ωt, zt, j), zt)

dF(cit|ωt, zt)dG(zt|zt−1). (94)
As is now common practice in the empirical auctions literature (Guerre, Perrigne, and
Vuong (2000)), JP show that bidders’ costs can be recovered by inverting the first order
condition associated with the optimal bid. Let Gi(·|ωt, zt) be the distribution of bids
submitted by bidder i conditional on the state variables and g
i
(·|ωt, zt) be the density
function. Let
h
i
(·|ωt, zt) = g
i
(·|ωt, zt)
1 − Gi(·|ωt, zt)
denote the associated hazard function, and note that
∂P r(i wins|bit, ωt, zt)
∂bi,t
= −
X
j6=i
h
j
(bit|ωt, zt)P r(i wins|bit, ωt, zt)
while
∂P r(j wins|bit, ωt, zt)
∂bi,t
= h
j
(bit|ωt, zt)P r(i wins|bit, ωt, zt).
Using these expressions, the first order condition for optimal bids yields the equation
bit = cit+
1
P
j6=i
h
j (bit|ωt
, zt)
−β
X
j6=i
h
j
(bit|ωt, zt)
P
l6=i
h
l(bit|ωt
, zt)

Vi(Γ(ωt, zt, i), zt)−Vi(Γ(ωt, zt, j), zt)

.
(95)
The optimal bid equals the cost plus a markup that has two terms. The first term reflects
competition in the current auction. The second term accounts for the incremental effect
on future profits of firm i winning today’s auction.
Since the first order condition is strictly increasing in c it can be inverted to obtain
c = φ(b|ωt, zt), (96)
111
where φ is a function of the observed bids, the hazard function of bids, h, the transition
function, Γ, and the continuation values, V . The transition function is a known function.
Since the bids, contract characteristics, and state variables are observed, the hazard
function of bids can be obtained from the data. Thus, if the continuation values were
known, then the relationship in (96) could be used to infer bidders’ costs. Hence, as in
the examples above, in order to estimate the parameters of the cost distribution we need
first to obtain estimates of the continuation values.
3.7.1 Estimating Continuation Values.
In order to estimate the continuation values, JP note that the continuation values can
be written as a function only of the distribution of bids. The easiest way to see this
is to inspect (93). The expected discounted value involves terms in the probability of
winning, which can be derived from the distribution of bids, and terms in the expected
markup. Equation (95) shows that the optimal markup is a function of the distribution
of bids and the continuation values. JP show that by combining these two equations it
is possible to write the continuation values as a function only of the distribution of bids.
The derivation is long so we omit it here and instead refer readers to the appendix of JP.
Proposition 1 in JP shows that equations (94) and (95) can be manipulated to obtain,
Vi(ωt, zt−1) = Z  Z
1
P
j6=i
h
j (·|ωt
, zt)
dG(i)(·|ωt, zt)
+ β
X
j6=i
"
P r(j wins|ωt, zt) + Z
h
i
(·|ωt, zt)
P
l6=i
h
l(·|ωt
, zt)
dG(j)(·|ωt, zt)
#
× Vi(Γ(ωt, zt, j), zt)

dG(zt|zt−1),
(97)
where the notation
G
(i)
(·) = Πk6=i
h
1 − G
k
(b|ωt, zt)
i
g
i
(b|ωt, zt).
The terms in square brackets in the second line of (97) sum to one and therefore can
be interpreted as transition probabilities. This interpretation leads to the following
construction. Assume that the state space is discrete and let Ai be a vector with one
element for each state representing the first term above,
Ai(s) = Z Z 1
P
j6=i
h
j (·|ωt
, zt)
dG(i)(·|ωt, zt)dG(zt|zt−1).
112
Next, construct the matrix Mi
such that each element (k, l) reflects the transition probabilities above,
Mi
k,l =
( h
P r(j wins|ωk, zl) + R h
i P
(·|ωk,zl)
l6=i
hl(·|ωk,zl)
dG(j)(·|ωk, zl)
i
P r(zl|zk), if ωl = Γ(ωk, zl, j)
0, otherwise.
Then the value function can be expressed as,
Vi = [I − βMi]
−1Ai (98)
The matrices Mi and Ai can be estimated using estimates of the bid distribution.
3.7.2 Estimating the Cost Distribution.
Once the continuation values are known, estimating the cost distribution is straightforward. There is a relationship between the cost distribution and the bid distribution that
is given by
F(c|ωt, zt) =G(b(c, ωt, zt)|ωt, zt)
=G(φ
−1
(c|ωt, zt)|ωt, zt)
(provided that φ is invertible). The function φ can be estimated using the first order
condition, (95), and the estimated continuation values. The estimated φ can then be
substituted into the estimated bid distribution in order to obtain an estimate of the cost
distribution.
3.8 Outstanding Issues.
The literature on structural estimation of dynamic games is relatively recent. As a result our focus has been on reviewing assumptions and techniques that make it feasible
to use the implications of dynamic games to make inferences on parameters of interest
to I.O. We have paid little attention to a host of related issues including; the asymptotic efficiency of alternative estimators, the small sample properties of those estimators,
identification in the absence of auxiliary information, and the likely validity of various
assumptions.
It is not our attention to minimize any of these issues. Indeed we think it important to
explore all of them, particularly the assumptions underlying the analysis. This includes
the behavioral assumptions and the assumptions regarding the selection of equilibria, as
113
well as more traditional assumptions on the properties of the unobservables in the model.
The simple fact is that we have little to report on most of these issues. There is however
one exception; problems that arise due to the presence of serially correlate unobserved
state variables. Since this is an issue that has appeared in several related literatures, we
do have some idea of how to deal with it in the context of estimating dynamic games.
3.8.1 Serially Correlated Unobserved State Variables.
In all of the examples above it is assumed that all of the states that are commonly known
to the agents are also observed by the econometrician. In many empirical applications
this assumption is questionable. For example, in many cases we might expect there to
be an aggregate shock to profits that is known to all of the firms, but not controlled for
by the econometrician. The models presented above can be modified to accommodate
these shocks if they are i.i.d. over time. However we would often expect aggregate shocks
to be serially correlated, just as most aggregate variables are. In that case, behavior in
previous periods would depend on previous realizations of the unobserved states, leading
to a correlation between today’s values of the observed and unobserved states.
The statement of the problems caused by unobserved serially correlated state variables
in dynamic models with discrete outcomes dates back at least to Heckman (1981). Pakes
(1993) reviews three possible solutions to the problem: (i) solving for the unobserved
states as a function of observables, (ii) simulating the model from a truly exogenous
distribution of initial conditions, and (iii) using the ergodic distribution to model the
long run relationship between the unobserved and observed states. With the advent of
longer panels there is also the possibility of using techniques that allow one or more
parameters to differ across markets in a panel of markets (say a market specific time
invariant profit parameter, or a separate initial condition for each market), and then
estimate those parameters pointwise.
The first case is quite promising in contexts where there is an observable continuous
response to the unobservable state. Then conditional on the parameter vector, there is
often a one to one relationship between the unobserved states and the observed states
and controls. Several papers in the literature on static demand system estimation (Berry
(1994), Berry, Levinsohn, and Pakes (1995), and Bajari and Benkard (2005)) have used
such a condition to recover serially correlated unobserved product characteristics using
data on quantities, prices, and observed product characteristics. Timmins (2002) uses a
similar procedure to control for the initial conditions in a single agent dynamic control
problem with unobserved state variables. Olley and Pakes (1996) use the implications of
a dynamic Markov Perfect equilibrium model to recover a serially correlated productivity
term. These methods could be used to recover the unobserved state variables prior to
114
the dynamic estimation, and then the unobserved state variables could be treated as
if they were observed in the dynamic estimation algorithm (at least up to estimation
error).
Things become more difficult when the unobserved states are not recoverable in this way.
In single-agent dynamic models, several papers (e.g., Pakes (1986) and Keane and Wolpin
(1997)) have used exogenous initial conditions to solve the problem of serially correlated
unobserved states. Starting with an exogenous initial distribution of states, the model
can be used to simulate the relationship between the observed and unobserved states in
future periods. However, while there may be many reasonable ways of modelling initial
conditions for a single agent (be it a firm or an individual), such conditions tend to be
less realistic for an industry, whose history is typically much longer than the available
data.
The third case is perhaps more appropriate for industry studies. Instead of using an
exogenous initial condition for the unobserved states at the time the the market starts
up, we assume that the draws on the joint distribution of the unobserved states and
the initial condition are draws from an invariant distribution. That distribution is then
estimated along with the other parameters of the problem. The rational here is that
if the markets in question have been in existence long enough, the joint distribution of
the initial condition and the unobserved state will not depend on the early years of the
industry’s evolution. Rather it will depend only on the limiting structure of the Markov
process generated by the nature of the dynamic game, a structure we can analyze.
Aguirregabiria, V. and P. Mira (2006) implement one version of this solution. They allow
for an unobserved fixed effect which varies across markets and assume both that the fixed
effect can only take on a finite number of values and that the transition probabilities
for the observed exogenous variables are independent of the values of the fixed effect.
They then solve for an invariant distribution of the state of the system and the fixed
effect, form the conditional distribution of the initial condition given the fixed effect, and
integrate out over possible values of the fixed effect. They report that allowing for the
fixed effect has a noticeable impact on their empirical results.
Of course if one has a reasonably long panel of markets one should be able to estimate the
fixed effect (or some other unobserved initial condition) pointwise; our fourth solution
possibility. In that case continuation values could be estimated as described above but
separately for each market in the data. The observations across markets could then be
pooled together in the second stage in order to estimate the structural parameters that
are the same across markets. This would lead to substantially higher estimation error
in the continuation values, and one might want to think hard about estimators that
115
would be designed to minimize the impact of these errors62. Some Monte Carlo work on
just how long a panel is likely to be required for this procedure to be fruitful would be
extremely helpful.
62Note that this process could also solve the multiple equilibrium problem that might exist across
markets.
116
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None
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