# Forecasting with Feedback††thanks: We thank Darrel Cohen, Graham Elliott,
Keri Hu, Navin Kartik, Julian Martinez-Iriarte, Katrin Rabitsch, Joel Sobel,
Ross Starr and Yuehui Wang for useful comments.

Robert P<EMAIL_ADDRESS>
Department of Economics
Central European University  Augusto Nieto-
<EMAIL_ADDRESS>
Department of Economics
Universidad Nacional de Tucuman

###### Abstract

Systematically biased forecasts are typically interpreted as evidence of
forecasters’ irrationality and/or asymmetric loss. In this paper we propose an
alternative explanation: when forecasts inform economic policy decisions, and
the resulting actions affect the realization of the forecast target itself,
forecasts may be optimally biased even under quadratic loss. The result arises
in environments in which the forecaster is uncertain about the decision
maker’s reaction to the forecast, which is presumably the case in most
applications. We illustrate the empirical relevance of our theory by reviewing
some stylized properties of Green Book inflation forecasts and relating them
to the predictions from our model. Our results point out that the presence of
policy feedback poses a challenge to traditional tests of forecast
rationality.

Keywords: feedback, biased forecasts, forecast rationality, forecast
evaluation, quadratic loss

JEL codes: C53, E37, E58, D82

##  1 Introduction

It is generally agreed that the main reason why government institutions and
private entities engage in forecasting is to improve decision-making.333White
(1966), Granger (1969), Pesaran and Skouras (2004), Granger and Machina
(2006), Patton and Timmermann (2007), Faust and Wright (2008), and a host of
other authors have made this point at least in passing. For example, many
people use weather forecasts in their daily lives to make decisions that range
from the trivial, e.g., whether to go hiking or to the movies, to the more
consequential, e.g., whether to evacuate before a storm, or how much to
irrigate a field. Weather forecasting is an example of a relatively simple
setting in which the decisions made on the basis of the forecast do not
directly affect the realized outcome. In the language of our paper, this
forecasting-decision environment is free of (forecast-induced) _feedback_.
When it comes to forecasting some economic variables, the setting is
frequently more interactive. Forecasting and decision-making may take place
within the same institution and, more importantly, the actions taken in
response to forecasts often have an impact on the very outcome that the
forecast was meant to predict. The following two examples illustrate this type
of environment.

Example 1: The marketing department of a firm produces a sales forecast for a
product line for the next fiscal quarter. The managers of the firm have a
minimum sales target that they wish to meet. If the sales forecast is
substantially below the target, management may decide to take corrective
action, such as increasing marketing efforts, offering discounts, etc. These
actions are aimed at, and may succeed in, increasing future sales. Hence, the
ex-post accuracy of the original forecast may depend on the decisions made in
response to it.

Example 2: Central bank staff forecasts serve a dual role. First, as with any
other forecast, they are meant to provide an accurate prediction of realized
inflation over some horizon. Second, they also serve as an input for setting
the policy rate in a way that guides the economy toward an intermediate or
long run inflation target. Thus, forecast performance is hard to judge without
accounting for the feedback from the forecast to realized inflation.

These examples highlight the feedback mechanism that is naturally present in
many economic forecasting problems. Some decision makers may follow mechanical
decision rules that use forecasts as inputs. Others may be more sophisticated,
trying to “invert” forecasts to extract private information known by
forecasters, and to act on this information in an optimal way. In either case,
if these actions affect the outcome, then one can expect forward-looking
forecasters to adjust their forecasts in anticipation of the decision maker’s
reaction. This type of environment, while common in economic applications, has
nonetheless received little attention in the economic forecasting literature.
Our research fits into this gap.

More specifically, the goal of this paper is to derive the statistical
properties of forecasts that are produced in environments with feedback
effects. To this end, we present a simple, static model of the interaction
between a forecaster and a decision maker (DM).444We will henceforth use the
pronoun ‘he’ to refer to the forecaster and ‘she’ to the decision maker. We
use our model to derive optimal (equilibrium) forecasts, and we provide
explicit expressions for the optimal bias and the coefficients of the
associated Mincer-Zarnowitz (MZ) regression function.555The Mincer and
Zarnowitz (1969) regression consists of regressing the realized outcome on the
forecast. Testing whether the slope is one and the intercept is zero is
traditionally interpreted as a “rationality test” in the forecast evaluation
literature. We concentrate, for simplicity, on the case in which there is a
single forecaster and a single DM.

Our theoretical model borrows its basic structure from communication games,
pioneered by Crawford and Sobel (1982), where the forecast is interpreted as a
message, the forecaster plays the role of the “sender” of the message, and the
DM the “receiver” of the message. More specifically, using his private
information about the state of the economy, the forecaster predicts the future
value of an economic variable with the objective of minimizing the mean
squared error (MSE) of the prediction. The DM is also endowed with quadratic
loss: her payoff depends on the squared distance between the realized outcome
and an exogenous target value. Thus, the main objective of the DM is to steer
the outcome toward the desired target, and she can do so by first observing
the forecast and then choosing an action that affects the outcome. The
forecaster, in turn, is forward-looking in that he tries to anticipate the
DM’s reaction and account for it in constructing the forecast. However, we
assume that the forecaster faces some uncertainty about the DM’s objective, so
he is not able to predict the DM’s reaction perfectly. This uncertainty,
coupled with the forecast-based policy feedback, are the main drivers of our
results.

Our setup departs from standard models of costless communication (e.g.,
Crawford and Sobel (1982)) in two ways. First, in our model the forecaster’s
message (i.e., the forecast) enters his loss function directly, making ours a
game of “costly talk,” as in Kartik et al. (2007). Second, we explicitly model
the mechanism by which the state of the economy and the policy action jointly
determine the realization of the outcome. This causality is absent in standard
communication games, where the realization of the target variable is
exogenous.

The main results from our model are as follows. First, we show that in the
presence of feedback, and provided that the forecaster faces a moderate amount
of uncertainty about the DM’s reaction to the forecast, the optimal
(equilibrium) forecast is biased, and the slope and intercept of the MZ
regression differ from one and zero, respectively.666If there is no such
uncertainty, the forecaster can perfectly anticipate the feedback and correct
for it. If there is too much uncertainty, then an equilibrium does not exist.
Second, we provide analytical formulas for the bias and the MZ regression
coefficients in terms of the mean and variance of the uncertain parameter in
the DM’s reaction function. Finally, we provide results in case the DM follows
a mechanical policy rule rather than being fully strategic.

The mechanism that is behind the results described above is a straightforward
bias-variance tradeoff. When the forecast is used for policy, the forecaster
recognizes that the outcome is a function of the forecast and tries to use his
knowledge about the DM’s reaction to make a more accurate prediction. However,
as we will show, the uncertain parameter in the DM’s reaction function
interacts with the forecast in a way so that a forecast with a larger absolute
value increases the volatility of the outcome, which, in turn, increases the
forecaster’s expected loss. This tradeoff is resolved optimally by choosing a
forecast that is lower (in absolute value) than the unbiased prediction based
on the DM’s expected reaction function. That is, the forecaster mitigates the
volatility effect of his forecast by making it less sensitive to his
information than the unbiased forecast. This modification results in a
forecast with lower MSE but some bias.

#### Contributions to the literature

There is little work aimed at studying the properties of optimal forecasts
under feedback. In an early paper, Grunberg and Modigliani (1954) investigate
the existence of equilibrium forecasts in a non-strategic feedback model, but
do not characterize the properties of equilibrium forecast when they exist. In
a model of inflation forecasts and monetary policy, Bernanke and Woodford
(1997) study the existence of an informative rational expectations
equilibrium. They argue that it may not exist, but again do not derive the
properties of optimal forecasts in case there is an equilibrium. Faust and
Leeper (2005) focus on comparing the information content of conditional and
unconditional central bank forecasts for the public (the former type of
forecast assumes a fixed policy action, while the latter anticipates the
central bank’s optimal reaction). Our previous work, Lieli and Nieto-
Barthaburu (2020), deals with policy feedback in futures markets; there are
analogies between forecasts and equilibrium prices in this setting. On the
empirical side, Faust and Wright (2008) propose forecast efficiency tests for
forecasts conditioned on policy assumptions. Knüppel and Schultefrankenfeld
(2017) compare inflation forecasts from various central banks with different
assumptions about future monetary policy. But neither paper gives a
theoretical account of the feedback mechanism and its consequences.

Demonstrating that feedback can generate biased forecasts under quadratic loss
and rational behavior is a significant conceptual contribution to the forecast
evaluation literature. It is well known that in the absence of feedback, the
MSE-optimal forecast is the conditional mean; therefore, the optimal forecast
is unbiased, errors are uncorrelated with the forecaster’s information, the MZ
slope is unity, etc. The literature has a long history of interpreting
violations of these canonical optimality properties as evidence against
forecasters’ rationality; see, e.g., Mincer and Zarnowitz (1969), Zarnowitz
(1985), Keane and Runkle (1989, 1990), Romer and Romer (2000), Croushore
(2012), Patton and Timmermann (2012), Rossi and Sekhposyan (2016), among many
others. Another strand of the forecasting literature raises the caveat that
forecasts can be optimally biased under asymmetric loss, as in Granger (1969),
Elliott et al. (2005, 2008), Patton and Timmermann (2007), Capistran (2008).
Our model goes beyond these traditional narratives, and highlights a specific
mechanism—feedback—that can rationally explain such patterns and account for
the factors that drive them. We are not aware of other theoretical models that
do so.

In Section 2, as motivation for our subsequent analysis, we will present some
stylized facts related to apparent inefficiencies in Green Book forecasts of
inflation. In that section we also discuss the limitations of preference or
irrationality based explanations of these facts. In Section 3 we present our
feedback model and provide a thorough discussion of the underlying
assumptions. In Section 4 we derive and state our core results on the
statistical properties of optimal forecasts under feedback. Section 5 revisits
the stylized facts related to Green Book forecasts to interpret them through
the lens of the model. In Section 6 we discuss an extension of the main model
to conditional forecasts. We summarize and state our conclusions in Section 7.

##  2 Empirical motivation

In this section we present some stylized properties of Green Book (GB)
inflation forecasts produced by the Federal Reserve staff over several
decades.777The Green Book is now the Teal Book, but we keep the traditional
name. There is little question that feedback should be a relevant
consideration in evaluating the properties of GB forecasts. The Federal Open
Market Committee uses these forecasts in formulating monetary policy, which
then affects realized inflation in the future (among other things). We
demonstrate some remarkable patterns that our theory can speak to, but have
been attributed to preferences or irrationality in the existing literature. We
highlight and discuss the following two facts:

  1. (i)

GB forecasts show systematic bias over extended periods, and the sign of the
bias shifts over time.

  2. (ii)

The statistical relationship between realized inflation and GB forecasts, as
described by the MZ regression, changes gradually over time from the mid-1970s
to the mid-2010s (and the overall change is quite radical if we compare the
first half of the time period with the second).

In their influential paper, Romer and Romer (2000) use MZ regressions to test
the rationality (unbiasedness) of GB forecasts, and find no evidence against
it on a sample from 1965 to 1991. However, in a follow-up study Capistran
(2008) convincingly argues that the non-rejection of rationality by _ibid._ is
due to their overlooking a structural break in the data, where the bias
changes sign. Using an extended sample, the top panel of Figure 1 visually
confirms this fact. In particular, the figure shows the 40-quarter moving
average of the errors associated with the 4-quarter-ahead GB inflation
forecasts, where we use the latest revision of the GDP deflator as the measure
of realized inflation. As it can be seen, persistent bias is a salient feature
of GB forecasts. Furthermore, Figure 1 shows that the sign of the bias has
shifted again since the latter study.

Figure 1: Properties of GB inflation forecasts: bias, MZ slope and MZ
R-squared

To our knowledge, stylized fact (ii) has not been documented in the literature
in an explicit way. The middle panel of Figure 1 depicts the slope coefficient
from a rolling-window MZ regression of realized inflation on the GB forecast
over the sample period 1974:q2 through 2016:q4. The forecast horizon is 4
quarters and the window is 40 quarters long. Realized inflation is again taken
to be the last revision. The evolution of the slope coefficient is nothing
less than striking. It hovers around one from the mid-1980s to the early 1990s
and, after a temporary drop, it again returns to unity by the end of the
decade. However, the relationship between forecast and realization shifts
completely in the early 2000s. The slope coefficient dips into significantly
negative territory by the middle of the 2000s and then remains
indistinguishable from zero in the last quarter or so of the sample period.
The bottom panel shows the time path of the corresponding R-squared statistic
and demonstrates how the explanatory power of the MZ regression evaporates
over time.888These facts are robust to using the second release of GDP
inflation as the dependent variable. The only difference is that in the last
part of the sample the MZ slope returns into positive territory but stays
small and statistically insignificant.

While fact (ii) has not been presented in the literature in this form, it is
consistent with some reported findings. For example, using an efficiency test
applicable in the presence of structural instabilities, Rossi and Sekhposyan
(2016) conclude:

> “It appears that the [Green Book] forecasts deteriorate over the 1990s and
> rationality tends to recover by the 2000s. However, for almost all forecast
> horizons, rationality breaks down again around 2005.” (p. 526)

Thus, _ibid_. also confirm the shifting nature of GB forecasts using formal
statistical tests, but they interpret the result as a breakdown of
rationality.

Capistran (2008) attributes the bias to asymmetries in the Fed’s loss
function, and the changes in the sign of the bias to changes in the direction
of these asymmetries across monetary policy regimes. While this is certainly
possible, the problem with appealing to preference shifts in explaining fact
(i) is that this argument can be used to rationalize, ex-post, any change in
behavior. Unless one specifies the factors that drive the preference shift,
this explanation has little predictive power for the future. On the other
hand, labeling fact (ii) as a rationality breakdown depends on the assumption
of quadratic loss, and also lacks predictive power unless one specifies
concrete ways in which behavior might be irrational.

The thesis of this paper is that the patterns presented above are consistent
with rational forecaster behavior under feedback. More specifically, our model
shows that such interaction between the forecaster and the policy maker can
cause systematic positive or negative bias in the forecasts, depending on
economic conditions, and changes in the nature of this interaction can alter
the relationship between forecasts and realizations, as captured by the MZ
regression. The model does not need asymmetric preferences to generate optimal
bias, and is specific about the determinants of forecast properties. We will
revisit facts (i) and (ii) in Section 5 and illustrate how our model can be
used to interpret them.999We do not claim that our model provides an exclusive
or exhaustive explanation of these facts. Inflation forecasting and monetary
policy are complex processes, and as such there are many other relevant
factors in the determination of the realized forecast error. The point is that
our model highlights a specific mechanism that can generate similar patterns
and can be used to derive empirical predictions.

##  3 The model

We present our model in three steps. First, we describe the general
forecasting environment. Second, we model the DM’s reaction to the forecast.
Third, we discuss in detail the assumptions that we build into the first two
steps.

###  3.1 The forecasting environment

We consider the problem of a forecaster who sets out to predict an outcome
$y\in\mathbb{R}$ in an environment where the forecast $f$ may also affect the
realized value of $y$. The forecaster’s information set consists of a private
signal $\theta\in\mathbb{R}$, which represents the state of the economy, and
is a determinant of $y$. In addition, there is a decision maker (DM) who does
not directly observe $\theta$ and relies on the forecast to take an action
$a\in\mathbb{R}$, which then also affects the outcome. In particular, we
assume that $y$ depends on the signal, the action, and a random error
$\epsilon$ in a linear way:

$\displaystyle y=\theta+a+\epsilon.$ (1)

The error $\epsilon$ is assumed to have zero mean and finite variance
$\sigma^{2}>0$, and to be unforecastable in the sense of being independent of
$\theta$. The linear specification implies that the DM’s action shifts the
conditional mean of $y$ given $\theta$, but not the conditional variance or
higher moments.

The forecaster is endowed with a standard quadratic loss function

$L(y,f)=(y-f)^{2},$

and his goal is to construct a forecast that minimizes expected loss (mean
squared error) conditional on the observed value of $\theta$. Thus, if the
forecaster expects the DM to use the reaction function $a=a(f)$, the
forecaster’s problem is to solve

$\min_{f}E[L(y,f)|\,\theta]=\min_{f}E\big{\\{}[\theta+a(f)+\epsilon-f]^{2}\mid\theta\big{\\}}.$
(2)

The solution of problem (2) defines the optimal forecast $f^{*}(\theta)$ under
the DM’s assumed policy or reaction function.

The fact that we allow the DM to choose the action based on the observed
forecast of $y$ introduces the feedback effect that is the focus of our paper.
Nevertheless, our framework also nests the traditional model of optimal
forecasting, in which the outcome is assumed to be exogenous with respect to
the forecast. Indeed, one can eliminate feedback in problem (2) by setting
$a(f)$ to a constant function, which can be absorbed into the state $\theta$.
It is well known that the optimal forecast is then given by the conditional
mean, i.e., $f^{*}(\theta)=E(y|\,\theta)=\theta$; hence, in this case the
optimal forecast is unbiased and the associated forecast errors are
uncorrelated with the forecaster’s information. We will show that this is
generally not the case in the presence of feedback. Using squared-error loss
makes it clear these canonical optimality properties can be violated without
introducing preferences for over- or underprediction into the model.

It is of course not possible to study problem (2) in full generality without
being more specific about the form of the reaction function $a(f)$ and where
it comes from. This is the problem we address next.

###  3.2 The decision maker’s reaction function and the resulting forecasting
problem

We equip the DM with a microfounded reaction function. The main building
blocks of this foundation are the DM’s objective, and her expectations about
how the forecast is constructed. In our general setting, the DM is allowed to
be non-strategic in that these expectations — essentially, her interpretation
of the forecast — may not be correct.101010In other words, the DM’s conjecture
about the form of the forecast $f(\theta)$ may give rise to a reaction
function $a(f)$ that induces the forecaster to use some other forecasting
rule. Studying this type of behavioral feedback is still insightful and helps
derive the subsequent results under fully rational (strategic) feedback by the
DM.

#### The DM’s objective

We assume that the DM has a target value $y^{T}$ for the outcome, which is
common knowledge, and which she tries to achieve by manipulating $a$. The DM’s
loss for not hitting the target is a quadratic function of the control error
$y-y^{T}$. In addition, the DM faces a quadratic adjustment cost from setting
the policy variable to a nonzero value. Hence, the DM’s total ex-post loss
from her control problem has the form

$\displaystyle(y-y^{T})^{2}+ta^{2},$

where $t\in(-1,\infty)$ is the DM’s adjustment cost parameter. A higher
(positive) value of $t$ will dissuade the DM from taking extreme actions to
hit the target. Conversely, negative values of the cost parameter correspond
to a DM who is prone to overreaction, e.g., for political reasons or because
of using the policy action $a$ to fulfill other objectives besides controlling
the outcome $y$.111111For example, in the context of monetary policy, higher
values of $t$ can be interpreted as describing a more “dovish” policymaker,
while lower values can be interpreted as an indication of a more “hawkish”
policymaker. Furthermore, some central banks have an explicit dual mandate.
For example, the Federal Reserve may under- or overreact to inflationary
pressures due to the tradeoff involved in also pursuing an employment
objective. Importantly, we assume that the value of $t$ is randomly drawn, by
“nature,” from a distribution that is common knowledge. The realized value of
$t$ is then observed by the DM, but not by the forecaster, who only knows the
distribution of $t$. (In the language of economic theory, $t$ is the DM’s
private information, i.e., her type.) We further assume that $t$ is drawn
independently of $(\theta,\epsilon)$.

The DM’s goal is to choose the action $a$ to minimize expected loss
_conditional on the information conveyed by the forecast_. That is, for any
given value of the forecast $f$, the DM chooses an action $a$ to solve

$\displaystyle\min_{a}\big{\\{}E[(y-y^{T})^{2}|f]+ta^{2}\big{\\}}=\min_{a}\big{\\{}E[(\theta+a+\epsilon-y^{T})^{2}|f]+ta^{2}\big{\\}},$

where the expectation is with respect to the distribution of
$(\theta,\epsilon)$ given $f$. The following lemma characterizes the solution
to this problem.

###### Lemma 1

The optimal action of the policymaker is given by

$\displaystyle
a^{*}(f)=\frac{1}{1+t}[y^{T}-E(\theta|f)]:=x[y^{T}-E(\theta|f)],$ (3)

where $x=1/(1+t)>0$.

The intuition behind Lemma 1 is as follows. The DM wants to steer the outcome
$y$ towards her target $y^{T}$. To do so, she must correct exogenous
deviations from her target caused by the state. Since she has some information
about the state given by the forecast, she uses it to compute the optimal
policy which, absent adjustment costs, is equal to $y^{T}-E(\theta|f)$.
However, given the cost of policy $ta^{2}$, she under- or over-adjusts the
outcome by a factor $\frac{1}{1+t}\in(0,\infty)$.121212Decision rules of the
form (3) appear in applications. For example, in the monetary policy
literature such reaction function is known as a “Taylor rule” (Taylor (1993)).
In our model such Taylor rule behavior is a rational response of the DM to
information revealed by the forecast.

#### The DM’s expectations

In order to implement policy rule (3), the DM must make a conjecture about the
form of the functional relationship between $\theta$ and $f$, and use it to
compute the expectation $E(\theta|f)$. Suppose that the DM conjectures that
the forecast is linear in $\theta$; specifically, let $f=b+c\cdot\theta$,
$c\neq 0$. Then the conditional expectation in (3) can be computed as

$\displaystyle E(\theta|f)=\frac{f-b}{c}.$

Given the way in which the DM forms expectations, her optimal action (3)
becomes

$\displaystyle a^{*}(f)=x\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right).$ (4)

We will shortly see that the assumption of a linear forecasting rule is self-
confirming.

At this point we do not require that the DM-conjectured intercept and slope be
equal to the actual intercept and slope used by the forecaster. Instead, in
the general case we allow for the DM to be mistaken about the forecaster’s
strategy. This generality helps accommodate applications in which policy
follows a rigid, mechanical rule. The special case in which the DM’s
conjecture is correct constitutes an _equilibrium_ in our model, which we
formally define in Section 4.

Example 3: If the DM sets $c=1$ and $b=0$, the reaction function (4) reduces
to the particularly simple Taylor rule

$a^{*}(f)=x(y^{T}-f).$

An interpretation that rationalizes this reaction function is that the DM
takes the forecast at “face value” and considers it, somewhat naively, as the
state or, equivalently, as the expected outcome given no action ($a=0$).
Alternatively, this may be a “pure” behavioral rule without explicit
justification. Either way, as we will show, the optimal forecast under this
reaction function is not $f(\theta)=\theta$. Therefore, the DM’s conjecture
would not be part of an equilibrium.

#### The forecaster’s problem

We now return to the forecaster’s problem given the DM’s reaction function
(4). As discussed above, the realized value of $x$ is the DM’s private
information, and the forecaster knows only the distribution of $x$, which is
independent of $(\theta,\epsilon)$. However, we assume that the DM’s
conjecture of $(b,c)$ is known to the forecaster; see Section 3.3 below for a
detailed justification of these modeling choices. Thus, under the conditions
stated above, the forecaster’s problem (2) becomes

$\displaystyle\min_{f}E\left[\left(\theta+a^{*}(f)+\epsilon-f\right)^{2}|\theta\right]=$
$\displaystyle\min_{f}E\left\\{\left[\theta+x\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)+\epsilon-f\right]^{2}\Bigg{|}\,\theta\right\\},$
(5)

where the expectation is taken w.r.t. the distribution of $(x,\epsilon)$
conditional on $\theta$. Given the maintained independence assumptions, this
coincides with the marginal distribution of $(x,\epsilon)$ for any given value
of $\theta$. We present and study the solution to problem (5) in Section 4.

###  3.3 Discussion of the modeling assumptions

#### Outcome determination

A scalar state variable and a linear outcome equation are simplifications that
allow us to make our points in a clear and tractable way. In real world
applications, such as inflation forecasting, the forecaster’s information set
will typically contain observations on many variables. Nevertheless, one might
think of $\theta$ as a “sufficient statistic” of the forecaster’s information
or some function of the raw covariates that compresses information (such as,
say, a principal component). Moreover, linearity, coupled with the
independence of $\theta$ and $\epsilon$, imply that the DM’s action has a
direct impact only on the conditional mean $E(y|\,\theta)$ and not on higher
moments. Nonetheless, the feedback mechanism described in our model will be
present qualitatively even if policy affects the distribution of the outcome
in a more complex way.

#### Forecaster’s preferences

Our choice of the forecaster’s quadratic loss function is deliberate. In the
absence of feedback effects, the MSE-optimal forecast is unbiased, and
forecast errors are unpredictable given the forecaster’s information. We will
show that these properties generally break down under feedback. Using squared-
error loss makes it clear that it is the feedback mechanism itself which
generates the biases and other non-standard properties of optimal forecasts in
our model.

#### Forecaster’s information

Two key assumptions in the model are that $(i)$ the state is the forecaster’s
private information, and $(ii)$ the state cannot be communicated to the DM
directly, only through the forecast. We believe that these assumptions capture
the main features of forecaster-DM interactions in practice. First, DMs
typically rely on advisers because, on their own, they lack the capability or
resources to acquire and process all the information relevant to their
decisions. Second, for similar reasons, the advice given by experts often
comes in the form of summary statistics and projections of key outcomes rather
than raw data. Indeed, forecasting would be a meaningless activity if
constraints justifying $(ii)$ did not exist in the real world.

#### DM’s objective and policy function

Quadratic loss ensures that the DM has an incentive to hit the target on
average, and the assumption of a convex adjustment cost is standard throughout
the economics literature. The independence of the parameter $x$ and $\theta$
means that the strength of the policy reaction to the forecast is independent
of economic conditions. More generally, it is conceivable that the mean or
variance of $x$ could depend on $\theta$; for example, the DM might react more
aggressively in crisis periods, etc. However, there are many possible ways to
model this dependence, and it is not obvious what are the salient facts (if
any) that should be captured by this additional feature. Therefore, in an
abstract model such as ours, independence is a natural benchmark.

A more general interpretation of $x$ in (3), which can be divorced from its
microfoundation, is that it represents policy uncertainty due to DM-specific
idiosyncratic factors at the time of the decision. It is reasonable to assume
that such information (at least in part) is private to the DM.

#### Forecaster’s expectations about policy

We assume throughout that the forecaster knows the reaction function
$a^{*}(f)$ that governs the DM’s behavior up to the uncertain parameter $x$.
The introduction of this additional uncertainty into the forecaster’s problem
(2) serves two related purposes. First, it captures the fact that in many
applications the forecaster is not fully informed about the feedback mechanism
that he faces. Second, we will show below that uncertainty about the reaction
function is a necessary condition for feedback to induce bias in the optimal
forecast.

It may seem contradictory that $x$ is the DM’s private information while her
conjecture about $(b,c)$ is not. This modeling choice can be rationalized in
several ways. First, some forecasters, such as those in a central bank, may
interact with the DM on a regular basis, and know a lot, but not everything,
about their reaction function. As explained above, we use $x$ in our model to
represent this residual uncertainty about policy. Second, some DMs may operate
under set rules or institutional constraints to interpret forecasts (i.e.,
choose $b$ and $c$) in a certain way, such as in Example 3.2. Third, the
assumption is certainly true in equilibrium, where, by definition, the DM’s
conjecture about $b$ and $c$ coincides with the values actually used by the
forecaster. Finally, forecasts derived under a given $(b,c)$ vector can be
regarded simply as technical objects that are conditional on the assumed value
of $(b,c)$. Even if one questions the empirical relevance of these conditional
forecasts, they are still helpful in deriving results about equilibrium
forecasts with a fully strategic DM.

#### Forecaster’s inability to commit

In our model we assume that the forecaster lacks the power to commit to a
fixed forecasting function. We consider this to be a natural assumption, given
that the state is not observable directly by the DM, even ex-post. If the
forecaster could commit to a set forecasting rule, he could do better than in
the no-commitment equilibrium. Such case can be analyzed as a simple extension
of our model.

##  4 Results

###  4.1 Optimal forecasts under feedback

Two key parameters in the determination of the optimal forecast are the mean
and variance of the “strength of policy” variable $x$. Hence, we introduce
notation for these parameters: let $\mu:=E(x)>0$ and $\tau^{2}:=var(x)$. The
following proposition characterizes the optimal forecast, that is, the
forecast that minimizes (5), in terms of these parameters.

###### Proposition 1

Given the reaction function (4), and the forecaster’s knowledge of $b$ and
$c$, the optimal forecast is

$\displaystyle f^{*}=d^{*}+e^{*}\cdot\theta,$

where

$\displaystyle
d^{*}=c\left[\frac{\tau^{2}+\mu(\mu+c)}{\tau^{2}+(\mu+c)^{2}}\right]\left(y^{T}+\frac{b}{c}\right),$
(6)

and

$\displaystyle e^{*}=c\frac{\mu+c}{\tau^{2}+(\mu+c)^{2}}.$ (7)

#### Remarks

  1. 1.

Proposition 1 shows that the DM’s conjecture of a linear forecast function $f$
is self-confirming, i.e., that given such a conjecture, the optimal forecast
$f^{*}$ is indeed a linear function of $\theta$. Nevertheless, as discussed
above, the DM’s guesses about the slope and the intercept of $f^{*}$ are
allowed to be incorrect. (In fact, it is clear from formulas (6) and (7) that
this is generally so for arbitrary $b$ and $c$.) The function $f^{*}$ gives
the optimal forecast conditional on the DM’s beliefs about forecaster
behavior—regardless of whether these beliefs are correct or not.

  2. 2.

The forecast slope in Proposition 1 is decreasing in the variance of $x$ (in
absolute value). That is, the more uncertain the policy reaction to the
forecast is, the less responsive is the forecast to the state. The intuition
is the following. Since the DM’s action is a function of the forecast, the
forecaster correctly anticipates the outcome itself to be a function of the
forecast. When uncertainty about the DM’s reaction increases (i.e., when
$\tau^{2}$ increases), any given forecast function introduces more volatility
into $y$, which is a component of the forecaster’s MSE loss. However, the
forecaster can dampen this volatility by making the forecast less responsive
to the state. As we more formally argue in Section 4.3, the optimal choice of
the forecast slope is smaller in magnitude than that of the unbiased forecast.
Thus, the forecaster accepts some bias in exchange for an overall reduction in
MSE.

As a simple application of Proposition 1, consider a DM acting according to
the Taylor rule of Example 3.2.

Example 4: If $c=1$ and $b=0$, i.e., $a^{*}(f)=x(y^{T}-f)$, then
$f^{*}=d^{*}+e^{*}\cdot\theta$ with

$e^{*}=\frac{\mu+1}{\tau^{2}+(\mu+1)^{2}}\;\text{ and
}\;d^{*}=\frac{\tau^{2}+\mu(\mu+1)}{\tau^{2}+(\mu+1)^{2}}y^{T}.$

Thus, we see that the optimal forecast under the Taylor rule is not
$f(\theta)=\theta$; instead, as explained above, the slope is attenuated
toward zero, and the intercept is a fraction of the target. The case in which
$\tau^{2}=0$ is also instructive. The optimal forecast is then given by
$f^{*}(\theta)=\frac{1}{\mu+1}\theta+\frac{\mu}{\mu+1}y^{T}$. When the
forecaster learns $\theta$, he recognizes that it will have both a direct
effect on the outcome and an indirect one through the DM’s reaction to the
forecast. The optimal forecast accounts for both effects. More formally, the
conditional mean of the outcome is $E(y|\,\theta)=\theta+\mu(y^{T}-f)$. The
optimal forecast arises by setting this equal to $f$ and then solving for it,
i.e., in this case the optimal forecast is unbiased. The case when
$\tau^{2}>0$ is more complex, in that the forecaster also cares about how much
volatility is introduced into the outcome through the term $-xf$.

In Example 4.1 the DM is not strategic, in the sense that her interpretation
of the forecast is not consistent with how it is actually produced. We can use
Proposition 1 to derive the optimal forecast under a fully rational DM’s
reaction function. A fully rational DM conjectures the use of an equilibrium
forecast, which we formally define as follows.

###### Definition 1

An equilibrium forecast is a forecast function
$f^{\dagger}(\theta)=b^{\dagger}+c^{\dagger}\cdot\theta$ such that if the DM
conjectures the use of $f^{\dagger}(\theta)$, then it is optimal for the
forecaster to use $f^{\dagger}(\theta)$. More formally, if the DM sets
$b=b^{\dagger}$ and $c=c^{\dagger}$ in (4), then $d^{*}=b^{\dagger}$ and
$e^{*}=c^{\dagger}$ in Proposition 1.

The equilibrium concept that underlies Definition 1 is pure-strategy Perfect
Bayesian Equilibrium (PBE). In equilibrium the forecaster and the DM mutually
best-respond to each other, and hold correct (i.e., rational) expectations.
The following corollary to Proposition 1 shows that, under some conditions,
there are two linear PBE in our model, and it characterizes the equilibrium
forecasts when they exist.

###### Corollary 1

For $\tau^{2}\leq\frac{1}{4}$ there exist two PBE with equilibrium forecasts
$f^{\dagger}_{i}=b_{i}^{\dagger}+c_{i}^{\dagger}\cdot\theta$, $i=1,2$. The
coefficients are as follows.

  1. 1.

Slopes:

$\displaystyle
c^{\dagger}_{1}=\frac{1}{2}-\mu+\frac{\sqrt{1-4\cdot\tau^{2}}}{2},$ (8)

and

$\displaystyle
c^{\dagger}_{2}=\frac{1}{2}-\mu-\frac{\sqrt{1-4\cdot\tau^{2}}}{2},$

provided that $c^{\dagger}_{1}\neq 0$ and $c^{\dagger}_{2}\neq 0$.

  2. 2.

Intercepts:

$\displaystyle b_{i}^{\dagger}=\frac{k_{i}}{1-k_{i}}c_{i}^{\dagger}\cdot
y^{T},$ (9)

where

$\displaystyle
k_{i}=\frac{\tau^{2}+\mu(\mu+c_{i}^{\dagger})}{\tau^{2}+(\mu+c_{i}^{\dagger})^{2}}.$

#### Remarks

  1. 1.

The equilibrium forecast slopes are derived from equation (7) by substituting
the common value $c^{\dagger}$ for $c$ and $e^{*}$ and then solving for it.
The equilibrium intercepts are then obtained analogously from equation (6).

  2. 2.

The equilibria are one-to-one functions of the state; hence, a rational DM can
learn the value of the state from the forecast. (In the language of economic
theory, the equilibria are _separating_ or fully revealing.)

  3. 3.

If $\tau^{2}>\frac{1}{4}$ an equilibrium does not exist. Indeed, it is easy to
verify that if $\tau^{2}>\frac{1}{4}$, then
$\frac{\mu+c}{\tau^{2}+(\mu+c)^{2}}<1$ for all values of $c$ and $\mu$. Hence,
equation (7) implies that for any slope $c$ conjectured by the DM, the
forecaster will always want to use a slope that is attenuated toward zero
compared with the DM’s conjecture.

  4. 4.

If $\operatorname{\emph{supp}}(x)\subset[0,1]$, the condition
$\tau^{2}\leq\frac{1}{4}$ is always satisfied. In particular,

$\displaystyle\tau^{2}=E(x^{2})-\mu^{2}\leq\mu-\mu^{2}=\mu(1-\mu)\leq\frac{1}{4},$

where we have used the fact that $E(x^{2})\leq E(x)$ for a random variable
with support on $[0,1]$. More generally, if $x$ is allowed to take values
outside $[0,1]$, then $\tau^{2}\leq\frac{1}{4}$ is not guaranteed.

#### Equilibrium selection

In order to use the model for interpreting observed forecast patterns, we need
to take a stance on which equilibrium provides more plausible predictions
empirically. We argue that it is the first one, for the following reasons:

  1. 1.

The second equilibrium gives rise to a forecast that decreases in $\theta$ for
a larger range parameter values $(\mu,\tau^{2})$ than the first one. In
particular, $f^{\dagger}_{2}(\theta)$ decreases in $\theta$ when $\tau^{2}$ is
sufficiently close to zero, i.e., when there is little uncertainty. As the
outcome depends positively on $\theta$, a forecast function with a negative
slope is rather counterintuitive — it is akin to having an inflation forecast
that decreases in measures of how overheated the economy is.

  2. 2.

For a more formal argument, consider the limit of the two equilibria as
uncertainty about $x$ vanishes, i.e., $\tau^{2}\rightarrow 0$ (and so
$x\rightarrow_{p}\mu$). As the DM learns the true value of $\theta$ from the
forecast in both cases, the optimal action, in the limit, is
$a^{*}=\mu(y^{T}-\theta)$ and the resulting outcome is
$y=\theta+\mu(y^{T}-\theta)+\epsilon$. The “natural” optimal forecast of this
outcome is $\theta+\mu(y^{T}-\theta)$, which is indeed the limit of
$f_{1}^{\dagger}(\theta)$ as $\tau^{2}\rightarrow 0$. But how can $\lim
f^{\dagger}_{2}(\theta)$ also be an equilibrium forecast? It can be verified
that $\lim f^{\dagger}_{2}(\theta)=(1+\mu)y^{T}-\mu\theta$. Thus, expressed in
terms of the value of the forecast, the DM reaches the optimal action by
adopting the decision rule $a^{*}(f)=f-y^{T}$, as prescribed by (4). But if
the DM uses this rule, then the forecaster’s objective (5) becomes completely
independent of the forecast. So, he might as well report
$f(\theta)=(1+\mu)y^{T}-\mu\theta$, which, in turn, justifies the DM’s
decision rule. This is an odd case, in the sense that forecaster behavior in
equilibrium rests on the fact that the DM chooses an action that makes the
forecaster completely indifferent among all forecasts.131313When $\tau^{2}$ is
exactly zero, there exist many more equilibria that rely on the forecaster’s
indifference, including a family of pooling equilibria where the forecast is
completely uninformative, i.e., a constant function of $\theta$. Since we
expect that in practice there is at least some uncertainty about the DM’s
reaction, we do not deem those equilibria as relevant. Furthermore, that sort
of forecast do not match observed behavior.

For these reasons, we will choose the first PBE, with slope (8), as our
preferred prediction about behavior in the forecasting game. From now on, we
will refer to that equilibrium when deriving the statistical properties of
forecasts with feedback.

###  4.2 The statistical properties of optimal forecasts under feedback

The following proposition characterizes the statistical properties of optimal
forecasts in our model, conditional on the DM believing that the forecast
function is given by $f(\theta)=b+c\cdot\theta$, $c\neq 0$.

###### Proposition 2

Let $f^{*}=f^{*}(\theta)$ denote the optimal forecast described in Proposition
1. Then:

  1. 1.

The optimal forecast is biased, and the bias conditional on the state is given
by

$\displaystyle
E(y-f^{*}|\,\theta)=\frac{\tau^{2}}{\tau^{2}+(\mu+c)^{2}}(\theta-cy^{T}-b).$
(10)

  2. 2.

The Mincer-Zarnowitz regression function of the optimal forecast is given by

$\displaystyle
E(y|f^{*})=-\frac{\tau^{2}}{\mu+c}\left(y^{T}+\frac{b}{c}\right)+\frac{\tau^{2}+c(\mu+c)}{c(\mu+c)}f^{*}.$
(11)

#### Remarks

  1. 1.

Proposition 2 shows that, in general, the statistical properties of optimal
forecasts in our model differ from those of optimal forecasts in feedback-free
environments. In particular, the optimal forecast is biased (despite the
forecaster’s quadratic loss), and the slope of the MZ regression is different
from one, implying that the forecast error $y-f^{*}$ is correlated with
$f^{*}$. We emphasize that these properties hold even when the DM is fully
rational (see Corollary 2 below).

  2. 2.

The driving force behind the results in Proposition 2 is a combination of two
mechanisms: the _feedback effect from forecasts to realizations_ , coupled
with the forecaster’s _uncertainty about policy strength_ , i.e., uncertainty
about $x$. Indeed, both uncertainty about the DM’s action _and_ feedback from
forecasts to policy are needed for the optimal forecast to be biased, as we
argue in the next two remarks.

  3. 3.

To assess the role of uncertainty, one can take the limit $\tau^{2}\to 0$ in
equations (10) and (11). It is immediately seen that even in the presence of
feedback, when uncertainty about $x$ vanishes, $(i)$ bias vanishes; $(ii)$ the
slope of the MZ regression function converges to 1, and $(iii)$ the intercept
converges to 0. Intuitively, when $\tau^{2}=0$, the DM’s optimal action (4)
becomes a fixed, known function of $f$. Hence, a forward-looking forecaster
will perfectly anticipate the feedback effect and compensate for it to
minimize the MSE of his forecast. Therefore, if there is feedback but no
uncertainty about the DM’s action, the properties of optimal forecast in our
model coincide with those of feedback-free predictions.

  4. 4.

To see the effect of policy uncertainty without feedback, suppose that the DM
does not rely on the forecast to take action. In this case, the DM’s optimal
action is $a^{*}(f)=x[y^{T}-E(\theta)]$, that is, she chooses an adjustment
with respect to her unconditional expectation of the state. In turn, the
forecaster’s problem is

$\displaystyle\min_{f}E[(y-f)^{2}|\,\theta]=\min_{f}E\\{[\theta+x(y^{T}-E(\theta))+\epsilon-f]^{2}|\,\theta\\}.$

This is then a perfectly standard prediction problem whose solution is the
conditional mean $f^{*}(\theta)=E(y|\,\theta)=\theta+\mu[y^{T}-E(\theta)]$.
Thus, in this feedback-free scenario uncertainty about the action will add to
the volatility of forecast errors, but will not cause bias.

As indicated above, the statistical properties of equilibrium forecasts can
easily be derived from Proposition 2 by requiring that DM’s conjectures are
correct. In Corollary 2 we state these properties for the preferred (first)
equilibrium forecast. In addition, Corollary 2 also specializes Proposition 2
to the case of the simple Taylor rule presented in Example 3.2.

###### Corollary 2

  1. 1.

Suppose that $\tau^{2}\leq\frac{1}{4}$, and
$\mu\neq\frac{1}{2}+\frac{\sqrt{1-4\tau^{2}}}{2}$ (so an equilibrium exists).
The conditional bias and MZ regression function of the equilibrium forecast
are given by

    1. (a)

$E(y-f^{\dagger}|\,\theta)=\frac{2\tau^{2}}{1+\sqrt{1-4\tau^{2}}}(\theta-y^{T})$;

    2. (b)

$E(y|f^{\dagger})=\frac{\tau^{2}}{\tau^{2}-(1-\mu)\left(\frac{1}{2}+\frac{\sqrt{1-4\cdot\tau^{2}}}{2}\right)}y^{T}+\frac{(1-\mu)\left(\frac{1}{2}+\frac{\sqrt{1-4\cdot\tau^{2}}}{2}\right)}{(1-\mu)\left(\frac{1}{2}+\frac{\sqrt{1-4\cdot\tau^{2}}}{2}\right)-\tau^{2}}f^{\dagger}.$

  2. 2.

If the DM adopts the simple Taylor rule $a(f)=x(y^{T}-f)$, then the
conditional bias and MZ regression function of the optimal forecast are given
by

    1. (a)

$E(y-f^{*}|\,\theta)=\frac{\tau^{2}}{\tau^{2}+(\mu+1)^{2}}(\theta-y^{T})$;

    2. (b)

$E(y|f^{*})=-\frac{\tau^{2}}{\mu+1}y^{T}+\frac{\tau^{2}+\mu+1}{\mu+1}f^{*}$.

Figure 2: The MZ slope of the equilibrium forecast as a function of $\tau^{2}$
for various values of $\mu$ (functions truncated at 2 and $-2$) Figure 3: The
MZ slope of the equilibrium forecast as a function of $\tau^{2}$ for various
values of $\mu$ (functions truncated at $2$ and $-2$; $\mu$-axis also
truncated)

#### Remarks

  1. 1.

In general, the MZ intercept and slope are highly nonlinear functions of $\mu$
and $\tau^{2}$, as shown by Figures 2 and 3.

  2. 2.

When $\mu=1$, the MZ slope of the equilibrium forecast is zero, and the MZ
intercept is $y^{T}$. To gain some intuition for these results, we need to
reinterpret the MZ regression function in the context of feedback. In
particular, when the DM chooses her policy actions as a function of the
forecast, the outcome itself becomes a function of the forecast. Therefore,
the MZ regression is no longer a purely predictive relationship; it becomes
“causal” in the sense that different values of the announced forecast will
lead to different realizations of the outcome (on average). Now, in
equilibrium, a rational DM learns the exact value of the state from the
forecast and uses it to adjust the outcome toward the target. When $\mu=1$,
there is full adjustment on average, i.e., the average value of $y$ will be
equal to $y^{T}$ for any value of the forecast. That is, $E(y|f^{*})=y^{T}$
for all $f^{*}$. Note however, that this result is rather fragile in the sense
that even a relatively small deviation of $\mu$ from unity can induce a large
positive or negative MZ slope.

###  4.3 Bias-variance analysis of optimal forecasts under feedback

To grasp the intuition of why the forecaster chooses to optimally bias his
forecast when there is feedback and uncertainty about policy, consider the
following decomposition of the forecaster’s MSE objective into a conditional
variance term and a conditional bias term:

$\displaystyle E[(y-f)^{2}|\theta]$ $\displaystyle=$ $\displaystyle
E\Big{\\{}\big{[}y-E(y|\theta)\big{]}^{2}\big{|}\,\theta\Big{\\}}+[E(y|\theta)-f]^{2}$
$\displaystyle=$ $\displaystyle Var(y|\theta)+bias^{2}(f|\theta).$

Given the outcome equation (1), the DM’s reaction function (4), and the mutual
independence of $\theta$, $\epsilon$ and $x$, it is easy to see that

$\displaystyle
Var(y|\,\theta)=Var\left(\theta+a^{*}(f)+\epsilon\Big{|}\,\theta\right)=Var\left[x\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)\Big{|}\,\theta\right]+\sigma^{2}$
$\displaystyle\quad=Var(x|\,\theta)\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)^{2}+\sigma^{2}=\tau^{2}\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)^{2}+\sigma^{2}$
(12)

and

$\displaystyle
bias^{2}(f|\,\theta)=\left\\{E[\theta+a^{*}(f)+\epsilon\big{|}\,\theta]-f\right\\}^{2}=\left\\{\theta+E\left[x\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)\Big{|}\,\theta\right]-f\right\\}^{2}$
$\displaystyle\quad=\left\\{\theta+E(x|\,\theta)\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)-f\right\\}^{2}=\left\\{\theta+\mu\Big{(}y^{T}+\frac{b}{c}\Big{)}-\Big{(}\frac{\mu+c}{c}\Big{)}f\right\\}^{2}.$
(13)

The variance-bias tradeoff that the forecaster faces is clear from considering
equations (12) and (13). The forecaster could choose a forecast function
$f(\theta)$ that reduces bias to zero; in particular, the unbiased forecast
requires a slope of

$\displaystyle e=\frac{c}{\mu+c},$

and an intercept of

$\displaystyle
d=\left(\frac{c}{\mu+c}\right)\left[\mu\left(y^{T}+\frac{b}{c}\right)\right].$

However, the optimal forecast is different from the unbiased forecast because
the variance of the outcome, i.e., equation (12), also depends on the forecast
because of the feedback through $a^{*}(f)$. That is, the forecaster recognizes
that he has an impact on the volatility of the outcome, and that impact occurs
through policy actions that are taken in response to the announced forecast.
Therefore the forecaster, trying to balance the variance and bias effects of
the forecast, optimally chooses a prediction that is systematically biased.

##  5 Interpreting the properties of GB forecasts

We now revisit the stylized facts presented in Section 2 in light of the
predictions from our model, specifically the results in Corollary 2.

First, the model naturally explains the sign changes in the bias of GB
forecasts with sign changes in $y^{T}-\theta$, i.e., how economic conditions
relate to the Fed’s inflation target. The value of this prediction is somewhat
limited by the fact that our abstract model is not specific about the
empirical interpretation of $\theta$. Still, the model provides the insight
that changes in the inflation target alone, without substantial changes in
economic conditions, can affect both the size and the bias of GB forecasts.

Second, as Figures 2 and 3 show, the model is capable of generating a wide
range of values for the MZ slope depending on the underlying parameters $\mu$
and $\tau^{2}$. But more importantly, using the interpretation of these
parameters, one can tell various stories about the main stylized fact in the
middle panel of Figure 1 — the gradual flattening of the MZ slope by the end
of the 1990s, its turning negative in the early 2000s, and becoming
essentially zero after 2010.

The broad interpretation of the parameter $\mu$ in this context is how
aggressively the Fed pursues its inflation target on average. A value of $\mu$
close to one would indicate that the Fed is prepared to use whatever rate hike
is necessary to achieve its target over a short period, while a low value of
$\mu$ means that the Fed is very conscious of the costs of doing so and
prefers more cautious adjustments.141414Of course, our static model does not
explicitly account for time or any dynamic effects. Nevertheless, one could
imagine playing the forecasting game repeatedly over successive time periods,
under potentially changing conditions. On the other hand, the parameter
$\tau^{2}$ shows how consistently the Fed follows the “average” policy set by
the value of $\mu$. A larger value of $\tau^{2}$ could indicate that there is
a lot of disagreement in the FOMC, resulting in wider swings in the actions
taken. By contrast, a lower $\tau^{2}$ means that the Fed is very clearly
committed to a given monetary policy stance — there is not much uncertainty
about what they intend to do.

In light of these interpretations, one can formulate several hypotheses about
the evolution of the MZ slope.

  * (i)

As Figure 2 shows, if the Fed is initially reluctant to fight inflation
($\mu=0.1$), but then switches to a policy of aggressively fighting it ($\mu$
around 1), then we would expect to see a flattening of the MZ slope provided
that uncertainty is above a fairly low threshold (say, $\tau^{2}\geq 0.05)$.
Moreover, once $\mu$ is close to one, it is easy for the MZ slope to fluctuate
around zero and take on negative values as well (this happens when $\mu$
slightly below one).151515One caveat about this story is that it requires a
sudden large jump in $\mu$ from a value close to zero to a value close to 1.
Intermediate values of $\mu$, such as $\mu=0.5$ or $\mu=0.75$, can give rise
to very large positive or negative MZ slopes for a wide range of $\tau^{2}$.

  * (ii)

Figure 2 also shows that once $\mu$ is sufficiently close to 1, and there is
already a moderate level of uncertainty, then further increases in $\tau^{2}$
attenuate the MZ slope to zero.

  * (iii)

There can be other institutional reasons for the flattening of the MZ slope.
As Corollary 2 shows, if the Fed reacts to the forecast mechanically using a
simple Taylor rule, then the MZ slope is greater than 1 for all values of
$\mu$ and $\tau^{2}$. However, if the Fed starts behaving more strategically
(i.e., tries to recover the underlying information $\theta$), then the
forecast production process also changes, and the equilibrium forecast may
have a flatter slope (and may switch sign) for the _same_ values of $\mu$ and
$\tau^{2}$. For example, if $\mu=0.98$ and $\tau^{2}=0.1$, then the MZ slope
under the Taylor rule is 1.05, while the strategic slope is $-0.22$.

Of course, our simple model cannot and does not capture the monetary policy
and forecast production processes in all their complexity. The main point we
want to make here is that our model is capable of generating meaningful and
testable narratives about the observed properties of GB forecasts rather than
attributing them to (changing) preferences or irrationality. Nevertheless,
there is a substantial empirical literature on monetary policy in the US,
which documents several changes in the monetary policy regime from the 1970s
to the 2000s using dynamic macroeconomic models. While these models are much
more complex than ours, they presumably contain parameters that could be
mapped into plausible values for $\mu$ and $\tau^{2}$ over different time
periods, opening the door for a more formal empirical validation of the types
of hypotheses stated above. We leave this exercise for future research.

##  6 Extension: conditional forecasts

We consider an extension of our forecasting model to _conditional forecasts_ ,
i.e., forecasts made under a fixed (possibly counterfactual) value of the
policy action. For example, the staff of some central banks may produce a
collection of such forecasts under different policy scenarios. Assuming a
fixed policy action means that the forecaster does not try to anticipate the
response of the DM to his prediction. Hence, such forecasts will be produced
only if the forecaster is not fully rational, or if he follows an
institutional mandate to do so. This is an extension of our model in the sense
that all previous results assume a fully strategic forecaster.161616In the
language of monetary macroeconomics, all previous forecasts were
_unconditional_ ; see, e.g., Faust and Leeper (2005) or Knüppel and
Schultefrankenfeld (2017). We distinguish between two cases, depending on
whether the DM’s reaction to the forecast is unconstrained or is restricted to
a pre-specified menu of choices. It is the latter case in which the production
of conditional forecasts makes more sense.

###  6.1 Unconstrained DM

Denote by $a_{0}$ the assumed fixed policy action. Given this action, the
outcome would be

$\displaystyle y=\theta+a_{0}+\epsilon.$

It is easy to see then that an MSE-minimizer forecaster would choose

$\displaystyle f^{*}(\theta)=\theta+a_{0},$ (14)

which is just the expectation of $y$ conditional on $\theta$ and the action
being $a_{0}$. If the DM believes that the forecast is of the form
$f=b+c\cdot\theta$, her optimal reaction to it is given by the same formula as
before:

$\displaystyle a^{*}(f)=x\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right).$ (15)

In Proposition 3 we state the statistical properties of the conditional
forecast (14).

###### Proposition 3

  1. 1.

Conditional forecasts are biased, with conditional bias given by

$\displaystyle
E(y-f^{*}|\theta)=\mu\left(y^{T}+\frac{b}{c}\right)-\left(\frac{\mu+c}{c}\right)a_{0}-\frac{\mu}{c}\theta.$
(16)

  2. 2.

The MZ regression function of the the conditional forecast is

$\displaystyle
E(y|f^{*})=-a_{0}+\mu\left(y^{T}+\frac{b}{c}\right)-\left(\frac{\mu-c}{c}\right)f^{*}.$
(17)

Proposition 3 states the properties of conditional forecasts under any
assumption that the DM may make about the intercept and slope of the
forecasting function. The next corollary states those properties for two
special cases of interest: a fully rational DM, who inverts the conditional
forecast and chooses her action optimally given the state, and a DM who
follows a pure Taylor rule. In the notation of our model, the former uses the
correct forecast intercept and slope, $b=a_{0}$ and $c=1$ respectively, and
therefore chooses a policy action of the form $a=x(y^{T}-f+a_{0})$; the latter
assumes that $b=0$ and $c=1$, and therefore chooses action $a=x(y^{T}-f)$. The
expressions for the bias and MZ regression under a fully rational DM and a
pure Taylor rule DM follow directly from Proposition 3 by substituting the
corresponding values of $b$ and $c$ for each type of DM.

###### Corollary 3

  1. 1.

The bias and MZ regression function of a conditional forecast under a fully
rational DM are

    1. (a)

$E(y-f^{*}|\theta)=-a_{0}+\mu(y^{T}-\theta)$;

    2. (b)

$E(y|f^{*})=-(1-\mu)a_{0}+\mu y^{T}+(1-\mu)f^{*}$.

  2. 2.

The bias and MZ regression function of a conditional forecast under a pure
Taylor-rule policy are

    1. (a)

$E(y-f^{*}|\theta)=-(\mu+1)a_{0}+\mu(y^{T}-\theta)$;

    2. (b)

$E(y|f^{*})=-a_{0}+\mu y^{T}+(1-\mu)f^{*}$.

#### Remarks

  1. 1.

Unsurprisingly, conditional forecasts are biased both under the fully rational
response and the Taylor rule. The reason is that the forecaster conditions on
a counterfactual policy action, i.e., the assumed action generally differs
from the one that the DM ultimately implements in response to the forecast.
This leads to a mispecified model for the outcome determination. We emphasize,
however, that in constructing a conditional forecast the forecaster does not
face a bias-variance tradeoff, because he does not build the DM’s anticipated
reaction into the forecast. This is the reason why the bias formulas and the
MZ regression coefficients only depend on $\mu$ (the mean of $x$) but not
$\tau^{2}$ (the variance of $x$).

  2. 2.

When $\mu=1$, the MZ regression function under a fully rational policy
reaction has slope zero and intercept $y^{T}$. The intuition for this is
analogous to the case of an equilibrium forecast with $\mu=1$. As the
conditional forecast reveals $\theta$, a rational DM, on average, corrects for
the difference between $\theta$ and $y^{T}$ in full (because $\mu=1$). This
will produce realizations of $y$ that are, on average, equal to $y^{T}$.

  3. 3.

When $\mu=1$, the MZ regression function under a pure Taylor rule has slope
zero and intercept $-a_{0}+y^{T}$. A DM who follows a pure Taylor rule
correctly perceives (either just by chance or by some unmodelled reasoning)
that the slope of the forecast is 1, and therefore correctly designs her
policy to correct for the state. However, such a DM does not account for the
intercept used by the forecaster, coming from the assumed policy underlying
the forecast. Hence, this DM systematically misses the target by $-a_{0}$. An
alternative Taylor rule of the form $a=a_{0}+x(y^{T}-f)$ would be optimal for
the DM who receives a conditional forecast made under the counterfactual
policy $a_{0}$.

###  6.2 Constrained DM

We now consider a situation where the DM is restricted to a pre-specified,
fixed menu of choices. For simplicity, let the choice set consist of two
actions $\\{a_{0},a_{1}\\}$; e.g., hold the policy rate or cut it by 25 basis
points. The DM instructs the forecaster to produce a vector of conditional
forecasts $f=(f_{0},f_{1})$, where $f_{j}$ is to be constructed under the
assumption that $a=a_{j}$, $j=0,1$. A forecaster with quadratic loss optimally
reports $f_{j}(\theta)=\theta+a_{j}$, which is unbiased when $a=a_{j}$. The DM
will choose $a_{0}$ if and only if

$\displaystyle E[(\theta+a_{0}+\epsilon-y^{T})^{2}|\,f]+ta_{0}^{2}\leq
E[(\theta+a_{1}+\epsilon-y^{T})^{2}|\,f]+ta_{1}^{2}$
$\displaystyle\Leftrightarrow(f_{0}-y^{T})^{2}+\sigma^{2}+ta_{0}^{2}\leq(f_{1}-y^{T})^{2}+\sigma^{2}+ta_{1}^{2}$
$\displaystyle\Leftrightarrow(f_{0}-y^{T})^{2}-(f_{1}-y^{T})^{2}\leq
t(a_{1}^{2}-a_{0}^{2}).$

Thus, if $t=0$, the DM will simply choose the policy scenario for which the
conditional forecast is closest to the target in absolute value. More
generally, the cost of taking action is also taken into consideration.

In this situation one of the conditional forecasts, the one conditional on the
action actually taken, will be unbiased, and will have a MZ regression
function with an intercept equal to zero and a slope equal to one. On the
other hand, the conditional forecast based on the counterfactual action will
be biased. Hence, in evaluating conditional forecasts the extent to which the
assumed policy differs from the actually implemented one should be a primary
consideration.

##  7 Conclusion

We present and analyze a model of forecasting in the presence of policy
feedback. While empirically relevant, the forecasting literature has largely
overlooked this problem in evaluating forecasts. Our paper is an attempt to
bridge this gap, by clarifying the consequences of feedback for the properties
of forecasts produced under quadratic loss.

Our results show that when feedback effects are present, and there is some
degree of uncertainty about the decision maker’s response to the forecast, the
canonical properties of mean-square optimal forecasts break down. In
particular, forecasts in feedback-prone environments are biased, and forecast
errors are predictable given information possessed by the forecaster.
Furthermore, the biases become more pronounced when the strength of feedback
increases. These properties are in stark contrast to those under no feedback.

Given the simple structure of our model, we can explicitly show how the
forecast bias and the coefficients of the Mincer-Zarnowitz regression depend
on the mean and variance of the uncertain parameter in the decision maker’s
reaction function. These detailed results can be used to derive a range of
testable predictions about how the properties of feedback-prone forecasts may
change in different environments. We illustrate the use of these results by
interpreting the historical changes in the properties of Green Book inflation
forecasts through the lens of the model. A more formal application of the
model to Green Book forecasts is part of our future research agenda.

Our results have implications for the literatures concerned with forecast
rationality testing and loss function identification from observed data. In
particular, our model illustrates that forecast biases can arise from factors
other than irrationality or asymmetries in the loss function. Therefore, we
argue that forecast evaluation exercises need to either make the identifying
assumption of no-feedback (when reasonable), or to think carefully about the
feedback mechanism present in the application at hand.

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## Appendix: Proofs

### Proof of Proposition 1

To facilitate other derivations, it is useful to decompose the forecaster’s
conditional MSE (5) into a conditional variance and bias component. In
particular,

$E[(y-f)^{2}|\,\theta]=E\big{[}(y-E(y|\theta))^{2}\big{|}\,\theta\big{]}+(E(y|\theta)-f)^{2}=var(y|\theta)+bias^{2}(f|\theta),$

where

$y=\theta+x\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)+\epsilon.$ (18)

Equations (13) and (12) in the main text show that

$var(y|\,\theta)=\tau^{2}\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)^{2}+\sigma^{2},$
(19)

and

$E(y|\,\theta)-f=\theta+\mu\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)-f=\theta+\mu\left(y^{T}+\frac{b}{c}\right)-\left(\frac{\mu}{c}+1\right)f,$
(20)

where the variance formula relies heavily on the mutual independence of $x$,
$\theta$ and $\epsilon$.171717In the text we assume that $\epsilon$ is
independent of $\theta$ (after equation (1)) and that $(\theta,\epsilon)$ is
independent of $x$ (in Section 3.2). This implies mutual independence of $x$,
$\theta$ and $\epsilon$. Thus, the forecaster’s conditional MSE is given by

$\displaystyle var(y|\theta)+bias^{2}(f|\theta)$ $\displaystyle=$
$\displaystyle\tau^{2}\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)^{2}+\sigma^{2}+\left[\theta+\mu\left(y^{T}+\frac{b}{c}\right)-\left(\frac{\mu}{c}+1\right)f\right]^{2}$
(21)

Setting the partial derivative of (21) w.r.t. $f$ equal to zero results in the
FOC:

$\frac{\tau^{2}}{c}\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right)+\left(\frac{\mu}{c}+1\right)\left[\theta+\mu\left(y^{T}+\frac{b}{c}\right)-\left(\frac{\mu}{c}+1\right)f\right]=0.$

Solving this linear equation for $f$ yields
$f^{*}(\theta)=d^{*}+e^{*}\cdot\theta$, where the slope $e^{*}$ is given by
expression (7) and the intercept $d^{*}$ is given by expression (6).

### Proof of Corollary 1

As explained in Remark 1 after Corollary 1, we set

$c^{\dagger}=\frac{c^{\dagger}(\mu+c^{\dagger})}{\tau^{2}+(\mu+c^{\dagger})^{2}}.$

Dropping the dagger superscript and rearranging yields the following quadratic
equation in $c$:

$c^{2}+(2\mu-1)c+\mu(\mu-1)+\tau^{2}=0.$

The quadratic formula then gives:

$c_{1,2}=\frac{-(2\mu-1)\pm\sqrt{(2\mu-1)^{2}-4(\mu(\mu-1)+\tau^{2})}}{2}=\frac{1}{2}-\mu\pm\frac{1}{2}\sqrt{1-4\tau^{2}},$

as stated in equation (8). The equilibrium intercept is obtained by
substituting the common value $b^{\dagger}$ for $d^{*}$ and $b$ in equation
(6) and then solving equation

$b^{\dagger}=\frac{c\left(y^{T}+\frac{b^{\dagger}}{c}\right)(\tau^{2}+\mu(\mu+c))}{\tau^{2}+(\mu+c)^{2}}$

for $b^{\dagger}$. This yields expression (9) once $c$ is taken to be
$c^{\dagger}_{i}$.

### Proof of Proposition 2

Starting from the bias formula in equation (20) and substituting
$f^{*}=d^{*}+e^{*}\cdot\theta$ gives

$\displaystyle E(y|\,\theta)-f^{*}$ $\displaystyle=$
$\displaystyle\theta+\mu\left(y^{T}+\frac{b}{c}\right)-\left(\frac{\mu}{c}+1\right)(d^{*}+e^{*}\cdot\theta)$
$\displaystyle=$
$\displaystyle\mu\left(y^{T}+\frac{b}{c}\right)-\left(\frac{\mu}{c}+1\right)d^{*}+\left[1-e^{*}\left(\frac{\mu}{c}+1\right)\right]\theta.$

Substituting expression (7) for $e^{*}$ and expression (6) for $d^{*}$ yields

$\displaystyle E(y|\,\theta)-f^{*}$
$\displaystyle=\mu\left(y^{T}+\frac{b}{c}\right)-\left(\frac{\mu}{c}+1\right)c\frac{\tau^{2}+\mu(\mu+c)}{\tau^{2}+(\mu+c)^{2}}\left(y^{T}+\frac{b}{c}\right)+\left[1-\left(\frac{\mu}{c}+1\right)\frac{c(\mu+c)}{\tau^{2}+(\mu+c)^{2}}\right]\theta$
$\displaystyle=\left(y^{T}+\frac{b}{c}\right)\frac{-c\tau^{2}}{\tau^{2}+(\mu+c)^{2}}+\frac{\tau^{2}}{\tau^{2}+(\mu+c)^{2}}\theta$
$\displaystyle=\frac{\tau^{2}}{\tau^{2}+(\mu+c)^{2}}(\theta-cy^{T}-b),$

which is the result stated in Proposition 2 part 1.

Turning to the Mincer-Zarnowitz (MZ) regression, one can take the expectation
of (18) with respect to $(x,\epsilon)$, conditional on $f=f(\theta)$, to
obtain

$\displaystyle
E(y|f)=E(\theta|f)+\mu\left(y^{T}-\frac{f}{c}+\frac{b}{c}\right),$ (22)

where we used the fact that $f(\theta)$ and $(x,\epsilon)$ are independent
(since $\theta$ and $(x,\epsilon)$ are independent). Since in equilibrium the
forecast is $f^{*}=d^{*}+e^{*}\theta$, it follows that
$E(\theta|f^{*})=f^{*}/e^{*}-d^{*}/e^{*}$. Substituting into (22) gives

$\displaystyle E(y|f^{*})$ $\displaystyle=$
$\displaystyle\frac{f^{*}}{e^{*}}-\frac{d^{*}}{e^{*}}+\mu\left(y^{T}-\frac{f^{*}}{c}+\frac{b}{c}\right)=\left(\frac{1}{e^{*}}-\frac{\mu}{c}\right)f^{*}+\mu\left(y^{T}+\frac{b}{c}\right)-\frac{d^{*}}{e^{*}}$

Further substituting expression (7) for $e^{*}$ and expression (6) for $d^{*}$
yields

$\displaystyle E(y|f^{*})$ $\displaystyle=$
$\displaystyle\left(\frac{\tau^{2}+(\mu+c)^{2}}{c(\mu+c)}-\frac{\mu}{c}\right)f^{*}+\mu\left(y^{T}+\frac{b}{c}\right)-\frac{\left(y^{T}+\frac{b}{c}\right)(\tau^{2}+\mu(\mu+c))}{\mu+c}$
$\displaystyle=$
$\displaystyle\frac{\tau^{2}+c(\mu+c)}{c(\mu+c)}f^{*}+\left(y^{T}+\frac{b}{c}\right)\left(\mu-\frac{\tau^{2}+\mu(\mu+c)}{\mu+c}\right)$
$\displaystyle=$
$\displaystyle-\left(\frac{\tau^{2}}{\mu+c}\right)\left(y^{T}+\frac{b}{c}\right)+\frac{\tau^{2}+c(\mu+c)}{c(\mu+c)}f^{*},$

as stated in equation (11).

### Proof of Corollary 2

For the equilibrium bias and MZ regression, notice that the equilibrium
forecast slope and intercept $c^{\dagger}$ and $b^{\dagger}$ are as in
equations (8) and (9), respectively. Substituting the values of $c$ and $d$ in
the expressions for the optimal bias and Mincer-Zarnowitz regression (10) and
(11) for $c^{\dagger}$ and $b^{\dagger}$ yields the expressions of the
equilibrium bias and Mincer-Zarnowitz regressions in part 1 of Corollary 2.

Turning to the bias and MZ regression under a simple Taylor rule policy
$a(f)=x(y^{T}-f)$, note that this policy rule is equivalent to the DM
conjecturing a forecast intercept $b=0$ and slope $c=1$, and acting optimally
conditional on these conjectures. Substituting the values of $c$ and $d$ in
the expressions for the optimal bias and Mincer-Zarnowitz regression (10) and
(11) for $1$ and $0$ respectively yields the expressions of the bias and
Mincer-Zarnowitz regressions under a simple Taylor rule policy in part 2 of
Corollary 2.

### Proof of Proposition 3

To compute the conditional bias of conditional forecasts, simply substitute
the formula of the optimal conditional forecast (14) into the bias expression
(20) to yield

$E(y|\,\theta)-f^{*}=\theta+\mu\left(y^{T}+\frac{b}{c}\right)-\left(\frac{\mu}{c}+1\right)f^{*}=\theta+\mu\left(y^{T}+\frac{b}{c}\right)-\left(\frac{\mu}{c}+1\right)(\theta+a_{0}).$

Simple algebra shows that this expression simplifies to (16).

Turning to the MZ regression function, with a conditional forecast
$f^{*}=\theta+a_{0}$, we have that $E(\theta|f^{*})=f^{*}-a_{0}$. Substituting
into the formula for the MZ regression (22) yields

$\displaystyle
E(y|f)=f^{*}-a_{0}+\mu\left(y^{T}-\frac{f^{*}}{c}+\frac{b}{c}\right).$

Again, simple algebra shows that this expression simplifies to (17).