# Off-Balance Sheet Activities and Scope Economies in U.S. Banking††thanks:
Emir Malikov would like to acknowledge financial support from the Troesh
Center for Entrepreneurship & Innovation at UNLV.

Email<EMAIL_ADDRESS>(Zhang) and<EMAIL_ADDRESS>(Malikov).

Jingfang Zhang Auburn University Emir Malikov University of Nevada, Las Vegas

(November 10, 2021)

###### Abstract

Propelled by the recent financial product innovations involving derivatives,
securitization and mortgages, commercial banks are becoming more complex,
branching out into many “nontraditional” banking operations beyond issuance of
loans. This broadening of operational scope in a pursuit of revenue
diversification may be beneficial if banks exhibit scope economies. The
existing (two-decade-old) empirical evidence lends no support for such
product-scope-driven cost economies in banking, but it is greatly outdated
and, surprisingly, there has been little (if any) research on this subject
despite the drastic transformations that the U.S. banking industry has
undergone over the past two decades in the wake of technological advancements
and regulatory changes. Commercial banks have significantly shifted towards
nontraditional operations, making the portfolio of products offered by
present-day banks very different from that two decades ago. In this paper, we
provide new and more robust evidence about scope economies in U.S. commercial
banking. We improve upon the prior literature not only by analyzing the most
recent data and accounting for bank’s nontraditional off-balance sheet
operations, but also in multiple methodological ways. To test for scope
economies, we estimate a flexible time-varying-coefficient panel-data quantile
regression model which accommodates three-way heterogeneity across banks. Our
results provide strong evidence in support of significantly positive scope
economies across banks of virtually all sizes. Contrary to earlier studies, we
find no empirical corroboration for scope diseconomies.

Keywords: bank, cost subadditivity, nontraditional banking, off-balance sheet,
product scope, scope economies

JEL Classification: G21, L25, D24

##  1 Introduction

Just like in other industries, executive managers in banking must choose the
optimal scope of operations. Despite the long-lasting implications of this
strategic choice for firm performance, the dichotomy between operational
“focus” and breadth remains unsettled from the corporate strategy perspective.
The common arguments for limited-scope operations à la Skinner, 1974a ;
Skinner, 1974b usually feature cost and quality benefits associated with more
specialized expertise and tacit knowledge, lessened complexity, diminished
technological uncertainty, etc. On the other hand, there may be a strong
incentive to diversify revenue streams by broadening the firm’s product mix in
order to capitalize on potential scope-driven cost savings and thereby
increase firm value (see Panzar and Willig, , 1981; Rumelt, , 1982;
Villalonga, , 2004). When it comes to commercial banking, leveraging
operational scope and breadth thereof continues to play a vital role in
operations management.

The scope of bank operations has also been a subject of intense policy debate,
thereby expanding practical importance of understanding the relation between
operational scope and bank performance beyond industry managers and
stakeholders. Namely, the financial crisis of 2007–2008 and the ensuing Great
Recession turned attention of policy-makers and academics alike onto large
“too-big-to-fail” (TBTF) commercial banks and the serious systemic risks that
they pose. The emergence of behemoth banks due to deregulation as well as
technological innovations (including those in information technologies) has
given rise to concerns about the costs that such “systemically important
financial institutions” impose on the economy and fueled policy debates about
whether banks should be subject to size limitations, even including the talks
of break-up. These policy discussions have led to the enactment of new
financial regulations such as the Dodd–Frank Wall Street Reform and the
Consumer Protection Act of 2010 that seek to eliminate the TBTF doctrine by
setting restrictions on the scale and scope of bank operations. However, the
potential cost savings associated with operating at a large scale with a more
diversified scope of revenue-generating activities, which are to be forgone
owing to the new regulations, have been by and large neglected in these policy
discussions.

Large banks may derive such cost efficiency benefits from their ability to
offer financial services at lower average cost due to (i) “scale economies”
driven by the increasing returns to scale as well as (ii) their unique
position to innovate and expand the scope of offered financial products and
thereby economize costs (“scope economies”) via input complementarities and
positive spillovers (see Markides and Williamson, , 1994; Milgrom and Roberts,
, 1995) as well as, in the case of commercial banking, risk diversification
across different products (e.g., Rossi et al., , 2009). In theory, these cost
savings are passed onto customers in the form of lower net interest margins.
This raises an important policy and research question about significance of
the trade-off between lower systemic risk pursued by the newly enacted
regulations and the cost savings that banks may be forced to forgo as a
result. Both have non-negligible implications for consumer welfare. It is
therefore imperative to investigate the prevalence of scale and scope
economies in banking in order to not only shed light on potential unintended
consequences of the financial reforms already put in place but also to inform
future policies and regulations. This information also can help banks in
formulating optimal product-scope operational strategies.

While studies of scale economies in commercial banking are many, the attempts
to measure scope economies are however scant and outdated. The latter is
especially lacking given the introduction of many “nontraditional” financial
product innovations involving derivatives, securitization and mortgages by the
large banks in the past two decades that have allowed them to expand the scope
of their revenue-earning operations. The objective of this paper is to fill in
this gap.

Early studies of scale economies in banking date as far back as Berger et al.,
(1987), Mester,  (1987, 1992) and Hughes and Mester,  (1993, 1998) to name a
few, and with the passage of new financial reforms, this body of research has
only been growing. No matter the methods employed, most recent studies find
empirical evidence in support of the statistically significant increasing
returns to scale in the U.S. banking sector. Some find significant scale
economies mostly for large commercial banks (e.g., Wheelock and Wilson, ,
2012; Hughes and Mester, , 2013; Restrepo-Tobòn and Kumbhakar, , 2015); others
find economies of scale for medium and small banks as well (e.g., Malikov et
al., , 2015; Restrepo-Tobòn et al., , 2015; Wheelock and Wilson, , 2018).

With the sole exception,111To our knowledge, Yuan and Phillips,  (2008) who
explicitly recognize the role of nontraditional banking activities (namely,
insurance) is the only attempt at measuring scope economies in the U.S.
banking post 2000. Their analysis looks at a single nontraditional operation
and stops at 2005, which obviously excludes the most relevant period after the
structural-change-inducing financial crisis. there however have been virtually
no attempt to investigate product scope economies in banking over the past two
decades despite the drastic transformations that this sector has undergone
during that time. This perhaps can be attributed to the lack of empirical
evidence in support of statistically and/or economically significant scope
economies among U.S. commercial banks documented in the 1980s and 1990s; e.g.,
see Berger et al.,  (1987), Mester,  (1987), Hughes and Mester,  (1993),
Pulley and Braunstein,  (1992), Ferrier et al.,  (1993), Pulley and Humphrey,
(1993), Jagtiani et al.,  (1995), Jagtiani and Khanthavit,  (1996), Wheelock
and Wilson,  (2001). It makes scope economies in the present-day banking
sector be a seriously overlooked issue because the technological advancements
along with regulatory changes have restructured the U.S. banking industry
dramatically, especially since the passage of the Gramm–Leach–Bliley Act in
1999, which largely lifted the restrictions prohibiting the consolidation of
commercial banks, investment banks, securities firms and insurance companies.
U.S. banks have since experienced a drastic shift from traditional banking
activities (viz., issuance of loans) towards the nontraditional activities
such as investment banking, venture capital, security brokerage, insurance
underwriting and asset securitization (DeYoung and Torna, , 2013), and the
portfolio of products offered by the modern banks is very different from that
two decades ago, underscoring the importance of our study.

While nontraditional banking operations are usually associated with banks’ all
other non-interest fee-generating activities related to participating in
capital markets, the off-balance sheet banking represents one of the major
forms of such nontraditional activities. It chiefly consists of contingent
claims/contracts that involve obligations to lend or provide funds should the
contingency be realized and, unlike the traditional interest-income-centered
transactions, these off-balance sheet activities are not recorded on the
bank’s balance sheet (Hassan, , 1993; Hassan and Sackley, , 1994). For
example, an interest-earning loan is considered an asset on the bank’s balance
sheet, whereas a promise to make a loan is an off-balance sheet item since it
involves only a potential funding obligation in the future, albeit, for which
the bank earns a fee. Broadly, off-balance sheet items can be categorized into
four groups including guarantees, commitments, market-related activities, and
advisory or management functions (e.g., see Perera et al., , 2014). Such off-
balance sheet banking operations are well-documented to substantially
influence banks’ financial performance including profitability and risk
profiles (e.g., Stiroh, , 2004; Laeven and Levine, , 2007; Apergis, , 2014),
and omitting these revenue-earning operations in the analysis of banking
technology may lead to erroneous inference and conclusions due to
misspecification (see Clark and Siems, , 2002; Rime and Stiroh, , 2003; Casu
and Girardone, , 2005; Lozano-Vivas and Pasiouras, , 2010). When testing for
scope economies, we therefore recognize off-balance sheet operations as
another one of the bank’s revenue-generating outputs.

In this paper, we contribute to the literature by providing new and more
robust evidence about scope economies in U.S. commercial banking. We improve
upon the prior literature not only by analyzing the most recent and relevant
data (2009–2018) and accounting for bank’s nontraditional non-interest-
centered operations, but also in multiple methodological ways as follows. In a
pursuit of robust estimates of scope economies and statistical inference
thereon, we estimate a flexible, yet parsimonious, time-varying-coefficient
panel-data quantile regression model which accommodates (i) distributional
heterogeneity in the cost structure of banks along the size of their costs,
(ii) temporal variation in cost complementarities and spillovers due to
technological change/innovation, and (iii) unobserved bank heterogeneity
(e.g., latent management quality) that, if unaccounted, confounds the
estimates. Our analysis is structural in that we explicitly estimate a model
of bank cost structure which facilitates the measurement of counterfactual
costs necessary to test for scope economies.

By employing a quantile approach, we are able to capture distributional
heterogeneity in the bank cost structure. Unlike the traditional regression
models that focus on the conditional mean only, quantile regression provides a
complete description of the relationship between the distribution of bank
costs and its determinants. Since banks of varying size/scale are highly
heterogeneous in their operations (e.g., see Wheelock and Wilson, , 2012), it
is reasonable to expect that large- and small-scale banks exhibit different
scope-driven potential for cost saving (if any) and, therefore, there remains
much untapped benefit of examining scope economies in banking via quantile
analysis. Thus, contrary to all prior studies of scope economies in banking
which provide evidence solely for average costs via conventional conditional-
mean regressions, we focus our analysis on conditional quantiles of the bank
cost distribution, with the bank’s operating cost being a good proxy for its
size/scale. Not only does this approach enable us to accommodate potential
heterogeneity in the prevalence of scope economies among banks of different
sizes, but it is also more robust to the error distributions including the
presence of outliers in the data. Furthermore, it exhibits a useful
equivariance property thereby letting us avoid biases in the scope economies
computations that numerous earlier studies suffer from (to be discussed
later).

To operationalize our analysis, we employ the recently developed quantile
estimator (Machado and Santos Silva, , 2019) that we extend to allow temporal
variation of unknown form in the parameters in order to flexibly capture the
impact of technological innovations on bank operations and costs. Our
empirical results provide strong evidence in support of statistically
significant scope economies across banks virtually of all sizes in the U.S.
banking sector. Among banks between the bottom 10th and top 90th percentiles
of the cost distribution, 92% or more exhibit positive economies of scope. The
prevalence of significant scope economies in median banks is 99%. Even under
the alternative model specifications that produce smaller point estimates, the
evidence in support of scope economies in U.S. banking remains strong, with at
least 89% of mid-cost banks found to enjoy product-scope-driven cost savings.
We also find no empirical corroboration for scope diseconomies. Overall, our
findings are in stark contrast with earlier studies.

The rest of the paper unfolds as follows. Section 2 discusses the theoretical
framework. Section 3 describes our econometric model. Data are discussed in
Section 4, followed by Section 5 that reports the empirical results. We then
conclude in Section 6.

##  2 Theory of Multi-Product Costs

In order to test if there is an untapped cost savings potential for commercial
banks due to scope economies, we need to formally model their cost structure.
Following the convention in the banking literature, we do so using the dual
cost approach. Not only is this approach convenient because it facilitates the
direct measurement of the bank’s costs via the estimated dual cost function
necessary for testing for scope economies, but it also does not require the
use of input quantities during the estimation (unlike in the primal production
approach) which can lead to simultaneity problems since input allocations are
the bank’s endogenous decision whereas input prices are widely accepted as
being exogenously determined owing to competition in the factor market
including that for deposits.

A model of bank costs calls for specification of the outputs and inputs of
bank production. Given the bank’s core functions as a financial intermediary,
most studies in the literature adopt Sealey and Lindley, ’s (1977)
“intermediation approach” which focuses on the bank’s production of
intermediation services and the associated costs inclusive of both the
interest and operating expenses. In this paradigm, the revenue-generating
financial assets such as loans and trading securities are conceptualized as
outputs, whereas inputs are typically specified to include labor, physical
capital, deposits and other borrowed funds as well as equity capital (for an
excellent review, see Hughes and Mester, , 2015). Given the recent industry
trends and the growing importance of nontraditional income-earning activities
that banks engage in, we also include an output measure of non-interest off-
balance sheet income. Together with loans and securities, this makes a total
of $M=3$ outputs.

Concretely, we formalize the bank’s cost structure via the following multi-
product dual variable cost function:

$\mathcal{C}_{t}(\boldsymbol{Y},\boldsymbol{W},\boldsymbol{K})=\min_{\boldsymbol{X}\geq\mathbf{0}}\left\\{\boldsymbol{X}^{\prime}\boldsymbol{W}\
|\ (\boldsymbol{X},\boldsymbol{K})\ \text{can produce}\ \boldsymbol{Y}\
\text{at time}\ t\right\\},$ (2.1)

where the arguments of cost function $\mathcal{C}_{t}(\cdot)$ are the output
quantities $\boldsymbol{Y}\in\Re_{+}^{M}$, variable input prices
$\boldsymbol{W}\in\Re_{+}^{J}$ and fixed input quantities
$\boldsymbol{K}\in\Re_{+}^{P}$; and $\boldsymbol{X}\in\Re_{+}^{J}$ is the
vector of variable input quantities. Importantly, the cost function in (2.1)
is time-varying thereby accommodating the evolution of the bank cost structure
over time in the face of technological advancements and regulatory changes.

The multi-product firm’s cost structure is said to exhibit scope economies if
its average cost is decreasing in the number of outputs/operations (Panzar and
Willig, , 1981). Commercial banks may achieve such cost savings by spreading
fixed costs (e.g., branch costs and data processing costs) over the more
diversified output mix (fixed asset amortization) which now, more often than
not, includes nontraditional off-balance sheet operations. Scope economies may
also arise from positive spillovers via the (re)use of “public inputs” such as
client credit information and customer relations as well as intangible assets
including tacit knowledge and know-hows. Complementarities across different
products can play a big role too. For example, some off-balance sheet
operations such as loan commitments (which generate income for banks via fees)
essentially represent a technological expansion of traditional lending at a
little cost added. At the same time, they can help banks expand the scope of
their customer relationship with all the cost-saving informational gains that
come with it (Berger and Udell, , 1995; Das and Nanda, , 1999; Degryse and Van
Cayseele, , 2000). Banks can also reuse the information gathered when issuing
loans to reduce the searching or monitoring requirements of the off-balance
sheet activities.

To test for the potential for scope-driven cost savings, we use an expansion-
path measure of subadditivity of the bank’s cost function à la Berger et al.,
(1987), with the rationale being that subadditivity sheds light on scope
economies, the presence of which is a necessary condition for the former (see
Baumol et al., , 1982; Evans and Heckman, , 1984). Specifically, the
subadditivity measure relies on comparison of the costs of smaller multi-
output banks of differential degrees of specialization with the cost of a
larger, more diversified bank.222While preserving the equality of total output
quantities on both sides, of course. Intuitively, this approach zeroes in on
scope economies from a perspective of relative—as opposed to absolute—notion
of revenue diversification. Then, for some distribution weights
$0\leq\varpi_{m}^{\kappa}\leq 1$ such that
$\sum_{\kappa}\varpi_{m}^{\kappa}=1$ for all $m=1,2,3$ and
$\kappa\in\\{A,B,C\\}$, the bank is said to enjoy scope economies at time $t$
if

$\sum_{\kappa\in\\{A,B,C\\}}\mathcal{C}_{t}\big{(}\varpi_{1}^{\kappa}Y_{1},\varpi_{2}^{\kappa}Y_{2},\varpi_{3}^{\kappa}Y_{3}\big{)}-\mathcal{C}_{t}\big{(}Y_{1},Y_{2},Y_{3}\big{)}>0,$
(2.2)

where we have suppressed all arguments of the cost function besides outputs.

While the above methodology deviates from the conventional definition of scope
economies (Baumol et al., , 1982) which relies on the comparison of the cost
of producing outputs individually with the cost of their joint production,
whereby the bank is said to enjoy scope economies if
$\mathcal{C}_{t}(Y_{1},0,0)+\mathcal{C}_{t}(0,Y_{2},0)+\mathcal{C}_{t}(0,0,Y_{3})-\mathcal{C}_{t}(Y_{1},Y_{2},Y_{3})>0$,
it is both more realistic and robust. This is so because it does not require
computation of the counterfactual cost of producing each output separately by
a fully specialized single-output bank, which naturally suffers from
“excessive extrapolation” (Evans and Heckman, , 1984; Hughes and Mester, ,
1993) since the counterfactuals require extrapolation of the estimated multi-
output cost function to its boundaries corresponding to the non-existent
single-output specializations. Also, the conventional measure of scope
economies is just a special case of (2.2) with a pair of weights taking zero
values for each counterfactual bank.

To further avoid excessive extrapolation, we restrict the choice of
$\\{\varpi_{m}\\}$ to the “admissible region” defined by the two data-driven
constraints, following Evans and Heckman,  (1984). First, each counterfactual
bank is ensured to not produce less of each output than banks do in the
sample. That is, we require that $\varpi_{m}^{\kappa}Y_{m}\geq\min\\{Y_{m}\\}$
for all $m=1,2,3$ and $\kappa\in\\{A,B,C\\}$. The second constraint ensures
that each counterfactual bank does not specialize in either one of the outputs
to a greater extent than banks do in the sample. In other words, ratios of
output quantities for each counterfactual bank must fall in the range of such
ratios observed in the data, i.e., for any pair $Y_{m}$ and $Y_{m^{\prime}}$:

$\displaystyle\min\left\\{\frac{Y_{m}}{Y_{m^{\prime}}}\right\\}\leq\frac{\varpi_{m}^{\kappa}Y_{m}^{*}+\min\\{Y_{m}\\}}{\varpi_{m^{\prime}}^{\kappa}Y_{m^{\prime}}^{*}+\min\\{Y_{m^{\prime}}\\}}\leq\max\left\\{\frac{Y_{m}}{Y_{m^{\prime}}}\right\\},$
(2.3)

where $Y_{m}^{*}=Y_{m}-3\times\min\\{Y_{m}\\}$ for all $m=1,2,3$. Thus, we
examine the within-sample scope economies.

The quantitative measure of cost subadditivity $\mathcal{S}_{t}$ (in
proportions) is obtained by dividing the expression in (2.2) by
$\mathcal{C}_{t}(Y_{1},Y_{2},Y_{3})$:

$\mathcal{S}_{t}=\frac{\sum_{\kappa\in\\{A,B,C\\}}\mathcal{C}_{t}\Big{(}\varpi_{1}^{\kappa}Y_{1}^{*}+\min\\{Y_{1}\\},\varpi_{2}^{\kappa}Y_{2}^{*}+\min\\{Y_{2}\\},\varpi_{3}^{\kappa}Y_{3}^{*}+\min\\{Y_{3}\\}\Big{)}-\mathcal{C}_{t}\big{(}Y_{1},Y_{2},Y_{3}\big{)}}{\mathcal{C}_{t}\big{(}Y_{1},Y_{2},Y_{3}\big{)}},$
(2.4)

where the counterfactual costs under the summation operator have been
redefined in order to operationalize the first of the two constraints
characterizing the admissible region. Positive (negative) values of
$\mathcal{S}_{t}$ provide evidence of scope economies (diseconomies); while a
zero value suggests scope invariance of the bank’s cost structure.

Clearly however, the value of $\mathcal{S}_{t}$ depends on the choice of
distribution weights $\\{\varpi_{m}^{\kappa}\\}$. To test for scope economies,
we adopt a conservative approach to measuring cost subadditivity, whereby
$\\{\varpi_{m}^{\kappa}\\}$ are chosen such that the corresponding
$\mathcal{S}_{t}$ is the smallest. With this, “the” measure of cost
subadditivity (for each bank-year) is

$\displaystyle\mathcal{S}_{t}^{*}=\min_{\\{\varpi_{m}^{\kappa}\\}}\
\mathcal{S}_{t}\big{(}\varpi_{m}^{\kappa};\
m=1,2,3;\kappa\in\\{A,B,C\\}\big{)}\ .$ (2.5)

The rationale is as follows. If the smallest subadditivity measure is still
positive, then one can quite safely infer that scope economies are locally
significant over the bank’s feasible output space in a given year. Thus, the
main hypothesis of interest is as follows.

Hypothesis.—Consistent with scope economies at time $t$, the cost
subadditivity measure $\mathcal{S}_{t}^{*}>0$.

##  3 Empirical Model

We estimate the bank’s dual variable cost function $\mathcal{C}_{t}(\cdot)$ at
different conditional quantiles of costs. Let $C_{it}$ be the variable cost of
a bank $i=1,\dots,n$ in year $t=1,\dots,T$ and
$\boldsymbol{V}_{it}=(\boldsymbol{Y}_{it}^{\prime},\boldsymbol{W}_{it}^{\prime},\boldsymbol{K}_{it}^{\prime})^{\prime}$
be the vector of (strictly exogenous) cost-function regressors. We use lower
case of $C_{it}$ and $\boldsymbol{V}_{it}$ in the following to denote the log
transformations of the variables: e.g.,
$\boldsymbol{v}_{it}=\ln\boldsymbol{V}_{it}$. Letting the bank’s variable cost
structure be of the translog333Quadratic log-polynomial. form and described by
a location-scale model à la Koenker and Bassett,  (1982) extended to
accommodate bank fixed effects and time-varying coefficients, we have

$\displaystyle
c_{it}=\big{[}\beta_{0}+\beta_{0}^{*}L(t)\big{]}+\big{[}\boldsymbol{\beta}_{1}+\boldsymbol{\beta}_{1}^{*}L(t)\big{]}^{\prime}\boldsymbol{v}_{it}+\tfrac{1}{2}\big{[}\boldsymbol{\beta}_{2}+\boldsymbol{\beta}_{2}^{*}L(t)\big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\lambda_{i}+u_{it},$
(3.1)

with

$\displaystyle
u_{it}=\Big{(}\big{[}\gamma_{0}+\gamma_{0}^{*}S(t)\big{]}+\big{[}\boldsymbol{\gamma}_{1}+\boldsymbol{\gamma}_{1}^{*}S(t)\big{]}^{\prime}\boldsymbol{v}_{it}+\tfrac{1}{2}\big{[}\boldsymbol{\gamma}_{2}+\boldsymbol{\gamma}_{2}^{*}S(t)\big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\sigma_{i}\Big{)}\varepsilon_{it},$
(3.2)

where
$\left(\beta_{0},\boldsymbol{\beta}_{1}^{\prime},\boldsymbol{\beta}_{2}^{\prime},\beta_{0}^{*},\boldsymbol{\beta}_{1}^{*}{{}^{\prime}},\boldsymbol{\beta}_{2}^{*}{{}^{\prime}}\right)^{\prime}$
are unknown location-function coefficients;
$\left(\gamma_{0},\boldsymbol{\gamma}_{1}^{\prime},\boldsymbol{\gamma}_{2}^{\prime},\gamma_{0}^{*},\boldsymbol{\gamma}_{1}^{*}{{}^{\prime}},\boldsymbol{\gamma}_{2}^{*}{{}^{\prime}}\right)^{\prime}$
are unknown scale-function coefficients; and $\lambda_{i}$ and $\sigma_{i}$
are the unobserved bank-specific location and scale fixed effects,
respectively.

To allow for technological change in the bank cost structure, we borrow from
Baltagi and Griffin,  (1988) and introduce two scalar time indices $L(t)$ and
$S(t)$. Both time indices are unobservable and can be thought of as the
unknown functions of time. Such time indices are advantageous over simple
trends (including quadratic) in modeling temporal changes because they provide
richer variation in the measurement of technological change and much closer
approximation to observed temporal changes than do the simple time trends.
Note that index $L(t)$ enters the location function non-neutrally, shifting
not only the intercept $\beta_{0}+\beta_{0}^{*}L(t)$ but also the linear
$\boldsymbol{\beta}_{1}+\boldsymbol{\beta}_{1}^{*}L(t)$ and quadratic slopes
$\boldsymbol{\beta}_{2}+\boldsymbol{\beta}_{2}^{*}L(t)$, thereby allowing for
flexible locational shifts in the costs over time. Analogous scale changes
over time are allowed by means of $S(t)$. In all, by means of the time indices
in both the location and scale functions, we are able to accommodate temporal
changes in the entire conditional cost distribution.

Essentially, our model in (3.1)–(3.2) is a generalization of the popular
translog cost-function specification, where all parameters now vary with time,
the covariates affect not only the location (centrality) but also the scale
(variability) of the conditional cost distribution; and the bank fixed effects
are both location- and scale-shifting. The two equations together facilitate a
quantile analysis of the bank’s cost structure. Along the lines of Machado and
Santos Silva,  (2019) upon whom we build our estimation procedure, we assume
that (i) $\varepsilon_{it}$ is $i.i.d.$ across $i$ and $t$ with some cdf
$F_{\varepsilon}$; (ii) $\varepsilon_{it}\perp\boldsymbol{v}_{it}$ with the
normalizations that $\mathbb{E}\left[\varepsilon_{it}\right]=0$ and
$\mathbb{E}\left[|\varepsilon_{it}|\right]=1$; and (iii)
$\Pr\left[\big{[}\gamma_{0}+\gamma_{0}^{*}S(t)\big{]}+\big{[}\boldsymbol{\gamma}_{1}+\boldsymbol{\gamma}_{1}^{*}S(t)\big{]}^{\prime}\boldsymbol{v}_{it}+\tfrac{1}{2}\big{[}\boldsymbol{\gamma}_{2}+\boldsymbol{\gamma}_{2}^{*}S(t)\big{]}^{\prime}\times\right.$
$\left.\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\sigma_{i}>0\right]=1$.
Then, for any given quantile index $\tau\in(0,1)$, the $\tau$th conditional
quantile function of the log-cost $c_{it}$ implied by (3.1)–(3.2) is

$\displaystyle\mathcal{Q}_{c}\left[\tau|\boldsymbol{v}_{it}\right]=$
$\displaystyle\
\overbrace{\Big{[}{\beta}_{0}+{\gamma}_{0}{q}_{\tau}+{\beta}_{0}^{*}L(t)+{\gamma}_{0}^{*}S(t)q_{\tau}\Big{]}}^{t\text{-varying
quantile
intercept}}+\overbrace{\Big{[}\boldsymbol{\beta}_{1}+\boldsymbol{\gamma}_{1}q_{\tau}+\boldsymbol{\beta}_{1}^{*}L(t)+\boldsymbol{\gamma}_{1}^{*}S(t)q_{\tau}\Big{]}^{\prime}}^{t\text{-varying
linear quantile slopes}}\boldsymbol{v}_{it}\ +$ $\displaystyle\
\frac{1}{2}\underbrace{\Big{[}\boldsymbol{\beta}_{2}+{\boldsymbol{\gamma}}_{2}{q}_{\tau}+\boldsymbol{\beta}_{2}^{*}L(t)+\boldsymbol{\gamma}_{2}^{*}S(t)q_{\tau}\Big{]}^{\prime}}_{t\text{-varying
quadratic quantile
slopes}}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\underbrace{\Big{[}\lambda_{i}+\sigma_{i}q_{\tau}\Big{]},}_{\text{individual
quantile fixed effect}}$ (3.3)

where $q_{\tau}=F_{\varepsilon}^{-1}(\tau)$ is the (unknown) $\tau$th quantile
of $\varepsilon_{it}$.

The translog cost model in (3) is quantile-specific because all bracketed
“composite” coefficients vary not only with time but also with the cost
quantile $\tau$. Furthermore, the technological change in the cost frontier is
also quantile-specific thereby allowing for heterogeneous temporal shifts
across the entire cost distribution as opposed to a shift in the mean only.
The unobserved bank fixed effect inside the last brackets is also quantile-
specific. Thus, quantile model (3) can be rewritten compactly as

$\displaystyle\mathcal{Q}_{c}\left[\tau|\boldsymbol{v}_{it}\right]\equiv$
$\displaystyle\
\alpha_{0}(\tau,t)+\boldsymbol{\alpha_{1}}(\tau,t)^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\boldsymbol{\alpha_{2}}(\tau,t)^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\mu_{i,\tau},$
(3.4)

with the “alpha” coefficients corresponding to the bracketed expressions in
(3) and $\mu_{i}\equiv\lambda_{i}+\sigma_{i}q_{\tau}$.

We opt to begin with the location-scale model to derive the conditional
quantile function of interest in (3) as opposed to postulating a quantile
regression à la (3.4) prima facie because we seek to estimate these quantiles
indirectly. This is motivated by the presence of unobserved fixed effects in
the quantile model. Namely, since there is no known general transformation
that can purge unit fixed effects from the quantile model (owing to
nonlinearity of the quantile operator), in such a case the routine check-
function-based estimators proceed to directly estimate a vector of individual
effects by means of including a full set of unit dummies. However, as noted by
Koenker,  (2004), the introduction of a large number of unit fixed effects
significantly inflates the variability of estimates of the main parameters of
interest, i.e., the slope coefficients. Furthermore, the optimization of an
$L_{1}$-norm corresponding to the check-function-based estimators, when there
is a large number of binary variables and the associated parameters to be
estimated, is well-known to be computationally cumbersome and oftentimes
intractable in practice.444For instance, in our empirical application
$n>7,500$. The traditional solution to this assumes that unit fixed effects
are only location-shifting and regularizes these individual effects by
shrinking them to a common value (see Koenker, , 2004; Lamarche, , 2010), but
these estimators have gained little popularity in applied work largely because
of their complexity. While there is an alternative fixed-effect quantile
estimator proposed by Canay,  (2011) that requires no regularization and is
notably simpler to implement, it continues to assume that the unit fixed
effects have a pure location shift effect. Using the notation of (3.4), this
is tantamount to assuming that $\mu_{i,\tau}=\mu_{i}$ for all $\tau$.
Furthermore, none of these check-function-based estimators guarantee that the
estimates of regression quantiles do not cross, which is a pervasive but oft-
ignored problem in applied work. We therefore adopt the approach recently
proposed by Machado and Santos Silva,  (2019) that allows an easy-to-implement
indirect estimation of the quantile parameters via moments, where all
parameters are estimated based on the moments implied by the location-scale
model in (3.1)–(3.2). Besides its relative computational simplicity, this
approach is advantageous for its ability to control for unobserved unit
heterogeneity that is both location- and scale-shifting: the individual
effects are allowed to affect the entire distribution rather than just
shifting its location (therefore, $\\{\mu_{i,\tau}\\}$ are also quantile-
specific). Lastly but not least importantly, this moment-based approach can be
easily applied to nonlinear-in-parameters models (like ours is) and produces
non-crossing quantile regressions.

To operationalize the estimator, we model unobservable $L(t)$ and $S(t)$ via
discretization. For each $\kappa=1,\dots,T$, define the dummy variable
$D_{\kappa,t}$ that is equal to 1 in the $\kappa$th time period and 0
otherwise. Then, we discretize time indices as
$L(t)=\sum_{\kappa=2}^{T}\eta_{\kappa}D_{\kappa,t}$ and
$S(t)=\sum_{\kappa=2}^{T}\theta_{\kappa}D_{\kappa,t}$, where $L(1)=\eta_{1}=0$
and $S(1)=\theta_{1}=0$ are normalized for identification. Parameter
identification also requires that both $\beta_{0}^{*}$ and $\gamma_{0}^{*}$ be
normalized; we set $\beta_{0}^{*}=\gamma_{0}^{*}=1$. Under these identifying
normalizations, $\beta_{0}$, $\boldsymbol{\beta}_{1}$, $\gamma_{0}$ and
$\boldsymbol{\gamma}_{1}$ are naturally interpretable as “reference”
coefficients in time period $t=1$. Then, a feasible analogue of the $\tau$th
conditional cost quantile in (3) is given by

$\displaystyle Q_{c}\left[\tau|\boldsymbol{v}_{it}\right]=$ $\displaystyle\
\Big{[}{\beta}_{0}+{\gamma}_{0}{q}_{\tau}+\sum_{\kappa}(\eta_{\kappa}+\theta_{\kappa}q_{\tau})D_{\kappa,t}\Big{]}+\Big{[}\boldsymbol{\beta}_{1}+\boldsymbol{\gamma}_{1}q_{\tau}+\sum_{\kappa}(\boldsymbol{\beta}_{1}^{*}\eta_{\kappa}+\boldsymbol{\gamma}_{1}^{*}\theta_{\kappa}q_{\tau})D_{\kappa,t}\Big{]}^{\prime}\boldsymbol{v}_{it}\
+$ $\displaystyle\
\frac{1}{2}\Big{[}\boldsymbol{\beta}_{2}+{\boldsymbol{\gamma}}_{2}{q}_{\tau}+\sum_{\kappa}(\boldsymbol{\beta}_{2}^{*}\eta_{\kappa}+\boldsymbol{\gamma}_{2}^{*}\theta_{\kappa}q_{\tau})D_{\kappa,t}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\Big{[}\lambda_{i}+\sigma_{i}q_{\tau}\Big{]}.$
(3.5)

Two remarks are in order. First, the discretized parameterization of the
unknown $L(t)$ and $S(t)$ is akin to a nonparametric local-constant estimation
of these unknown functions of time with the bandwidth parameter being set to
0. Second, though it might appear at first that, when $L(t)$ and $S(t)$ are
modeled using a series of time dummies, we obtain the time-varying slope
coefficients on $\boldsymbol{v}_{it}$ by merely interacting the latter with
time dummies and adding them as additional regressors, this is not the case
here because time dummies are restricted to have the same parameters
$\\{\eta_{k}\\}$ and $\\{\theta_{k}\\}$ both when entering additively as well
as when interacting with $\boldsymbol{v}_{it}$. Thus, the location and scale
functions are not “fully saturated” specification but, in fact, are more
parsimonious nonlinear (in parameters) functions with much fewer unknown
parameters. In avoiding a fully saturated specification that is equivalent to
sample-splitting into cross-sections, we accommodate time-invariant bank fixed
effects.

###  3.1 Estimation Procedure

Although the estimation of (3) [or (3)] can be done in one step via nonlinear
method of moments, we adopt a multi-step procedure that is significantly
easier to implement. This is possible because the moments implied by model
(3.1)–(3.2) and its assumptions are sequential in nature. In other words, we
can first estimate parameters of the location function and then those of the
scale function in two separate steps. After that, based on the estimates of
these parameters, the third step is taken to estimate unknown quantiles and,
ultimately, recover time-varying quantile coefficients in (3). In what
follows, we briefly describe this procedure, with more details available in
Appendix A.

Step 1. We first estimate parameters of the location function. For ease of
notation, let $\boldsymbol{D}_{t}=[D_{2,t},\dots,D_{T,t}]^{\prime}$ and
$\boldsymbol{\eta}=[\eta_{2},\dots,\eta_{T}]^{\prime}$. Under the assumption
(ii), from (3.1) it follows that the conditional mean function of the log-cost
$c_{it}$ is

$\mathbb{E}\left[c_{it}|\boldsymbol{v}_{it},\boldsymbol{D}_{t}\right]=\beta_{0}+\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}+\Big{[}\boldsymbol{\beta}_{1}+\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{\beta}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\Big{[}\boldsymbol{\beta}_{2}+\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{\beta}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\lambda_{i},$
(3.6)

which can be consistently estimated in the within-transformed form via
nonlinear least squares after purging additive location fixed effects. Having
obtained the nonlinear fixed-effects estimates of the slope coefficients
$\big{(}\widehat{\boldsymbol{\eta}}^{\prime},\widehat{\boldsymbol{\beta}}_{1}^{\prime},\widehat{\boldsymbol{\beta}}_{1}^{*}{{}^{\prime}},\widehat{\boldsymbol{\beta}}_{2}^{\prime},\widehat{\boldsymbol{\beta}}_{2}^{*}{{}^{\prime}}\big{)}^{\prime}$,
we can then recover the location-shifting intercept ${\beta}_{0}$ and fixed
effects $\\{\lambda_{i}\\}$ under the usual $\sum_{i=1}^{n}\lambda_{i}=0$
normalization:

$\displaystyle\widehat{\beta}_{0}$
$\displaystyle=\frac{1}{nT}\sum_{i}\sum_{t}\Bigg{(}c_{it}-\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}-\Big{[}\widehat{\boldsymbol{\beta}}_{1}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}-\frac{1}{2}\Big{[}\widehat{\boldsymbol{\beta}}_{2}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)\Bigg{)},$
(3.7) $\displaystyle\widehat{\lambda}_{i}$
$\displaystyle=\frac{1}{T}\sum_{t}\Bigg{(}c_{it}-\widehat{\beta}_{0}-\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}-\Big{[}\widehat{\boldsymbol{\beta}}_{1}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}-\frac{1}{2}\Big{[}\widehat{\boldsymbol{\beta}}_{2}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)\Bigg{)}\
\forall i.$ (3.8)

Hence, the residual is
$\widehat{u}_{it}=c_{it}-\widehat{\beta}_{0}-\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}-\Big{[}\widehat{\boldsymbol{\beta}}_{1}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}-\frac{1}{2}\Big{[}\widehat{\boldsymbol{\beta}}_{2}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)-\widehat{\lambda}_{i}$.

Step 2. We then estimate parameters of the scale function. Based on the
assumptions (ii)–(iii), we have an auxiliary conditional mean regression:

$\mathbb{E}\left[|u_{it}||\boldsymbol{v}_{it},\boldsymbol{D}_{t}\right]=\gamma_{0}+\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}+\Big{[}\boldsymbol{\gamma}_{1}+\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{\gamma}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\Big{[}\boldsymbol{\gamma}_{2}+\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{\gamma}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\sigma_{i},$
(3.9)

where $\boldsymbol{\theta}=[\theta_{2},\dots,\theta_{T}]^{\prime}$ and which,
just like in the first step, we can estimate via nonlinear least squares after
within-transforming scale fixed effects out. This yields the estimates of the
scale-function slope coefficients
$\big{(}\widehat{\boldsymbol{\theta}}^{\prime},\widehat{\boldsymbol{\gamma}}_{1}^{\prime},\widehat{\boldsymbol{\gamma}}_{1}^{*}{{}^{\prime}},\widehat{\boldsymbol{\gamma}}_{2}^{\prime},\widehat{\boldsymbol{\gamma}}_{2}^{*}{{}^{\prime}}\big{)}^{\prime}$.
To recover the scale-shifting intercept ${\gamma}_{0}$ and fixed effects
$\\{\sigma_{i}\\}$, use $\sum_{i=1}^{n}\sigma_{i}=0$:

$\displaystyle\widehat{\gamma}_{0}$
$\displaystyle=\frac{1}{nT}\sum_{i}\sum_{t}\Bigg{(}|\widehat{u}_{it}|-\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}-\Big{[}\widehat{\boldsymbol{\gamma}}_{1}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}-\frac{1}{2}\Big{[}\widehat{\boldsymbol{\gamma}}_{2}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)\Bigg{)},$
(3.10) $\displaystyle\widehat{\sigma}_{i}$
$\displaystyle=\frac{1}{T}\sum_{t}\Bigg{(}|\widehat{u}_{it}|-\widehat{\gamma}_{0}-\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}-\Big{[}\widehat{\boldsymbol{\gamma}}_{1}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}-\frac{1}{2}\Big{[}\widehat{\boldsymbol{\gamma}}_{2}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)\Bigg{)}\
\forall i.$ (3.11)

Step 3. For any given quantile index $0<\tau<1$ of interest, we next estimate
the unconditional quantile of $\varepsilon_{it}$. From (3.2), we have the
conditional quantile function of $u_{it}$:

$\displaystyle\mathcal{Q}_{u}\left[\tau|\boldsymbol{v}_{it},\boldsymbol{D}_{t}\right]=\Big{(}\gamma_{0}+\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}+\Big{[}\boldsymbol{\gamma}_{1}+\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{\gamma}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\Big{[}\boldsymbol{\gamma}_{2}+\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{\gamma}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\sigma_{i}\Big{)}q_{\tau},$
(3.12)

and therefore we can estimate $q_{\tau}$ via the standard quantile regression
of $\widehat{u}_{it}$ from Step 1 on
$\big{(}\widehat{\gamma}_{0}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}+\big{[}\widehat{\boldsymbol{\gamma}}_{1}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{1}^{*}\big{]}^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\big{[}\widehat{\boldsymbol{\gamma}}_{2}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{2}^{*}\big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\widehat{\sigma}_{i}\big{)}$
from Step 2, with no intercept. With all unknown parameters now estimated, we
can construct the estimator of the feasible analogue of the $\tau$th
conditional quantile of the log-cost in (3).

For statistical inference, we use bootstrap. To correct for finite-sample
biases, we employ Efron, ’s (1982) bias-corrected bootstrap percentile
confidence intervals. Bootstrap also significantly simplifies testing because,
owing to a multi-step nature of our estimator, computation of the asymptotic
variance of the parameter estimators is not trivial. Due to the panel
structure of data, we use wild residual block bootstrap, thereby taking into
account the potential dependence in residuals within each bank over time.
Details are provided in Appendix B.

##  4 Data

The bank-level data come from the Reports of Condition and Income (the so-
called Call Reports) and the Uniform Bank Performance Reports (UBPRs). We
obtain annual year-end data for all FDIC-insured commercial banks between 2009
and 2018. As already discussed at length, we focus on the post-financial-
crisis period.

Consistent with the widely accepted Sealey and Lindley, ’s (1977)
“intermediation approach” to formalizing production in banking, we define the
bank’s cost-function arguments as follows. The two traditional interest-
income-centered outputs are $Y_{1}$ — total loans, which include real estate
loans, agricultural loans, commercial and industrial loans, individual
consumer loans and other loans, and $Y_{2}$ — total securities, which is the
sum of securities held-to-maturity and securities held-for-sale. These output
categories are conventional and the same as those considered by, e.g., Koetter
et al.,  (2012) and Wheelock and Wilson,  (2020). The third output included in
our analysis ($Y_{3}$) measures nontraditional off-balance sheet operations.
We use a sum of credit-equivalent measures of the bank’s various off-balance
sheet operations as a proxy for its involvement in nontraditional activities.
Namely, we convert off-balance sheet items into their credit equivalents which
we determine using credit conversion factors that account for the varying
credit risk of different nontraditional banking operations.555For example,
financial standby letter of credit and repo-style transactions have a credit
conversion factor of 1, whereas performance standby letters of credit have a
factor of 0.5. The conversion factors come from the Call Reports. This
facilitates comparability of (traditional) on- and (nontraditional) off-
balance sheet activities in the analysis of banking production, which makes it
a popular practice in the literature (e.g., Jagtiani and Khanthavit, , 1996;
Hughes and Mester, , 1998; Stiroh, , 2000; Clark and Siems, , 2002; Berger and
Mester, , 2003; Asaftei, , 2008; Hughes and Mester, , 2013; Wheelock and
Wilson, , 2020).

More concretely, following McCord and Prescott,  (2014) and the FFIEC 041
Reports, we compute $Y_{3}$ by summing credit-equivalent amounts of all off-
balance sheet items. For instance, in 2015–2018, Call Reports define these
items as off-balance sheet securitization exposures, financial standby letters
of credit, performance standby letters of credit and transaction-related
contingent items, commercial and similar letters of credit with an original
maturity of one year or less, retained recourse on small business obligations
sold with recourse, repo-style transactions, unused commitments excluding
unused commitments to asset backed commercial paper conduits, unconditionally
cancelable commitments, over-the-counter derivatives, centrally cleared
derivatives, and all other off-balance sheet liabilities.

We opt for the credit equivalent of off-balance sheet activities over another
popular alternative proxy for banks’ nontraditional operations based on net
non-interest income (e.g., DeYoung and Rice, , 2004; DeYoung and Torna, ,
2013; Lozano-Vivas and Pasiouras, , 2010; Davies and Tracey, , 2014; Malikov
et al., , 2015; Wheelock and Wilson, , 2012, 2018) because the latter can be
negative, which makes it an undesirable measure for one of the bank’s outputs
(see Hughes and Mester, , 1998). It is, perhaps, even less suitable a measure
for an “output” in the structural production analysis because of its
fundamental conceptual incongruity with how the bank’s other outputs are
measured following the convention in the literature: namely, it is based on a
“flow” (income) data whereas loans $Y_{1}$ and securities $Y_{2}$ are the
“stock” (asset) measures. No such issue arises when using credit equivalents
of off-balance sheet items. Having said that, we also redo our analysis using
this alternative income-based measure of nontraditional activities in one of
the robustness checks. In this case, following the literature, the $Y_{3}$
variable is measured using the total non-interest income (inclusive of the
income from fiduciary activities, securities brokerage, investment banking,
insurance activities, venture capital and the trading revenue) minus service
charges on deposit accounts.

The three variable inputs are $X_{1}$ — physical capital measured by fixed
assets, $X_{2}$ — labor, measured as the number of full-time equivalent
employees, and $X_{3}$ — total borrowed funds, inclusive of deposits and
federal funds. Their respective prices are $W_{1}$, $W_{2}$ and $W_{3}$, where
$W_{1}$ is measured as the expenditures on fixed assets divided by premises
and fixed assets, $W_{2}$ is computed by dividing salaries and employee
benefits by the number of full-time equivalent employees, and $W_{3}$ is
computed as the interest expenses on deposits and fed funds divided by the sum
of total deposits and fed funds purchased. Total variable cost $C$ is a sum of
expenses on $X_{1}$, $X_{2}$ and $X_{3}$.

Table 1: Data Summary Statistics Variables | Mean | 1st Qu. | Median | 3rd Qu.
---|---|---|---|---
$C$ | 13,602.44 | 2,344.10 | 4,612.10 | 10,066.33
$Y_{1}$ | 424,480.49 | 55,819.58 | 114,898.89 | 265,677.61
$Y_{2}$ | 118,621.10 | 13,161.49 | 31,803.01 | 77,360.61
$Y_{3}$ | 27,304.00 | 659.78 | 2,847.73 | 10,503.06
$W_{1}$ | 50.79 | 14.83 | 21.27 | 33.85
$W_{2}$ | 57.86 | 47.00 | 54.34 | 64.90
$W_{3}$ | 0.82 | 0.40 | 0.66 | 1.09
$K_{1}$ | 70,383.27 | 9,433.85 | 18,382.47 | 40,467.85
$K_{2}$ | 0.03 | 0.01 | 0.02 | 0.04
$K_{3}$ | 1.04 | 0.01 | 0.08 | 0.31
$C$ – total variable costs; $Y_{1}$ – total loans; $Y_{2}$ – total securities;
$Y_{3}$ – off-balance sheet output measured using credit equivalents; $W_{1}$
– price of physical capital; $W_{2}$ – price of labor; $W_{3}$ – price of
financial capital; $K_{1}$ – total equity; $K_{2}$ – the ratio of
nonperforming assets to total assets; $K_{3}$ –the ratio of loan loss
provisions to total assets. Variables $C$, $W_{1}$, $W_{2}$, $Y_{1}$, $Y_{2}$,
$Y_{3}$, and $K_{1}$ are in thousands of real 2005 USD. Variables $W_{3}$,
$K_{2}$ and $K_{3}$ are in %.

We also consider equity capital $K_{1}$ as an additional input. However, due
to the unavailability of the price of equity, we follow Berger and Mester,
(2003) and Feng and Serletis,  (2010) in modeling $K_{1}$ as a quasi-fixed
input. The treatment of equity as an input to banking production technology is
consistent with Hughes and Mester,  (1993, 1998) and Berger and Mester,
(2003) in that banks may use it as a source of loanable funds and thus as a
cushion against losses. By including equity $K_{1}$ in the cost analysis, we
are therefore also able to control for the bank’s insolvency risk along the
lines of Hughes and Mester, ’s (2003) arguments, whereby “an increase in
financial capital reduces the probability of insolvency and provides an
incentive for allocating additional resources to manage risk in order to
protect the larger equity stake” (p.314). In effect, conditioning the bank’s
cost on financial capital also allows controlling for quality of loans since
the latter is influenced by risk preferences: as Mester,  (1996) explains,
risk-averse bank managers may choose to fund their loans with higher equity-
to-deposits ratios (and thus less debt) than a risk-neutral bank would. In our
analysis, we also condition the bank’s cost on two other proxy measures of
output quality reflective of credit risk associated with the likelihood that
borrowers default on their loans and accrued interest by failing to make
payments as contractually obligated. We include two most commonly used
proxies: the ratio of nonperforming assets to total assets $K_{2}$ (e.g.,
Hughes and Mester, , 2013; Wheelock and Wilson, , 2018, 2020) and the ratio of
loan loss provision to total assets $K_{3}$ (e.g., see Laeven and Majnoni, ,
2003; Acharya et al., , 2006; Berger et al., , 2010).666Although we denote
these variables as “$K$,” we do not conceptualize them as the quasi-fixed
input quantities analogous to $K_{1}$. Obviously, banks’ expectations of
credit risk are unobservable but as noted by Berger et al.,  (2010), while the
former proxy is an ex-post measure of the actual incurred losses from lending,
the loan loss provisions can be interpreted as an ex-ante measure of the level
of expected losses and thus as a proxy for expected quality of assets.
Controlling for both when modeling bank costs is imperative because lower-
quality assets generally require more resources to manage a higher-level risk
exposure thereby raising the costs for banks (see Hughes and Mester, , 2013).
Following the literature, we define nonperforming assets as a sum of total
loans and lease financing receivables past due 30 days or more and still
accruing, total loans and lease financing receivables not accruing, other real
estate owned, and charge-offs on past-due loans and leases. The loss provision
is measured using the total provision for loan and lease losses.

We exclude observations that have negative/missing values for assets, equity,
output quantities and input prices, which are likely the result of erroneous
data reporting. This leaves us with an operational sample of 44,704
observations for 7,232 banks. We deflate all nominal variables to the 2005
U.S. dollars using the consumer price index. Table 1 provides summary
statistics for our main variables.

Figure 1: Output Share of Off-Balance sheet Activities over Time, in %: (a)
average, (b) select quantiles

Given our emphasis on accounting for banks’ off-balance sheet operations, of
particular interest is the nontraditional output. Although descriptive
statistics in Table 1 expectedly indicate that the volume of $Y_{3}$ is
significantly smaller than that of the two other traditional outputs, banks’
involvement in these off-balance sheet activities has, in fact, been steadily
expanding in recent years. To show this, we plot the average share of off-
balance sheet activities in bank’s total output $Y_{3}/(Y_{1}+Y_{2}+Y_{3})$ in
Figure 1(a), from where it is evident that the average share of nontraditional
outputs among U.S. commercial banks has been steadily increasing since 2011,
almost doubling from about 2% in 2009 to 3.7% in 2018. This is consistent with
the narrative that commercial banks in the U.S. are increasingly shifting
towards off-balance sheet banking.

Obviously, the level of involvement in such nontraditional activities varies
considerably across banks, and the rather modest average share of the off-
balance sheet activities plotted in Figure 1(a) does not provide a complete
picture of the growing prevalence of nontraditional activities in banks’
operations because it conceals the well-documented heterogeneity across
individual banks. For instance, some banks in our sample are highly
specialized in off-balance sheet activities, which account for about 70% of
their outputs. Therefore, we also examine an evolution of the off-balance
sheet share in banks’ output portfolio at different quantiles in the data,
with the particular focus on the upper tail of the distribution.

Figure 2: Distribution of the Output Share of Off-Balance Sheet Activities in
2009 vs. 2018

Figure 1(b) plots select upper quantiles of $Y_{3}/(Y_{1}+Y_{2}+Y_{3})$ over
the years, with the lines from bottom to top corresponding to the median,
0.75th, 0.90th, 0.99th and 0.995th quantiles. Two observations are in order
here. First, owing to the positive skew in the off-balance sheet share
distribution, the differences in banks’ involvement in nontraditional banking
are stark, with the output share ranging from about 1.8% at the median to
19.5% for banks at the top 0.995th quantile. This cross-bank heterogeneity is
expected as the choice to engage in nontraditional operations is associated
with various idiosyncratic characteristics of banks, including their asset
size (Rogers and Sinkey Jr, , 1999). Second, the rising share of off-balance
sheet activities is present across just about its entire distribution. The
latter is particularly evident in Figure 2 that plots this distribution at the
beginning (2009) and the end (2018) of our sample period. Altogether, these
data document the rising and heterogeneous level of involvement in
nontraditional off-balance sheet activities by U.S. banks in the post-crisis
period, not only corroborating the common argument that off-balance sheet
activities ought to be accounted in the analysis of banks but also
illustrating the importance of adequately accommodating vast heterogeneity
across banks in that analysis. We seek to address both these imperatives in
our paper.

##  5 Empirical Results

This section reports the results based on our time-varying-coefficient fixed-
effects quantile model of bank cost that explicitly accommodates three-way
heterogeneity across banks: (i) distributional heterogeneity, (ii) cross-time
heterogeneity and (iii) unobserved bank heterogeneity.

Although our analysis is at different quantiles of the bank’s cost, the
interpretation of distribution heterogeneity can be generalized and extended
to bank size because the bank’s operation cost is a good proxy for its
size/scale. To sufficiently capture distributional heterogeneity across banks,
we estimate our model for the $0.10$th, $0.25$th, $0.50$th, $0.75$th and
$0.90$th quantiles. The middle three quantiles shed light on the cost
structure of mid-size banks in the interquartile range of the conditional log-
cost distribution, whereas the more extreme $0.10$th and $0.90$th quantiles
provide evidence for the smaller and larger banks, respectively.

For inference, we use the 95% bias-corrected bootstrap percentile confidence
intervals: one- or two-sided, as appropriate. In what follows, we discuss our
main empirical results pertaining to scope economies. We then supplement that
discussion by also considering two other sources of potential cost savings in
banking, namely, scale economies and technological progress.

###  5.1 Scope Economies

As discussed in Section 2, we investigate the presence of scope economies by
using the expansion-path measure of cost subadditivity. Since we analyze bank
cost structure across the entire cost distribution as opposed to its first
moment (i.e., conditional mean), our cost subadditivity measure is not only
observation- but also cost-quantile-specific. When evaluating the formulae in
(2.4)–(2.5), we replace $\mathcal{C}_{t}(\cdot)$ with the exponentiated
quantile function of the log-cost $\mathcal{Q}_{c}(\tau|\cdot)$ since our cost
function estimation is for a conditional log-quantile. That is, for a given
quantile $\tau$, we compute the cost subadditivity measure as

$\small\mathcal{S}_{t}(\tau)=\frac{\sum_{\kappa}\exp\left[\mathcal{Q}_{c}\Big{(}\tau|\varpi_{1}^{\kappa}Y_{1}^{*}+\min\\{Y_{1}\\},\varpi_{2}^{\kappa}Y_{2}^{*}+\min\\{Y_{2}\\},\varpi_{3}^{\kappa}Y_{3}^{*}+\min\\{Y_{3}\\},t\Big{)}\right]-\exp\left[\mathcal{Q}_{c}\big{(}\tau|Y_{1},Y_{2},Y_{3},t\big{)}\right]}{\exp\left[\mathcal{Q}_{c}\big{(}\tau|Y_{1},Y_{2},Y_{3},t\big{)}\right]}.$
(5.1)

It is noteworthy that our use of quantiles offers another advantage over the
more traditional conditional-mean models whereby, owing to a “monotone
equivariance property” of quantiles, our estimates of $\mathcal{S}_{t}(\tau)$,
which are based on the level of cost, are immune to transformation biases due
to exponentiation of the estimated log-cost function. The same however cannot
be said about the estimates of scope economies in analogous conditional-mean
analyses. Specifically, to evaluate scope economies, most studies typically
exponentiate the predicted logarithm of bank cost from the estimated translog
conditional-mean regressions while ignoring Jensen’s inequality. Consequently,
their scope economies estimates are likely biased. To see this, let the
conventional fixed-coefficient translog cost regression be
$c=f(\boldsymbol{v})+\epsilon$ with $\mathbb{E}[\epsilon|\boldsymbol{v}]=0$,
and recall that upper/lower-case variables are in levels/logs. It then
trivially follows that
$\mathbb{E}[C|\boldsymbol{v}]=\exp\\{f(\boldsymbol{v})\\}\mathbb{E}[\exp\\{\epsilon\\}|\boldsymbol{v}]$
which generally diverges from $\exp\\{f(\boldsymbol{v})\\}$ by a
multiplicative function of $\boldsymbol{v}$. Since cost counterfactuals in
$\mathcal{S}_{t}(\tau)$ admit different “$\boldsymbol{v}$” values as
arguments, the cost subadditivity measure above will normally be biased and
need not have the same magnitude or even sign as the true quantity unless
$\exp\\{\epsilon\\}$ is mean-independent of $\boldsymbol{v}$ which is unlikely
to be true in practice, say, if $\epsilon$ is heteroskedastic. In the case of
quantile estimation, we however do not face such a problem owing to the
equivariance of quantiles to monotone transformations, viz.
$\mathcal{Q}_{C}[\tau|\boldsymbol{v}]=\mathcal{Q}_{\exp\\{c\\}}[\tau|\boldsymbol{v}]=\exp\\{\mathcal{Q}_{c}[\tau|\boldsymbol{v}]\\}$
(e.g., see Koenker, , 2005).

Table 2: Cost Subadditivity Estimates

Cost Point Estimates Inference Categories Quantiles ($\tau$) Mean 1st Qu.
Median 3rd Qu. $\mathbf{=0}$ $\neq 0$ $\mathbf{>0}$ $\leq 0$
$\mathcal{Q}(0.10)$ 0.138 0.078 0.125 0.181 9.76% 90.24% 92.04% 7.96% (0.058,
0.469) (0.023, 0.288) (0.048, 0.463) (0.082, 0.626) $\mathcal{Q}(0.25)$ 0.175
0.107 0.163 0.225 5.48% 94.52% 95.70% 4.30% (0.078, 0.598) (0.036, 0.361)
(0.067, 0.579) (0.106, 0.777) $\mathcal{Q}(0.50)$ 0.264 0.175 0.258 0.335
1.40% 98.60% 98.90% 1.10% (0.120, 0.937) (0.066, 0.549) (0.109, 0.873) (0.155,
1.185) $\mathcal{Q}(0.75)$ 0.388 0.259 0.394 0.496 0.45% 99.55% 99.50% 0.50%
(0.194, 1.205) (0.103, 0.683) (0.169, 1.113) (0.242, 1.582)
$\mathcal{Q}(0.90)$ 0.459 0.313 0.476 0.575 0.30% 99.70% 99.60% 0.40% (0.261,
1.164) (0.121, 0.671) (0.231, 1.036) (0.356, 1.567) The left panel summarizes
point estimates of $\mathcal{S}_{t}^{*}(\tau)$ with the corresponding two-
sided 95% bias-corrected confidence intervals in parentheses. Each bank-year
is classified as exhibiting scope economies [$\mathcal{S}_{t}^{*}(\tau)>0$]
vs. non-economies [$\mathcal{S}_{t}^{*}(\tau)\leq 0$] and scope invariance
[$\mathcal{S}_{t}^{*}(\tau)=0$] vs. scope non-invariance
[$\mathcal{S}_{t}^{*}(\tau)\neq 0$] using the corresponding one- and two-sided
95% bias-corrected confidence bounds, respectively. The right panel reports
sample shares for each category and for its corresponding negating
alternative. Percentage points sum up to a hundred within binary groups only.

Now, recall that $\mathcal{S}_{t}(\tau)$ depends on the choice of
$\\{\varpi_{m}^{\kappa}\\}$, which we circumvent by choosing weights that
yield the smallest cost subadditivity measure for a given cost quantile $\tau$
in the admissible region: $\mathcal{S}_{t}^{*}(\tau)$. Namely, for each fixed
cost quantile of interest, we perform a grid search over a permissible range
of weights in $[0,1]^{6}$ at the 0.1 increments. We do this for each bank in a
given year. Table 2 summarizes such point estimates of
$\mathcal{S}_{t}^{*}(\tau)$ for different quantiles of the conditional cost
distribution. (We caution readers against confusing quantiles of the
conditional cost distribution $\tau$, for which our bank cost function and the
cost subadditivity measure are estimated, with the quantiles of empirical
distribution of observation-specific $\mathcal{S}_{t}^{*}(\tau)$ estimates
corresponding to a given $\tau$.)

The two hypotheses of particular interest here are (i)
$\mathbb{H}_{0}:\mathcal{S}_{t}^{*}(\tau)\leq 0\ \text{v.}\
\mathbb{H}_{1}:\mathcal{S}_{t}^{*}(\tau)>0$ and (ii)
$\mathbb{H}_{0}:\mathcal{S}_{t}^{*}(\tau)=0\ \text{v.}\
\mathbb{H}_{1}:\mathcal{S}_{t}^{*}(\tau)\neq 0$. Both tests are essentially
the same, except for the one- or two-sided alternatives. Although the $(i,t)$
index on outputs is suppressed in (5.1), the tests are at the level of
observation (bank-year). In case of (i), rejection of the null would imply
that even the smallest subadditivity measure is statistically positive and
scope economies can thus be inferred to also be locally significant over the
bank’s output space in a given year. In case of (ii), failure to reject the
null would suggest that subadditivity measure is statistically
indistinguishable from zero, which is consistent with the bank’s cost
structure exhibiting local scope invariance.

Figure 3: The One-Sided 95% Lower Bounds (solid lines) of the Cost
Subadditivity Point Estimates (scatter points) Across Cost Quantiles

The right panel of Table 2 reports the results of these hypothesis tests.
Namely, for each cost quantile $\tau$, we classify banks in our data based on
the two dichotomous groups of categories: banks that exhibit scope economies
[$\mathcal{S}_{t}^{*}(\tau)>0$] vs. scope non-economies
[$\mathcal{S}_{t}^{*}(\tau)\leq 0$] and the banks whose cost structure that
exhibits scope invariance [$\mathcal{S}_{t}^{*}(\tau)=0$] vs. scope non-
invariance [$\mathcal{S}_{t}^{*}(\tau)\neq 0$].

Our results provide strong evidence in support of statistically significant
scope economies across banks virtually of all sizes in the U.S. banking
sector. For banks in the middle interquartile range of the cost—essentially,
size—distribution, at least 95.7% exhibit positive economies of scope. For the
top half of the distribution (median or higher), the prevalence of significant
scope economies is about 99%. Even at the very bottom of cost distribution
($\tau=0.1$) where the revenue diversification opportunities may not be as
abundant or easily accessible, our test results suggest that roughly 92% of
banks enjoy scope-driven cost savings and those, who do not, exhibit scope
invariance. Figure 3 provides a graphic illustration of these results. For
each considered cost quantile $\tau$, the figure shows a scatter-plot of the
$\mathcal{S}_{t}^{*}(\tau)$ estimates for each bank-year observation along
with the corresponding one-sided 95% lower confidence bound. Here, we sort
these estimates by their lower confidence bounds (solid line) and color them
based on whether they are significantly above 0 or not. From Figure 3, it is
evident that positive scope economies are ubiquitous and that their presence
is only growing with quantiles of the conditional variable-cost distribution
of banks (i.e., with the bank size).

As a robustness check, we re-estimate our model under alternative empirical
specifications of the cost-function variables. Namely, we consider a different
proxy for nontraditional operations used in the literature (net non-interest
income) as well as assess sensitivity of our findings to credit risk proxies
included in the analysis. Table 3 summarizes estimates of cost subadditivity
across these alternatives. Two observations are in order here. First, omitting
an ex-ante proxy for output quality (loan loss provisions) produces uniformly
larger point estimates of cost subadditivity. Consequently, the evidence in
favor of significantly positive scope economies is even stronger in the latter
case. Nonetheless, we continue to include this important control in our main
specification. Second, when using net non-interest income as a proxy measure
of nontraditional banking operations, we obtain smaller
$\mathcal{S}_{t}^{*}(\tau)$ estimates, with the largest differences seen at
the bottom tail of costs. But even then, the empirical evidence in support of
scope economies across banks is strong. For banks in the middle of the cost
distribution (at the conditional median), 89% exhibit significant economies of
scope. The prevalence of product-scope-driven cost savings is even more
pervasive ($\geq 97.6\%$) for larger banks at higher quantiles. In the case of
smaller-scale banks at the bottom 0.10th and 0.25th quantiles, the share of
banks that enjoy scope economies—while smaller—is nonetheless non-negligible,
ranging between 41–65% and 59–82%, respectively, depending on the credit risk
proxies included in the analysis. The cost structure of the remaining banks is
scope-invariant. All in all, our findings of significant scope economies in
banking are robust to alternative specifications, and in what follows, we
therefore focus on the results from our main specification only.

Table 3: Cost Subadditivity Estimates: Robustness to Alternative Variable
Specifications

(I): Main Specification (II) (III) (IV) Cost Median Categories Median
Categories Median Categories Median Categories Quantiles ($\tau$) Est.
$\mathbf{=0}$ $\mathbf{>0}$ Est. $\mathbf{=0}$ $\mathbf{>0}$ Est.
$\mathbf{=0}$ $\mathbf{>0}$ Est. $\mathbf{=0}$ $\mathbf{>0}$
$\mathcal{Q}(0.10)$ 0.125 9.8% 92.0% 0.182 2.1% 98.3% 0.068 58.9% 40.8% 0.239
38.0% 65.0% $\mathcal{Q}(0.25)$ 0.163 5.5% 95.7% 0.256 0.6% 99.4% 0.120 41.6%
59.1% 0.282 20.3% 82.3% $\mathcal{Q}(0.50)$ 0.258 1.4% 98.9% 0.429 0.1% 99.7%
0.226 11.4% 89.0% 0.346 3.3% 97.5% $\mathcal{Q}(0.75)$ 0.394 0.5% 99.5% 0.543
0.1% 99.7% 0.356 2.6% 97.6% 0.409 1.0% 99.2% $\mathcal{Q}(0.90)$ 0.476 0.3%
99.6% 0.589 0.0% 99.7% 0.427 1.7% 98.5% 0.447 0.8% 99.4% Nontraditional Output
Measure: Credit Equivalents $\checkmark$ $\checkmark$ Net Non-Interest Income
$\checkmark$ $\checkmark$ Credit Risk Proxies: Nonperforming Assets
$\checkmark$ $\checkmark$ $\checkmark$ $\checkmark$ Loan Loss Provision
$\checkmark$ $\checkmark$ Reported are the median point estimates of
$\mathcal{S}_{t}^{*}(\tau)$ and shares of the sample for which the estimates
are statistically $>0$ (i.e., a bank-year is classified as exhibiting scope
economies) and statistically not different from $0$ (i.e., a bank-year is
classified as exhibiting scope invariance) at the 95% level. Because the two
hypotheses are tested separately, percentage points need not sum up to a
hundred. Specification (I) is our main specification, the complete results for
which are reported in Table 2.

A finding worth emphasizing here is that, having accounted for three-way
heterogeneity across banks in a pursuit of robust estimates of bank cost
subadditivity, we find no empirical evidence in support of scope diseconomies.
This is in stark contrast with earlier studies of scope economies in U.S.
banking (e.g., Berger et al., , 1987; Mester, , 1987; Hughes and Mester, ,
1993; Pulley and Braunstein, , 1992; Ferrier et al., , 1993; Pulley and
Humphrey, , 1993; Jagtiani et al., , 1995; Jagtiani and Khanthavit, , 1996;
Wheelock and Wilson, , 2001). Besides our reliance on the more robust
estimation methodology, the qualitative differences between our and prior
findings can also be attributed to fundamental changes that the banking sector
has undergone in the past two decades characterized by the growing importance
of nontraditional banking operations propelled by the financial product
innovations.

Figure 4: Kernel Densities of Cost Subadditivity Estimates Across Cost
Quantiles

Although, the subadditivity measure does not directly quantify the magnitude
of scope economies in the conventional interpretation of the latter, the value
of its point estimates can still provide useful insights into the
diversification-driven cost savings. Recall that $\mathcal{S}_{t}^{*}(\tau)$
compares the cumulative cost of multiple smaller banks of higher degrees of
relative output specialization with the cost of a larger, more relatively
diversified bank. Essentially, the subadditivity measure sheds light on scope
economies from a perspective of relative—as opposed to absolute—notion of
revenue diversification. Measured is the reduction in bank cost (in
proportions) afforded by achieving lower specialization in any one output.
From the left panel of Table 2, the mean estimates of cost subadditivity
ranges from 0.138 to 0.459 depending on the conditional cost quantile. This
suggests, on average, the potential for a 14–46% cost saving if the bank
“rebalances” its joint production of loans, securities and off-balance sheet
outputs. We also find that the magnitude of diversification-driven economies
increases as one moves from the bottom to top of the bank cost distribution,
thereby suggesting that larger banks (higher $\tau$) may economize cost better
compared to those of smaller size in the lower end of the cost distribution.

For a more holistic look at the empirical evidence of scope economies across
different quantiles of the bank cost distribution, we also provide kernel
density plots of the $\mathcal{S}_{t}^{*}(\tau)$ estimates in Figure 4. It
enables us to compare distributions of the cost subadditivity estimates as
opposed to merely focusing on marginal moments. Consistent with our earlier
discussion, these plots indicate that large-scale banks lying in the upper
quantiles of the cost distribution appear to enjoy bigger diversification-
driven cost economies than those in the lower cost quantiles. To support this
visual evidence, we formally test for the (first-order) stochastic dominance
of scope economies exhibited by banks in the top cost quantiles over those
exhibited by those in the bottom quantiles. We utilize a generalized
Kolmogorov-Smirnov test proposed by Linton et al.,  (2005) which permits
testing dominance over multiple variables (in our case, more than two cost
quantiles) and allows these variables to be estimated latent quantities as
opposed to observables from the data and to also share dependence (in our
case, the dependence is due to common parameter estimates used to construct
quantile coefficients). Specifically, let $F_{\tau}(\mathcal{S})$ represent
the cumulative distribution functions of the $\mathcal{S}_{t}^{*}(\tau)$
estimates for a given cost quantile $\tau$. We then form the null hypotheses
that diversification-driven scope economies exhibited by banks in the lower
quantiles of the cost distribution are stochastically dominated by those in
the upper quantiles of the cost distribution. More formally, for any cost
quantile of interest $\overline{\tau}\in\mathbb{T}$ with
$\mathbb{T}=\\{0.10,0.25,0.50,0.75,0.90\\}$, we are interested in

$\displaystyle\mathbb{H}_{0}:\min_{\tau\neq\overline{\tau}\in\mathbb{T}}\sup_{\mathcal{S}\in\mathbb{S}}\left[F_{\tau}(\mathcal{S})-F_{\overline{\tau}}(\mathcal{S})\right]\leq
0\ \ \text{v.}\ \
\mathbb{H}_{1}:\min_{\tau\neq\overline{\tau}\in\mathbb{T}}\sup_{\mathcal{S}\in\mathbb{S}}\left[F_{\tau}(\mathcal{S})-F_{\overline{\tau}}(\mathcal{S})\right]>0.$

We use the sub-sampling procedure suggested by Linton et al.,  (2005) to
perform the test.777We employ $199$ equidistant sub-sample sizes
$B_{n}=\\{b_{1},\dots,b_{199}\\}$, where $b_{1}=[\log\log N]$,
$b_{199}=[N/\log\log N]$ with $N=nT$ being the sample size. For each sub-
sample size, we get a $p$-value. The reported is the mean of these 199
$p$-values.

Table 4: Stochastic Dominance of Scope Subadditivity Across Cost Quantiles

$\\{\mathcal{Q}({0.75}),\dots,\mathcal{Q}({0.10})\\}$
$\\{\mathcal{Q}({0.50}),\mathcal{Q}({0.25}),\mathcal{Q}({0.10})\\}$
$\\{\mathcal{Q}({0.25}),\mathcal{Q}({0.10})\\}$ $\mathcal{Q}({0.10})$
$\mathcal{Q}({90})$ 0.578 0.578 0.739 0.970 $\mathcal{Q}({75})$ 0.894 0.970
0.970 $\mathcal{Q}({50})$ 0.970 0.970 $\mathcal{Q}({25})$ 0.784 Reported are
the $p$-values.

Table 4 reports $p$-values for the tests of dominance of
$\mathcal{S}_{t}^{*}(\tau)$ from the “row” quantile over a multi-quantile set
of $\mathcal{S}_{t}^{*}(\tau)$ from the “column” quantiles. All $p$-values are
safely greater than the conventional 0.05 level, and we fail to reject the
nulls. Combined with the visual evidence from Figure 4, we can therefore infer
that bigger banks in the higher quantiles of the cost distribution exhibit
larger scope economies than do smaller banks from the lower cost quantiles for
the entire set of observable output mixes. Relatedly, of interest is the
relation between the scope economies magnitude and the degree of bank’s
specialization in nontraditional products. To examine this, for each cost
quantile $\tau$ that we consider in our analysis, we run a least-absolute-
deviation regression of the $\mathcal{S}_{t}^{*}(\tau)$ estimates on the share
of off-balance sheet activities in bank’s total output
$Y_{3}/(Y_{1}+Y_{2}+Y_{3})$. Their median associations are all significant and
monotonically increasing with cost quantile: $-0.31$, $-0.29$, $-0.09$, $0.35$
and $0.41$ for $\tau=0.10,0.25,0.50,0.75,0.90$, respectively. This suggest
that, among larger banks (higher $\tau$) those who engage in off-balance sheet
banking more heavily tend to enjoy scope economies of greater degrees. In
contrast, for smaller banks (lower $\tau$), pivoting off balance sheet is
associated with reduced scope-driven cost savings, plausibly because of their
limited capabilities to capitalize on cross-output spillovers and input
complementarities at smaller operations scales.

Figure 5: Evolution of Cost Subadditivity

Lastly, we take a look at the evolution of scope economies. Figure 5 documents
how distributions of the cost subadditivity estimates shifted over time.
Plotted are the box-plots of $\mathcal{S}_{t}^{*}(\tau)$ across five
considered cost quantiles $\tau$ for each year $t$. The data suggest a
divergence in the degree of cost subadditivity between smaller (lower cost
quantiles) and larger (higher cost quantiles) banks over time which, however,
started reverting in the last years of the sample. We further observe that,
while positive and significant throughout, the magnitude of a cost-saving
potential associated with the product-scope diversification picked around
2013–2014 and has since been in a steady decline, across all cost quantiles.

###  5.2 Scale Economies

We complement our analysis of the scope-driven cost savings in the U.S.
banking with the examination of economies of scale. Scale economies are said
to exist if the banks’ average cost declines with equiproportional expansion
of its outputs (i.e., with the increase in scale of production). As discussed
in the introduction, the latter has been a subject of particular academic
interest in face of the post-crisis regulatory reforms in the banking sector.

Our returns to scale measure takes into account quasi-fixity of the equity
input per Caves et al.,  (1981):

$\mathcal{R}_{t}(\tau)=\big{(}1-\partial\mathcal{Q}_{c}(\tau|\cdot)/\partial
k_{1}\big{)}\Big{/}\sum_{m}\partial\mathcal{Q}_{c}(\tau|\cdot)/\partial
y_{m},$ (5.2)

where we replaced the usual $\log\mathcal{C}_{t}(\cdot)$ with the quantile
function of the log-cost $\mathcal{Q}_{c}(\tau|\cdot)$ in the formula since
our cost function estimation is for a conditional quantile. The measure of
returns to scale is therefore both observation- and cost-quantile-specific.

Just like in the case of scope economies, for a given $\tau$, we are mainly
interested in the following two hypotheses: (i)
$\mathbb{H}_{0}:\mathcal{R}_{t}(\tau)\leq 1\ \text{v.}\
\mathbb{H}_{1}:\mathcal{R}_{t}(\tau)>1$ and (ii)
$\mathbb{H}_{0}:\mathcal{R}_{t}(\tau)=1\ \text{v.}\
\mathbb{H}_{1}:\mathcal{R}_{t}(\tau)\neq 1$. In case of (i), rejection of the
null would imply that the returns to scale statistically exceed 1 implying
increasing returns (IRS) and, thus, significant scale economies. In case of
(ii), failure to reject the null would suggest that returns to scale are
statistically indistinguishable from 1, which is consistent with the bank
exhibiting constant returns to scale (CRS) and, hence, scale invariance of
costs.

Table 5:  Returns to Scale Estimates

Cost Point Estimates Inference Categories, % Quantiles ($\tau$) Mean 1st Qu.
Median 3rd Qu. $\mathbf{=1}$ $\neq 1$ $\mathbf{>1}$ $\leq 1$
$\mathcal{Q}(0.10)$ 1.300 1.263 1.293 1.328 0.02 99.98 99.99 0.01 (1.263,
1.352) (1.226, 1.316) (1.257, 1.344) (1.286, 1.382) $\mathcal{Q}(0.25)$ 1.321
1.282 1.313 1.351 0.01 99.99 100.0 0.00 (1.282, 1.363) (1.243, 1.322) (1.276,
1.354) (1.306, 1.399) $\mathcal{Q}(0.50)$ 1.361 1.316 1.351 1.394 0.01 99.99
100.0 0.00 (1.319, 1.404) (1.276, 1.356) (1.31, 1.393) (1.347, 1.444)
$\mathcal{Q}(0.75)$ 1.405 1.352 1.392 1.441 0.00 100.0 100.0 0.00 (1.352,
1.457) (1.307, 1.398) (1.344, 1.443) (1.385, 1.500) $\mathcal{Q}(0.90)$ 1.430
1.373 1.416 1.469 0.01 99.99 100.0 0.00 (1.363, 1.491) (1.314, 1.421) (1.353,
1.469) (1.397, 1.533) The left panel summarizes point estimates of
$\mathcal{R}_{t}(\tau)$ with the corresponding two-sided 95% bias-corrected
confidence intervals in parentheses. Each bank-year is classified as
exhibiting IRS [$\mathcal{R}_{t}(\tau)>1$] vs. non-IRS
[$\mathcal{R}_{t}(\tau)\leq 1$] and CRS [$\mathcal{R}_{t}(\tau)=1$] vs. non-
CRS [$\mathcal{R}_{t}(\tau)\neq 1$] using the corresponding one- and two-sided
95% bias-corrected confidence bounds, respectively. The right panel reports
sample shares for each category and for its corresponding negating
alternative. Percentage points sum up to a hundred within binary groups only.

Table 5 summarizes point estimates of the returns to scale for all estimated
quantiles of the conditional cost distribution of banks. The right panel of
the table reports the results of the hypothesis tests. Namely, reported is the
breakdown of banks that exhibit IRS (scale economies) vs. non-IRS (scale non-
economies) and of banks that exhibit CRS (scale invariance) vs. non-CRS (scale
non-invariance).

The results in Table 5 provide overwhelming evidence of ubiquitous scale
economies in the banking sector, across all cost quantiles. The average point
estimates of returns to scale ranges from 1.30 to 1.43, with banks from the
higher quantiles of cost distribution exhibiting increasing returns to scale
of larger magnitudes compared to those from the lower quantiles. We find that
almost every single bank in our sample exhibits statistically significant
scale economies (IRS). These results suggest that, when the bank radially
expands the scale of its operation, its average variable cost decreases. These
findings are consistent with the prior results which however are almost
exclusively based on the analyses of bank costs at the conditional mean (e.g.,
Wheelock and Wilson, , 2012; Hughes and Mester, , 2013; Restrepo-Tobòn and
Kumbhakar, , 2015; Malikov et al., , 2015; Restrepo-Tobòn et al., , 2015;
Wheelock and Wilson, , 2018). Given that we find evidence of significant scale
economies along the entire cost distribution, our results provide the robust
assurance to these earlier findings reported in the literature.

###  5.3 Technological Change

We conclude our analysis of bank cost structure by examining temporal shifts
in the bank cost frontier in face of technological advancements as well as
regulatory changes in the industry in aftermath of the 2008 financial crisis.
A cost-diminishing technological change can provide another means for cost
savings.

Because we model temporal variation in the cost relationship using discretized
time indices, we replace the standard continuous measure of technical change
with a discrete dual measure of technological change at each cost quantile
$\tau$. Namely, from (3), we have

$\displaystyle-\mathcal{TC}_{t}(\tau)\equiv$ $\displaystyle\
\mathcal{Q}_{c}(\tau|\cdot,t)-\mathcal{Q}_{c}(\tau|\cdot,t-1)$
$\displaystyle=$ $\displaystyle\ \Delta L(t)+\Delta S(t)q_{\tau}\ +$
$\displaystyle\ \Big{[}\boldsymbol{\beta}_{3}\Delta
L(t)+\boldsymbol{\gamma}_{3}\Delta
S(t)q_{\tau}\Big{]}^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\Big{[}\boldsymbol{\beta}_{4}\Delta
L(t)+\boldsymbol{\gamma}_{4}\Delta
S(t)q_{\tau}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right),$
(5.3)

where $\Delta L(t)=L(t)-L(t-1)$ and $\Delta S(t)=S(t)-S(t-1)$, with its
feasible analogue given by

$\displaystyle-TC_{t}(\tau)=$ $\displaystyle\
\big{(}\eta_{\kappa}+\theta_{\kappa}q_{\tau}\big{)}D_{\kappa,t}-\big{(}\eta_{\kappa-1}+\theta_{\kappa-1}q_{\tau}\big{)}D_{\kappa-1,t-1}\
+$ $\displaystyle\
\Big{[}\big{(}\boldsymbol{\beta}_{3}\eta_{\kappa}+\boldsymbol{\gamma}_{3}\theta_{\kappa}q_{\tau}\big{)}D_{\kappa,t}-\big{(}\boldsymbol{\beta}_{3}\eta_{\kappa-1}+\boldsymbol{\gamma}_{3}\theta_{\kappa-1}q_{\tau}\big{)}D_{\kappa-1,t-1}\Big{]}^{\prime}\boldsymbol{v}_{it}\
+$ $\displaystyle\
\frac{1}{2}\Big{[}\big{(}\boldsymbol{\beta}_{4}\eta_{\kappa}+\boldsymbol{\gamma}_{4}\theta_{\kappa}q_{\tau}\big{)}D_{\kappa,t}-\big{(}\boldsymbol{\beta}_{4}\eta_{\kappa-1}+\boldsymbol{\gamma}_{4}\theta_{\kappa-1}q_{\tau}\big{)}D_{\kappa-1,t-1}\Big{]}^{\prime}\text{vec}\left[\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right].$
(5.4)

The first line in (5.3) corresponds to Hick-neutral component of technological
change, whereas the last two lines represent non-neutral change.

The point estimates of technological change at different cost quantiles are
summarized in Table 6. The right panel of the table reports results of a one-
sided test of $\mathbb{H}_{0}:TC_{t}(\tau)\leq 0\ \text{v.}\
\mathbb{H}_{1}:TC_{t}(\tau)>0$, i.e., a test of whether $TC_{t}(\tau)$ is
statistically positive implying that the bank enjoys technological progress
and, therefore, a ceteris paribus cost diminution over time.

Table 6: Technical Change Estimates

Cost Point Estimates Inference Categories, % Quantiles ($\tau$) Mean 1st Qu.
Median 3rd Qu. $\mathbf{=0}$ $\neq 0$ $\mathbf{>0}$ $\leq 0$
$\mathcal{Q}(0.10)$ –0.010 –0.026 –0.008 0.006 63.06 36.94 9.69 90.31 (–0.022,
0.004) (–0.039, –0.010) (–0.020, 0.006) (–0.007, 0.021) $\mathcal{Q}(0.25)$
–0.005 –0.019 –0.003 0.011 61.23 38.77 17.52 82.48 (–0.015, 0.007) (–0.031,
–0.007) (–0.013, 0.009) (0.000, 0.025) $\mathcal{Q}(0.50)$ 0.005 –0.008 0.007
0.021 53.71 46.29 36.91 63.09 (–0.004, 0.015) (–0.018, 0.001) (–0.002, 0.018)
(0.011, 0.037) $\mathcal{Q}(0.75)$ 0.015 0.002 0.017 0.031 47.22 52.78 54.33
45.67 (0.006, 0.026) (–0.008, 0.013) (0.008, 0.030) (0.019, 0.049)
$\mathcal{Q}(0.90)$ 0.021 0.007 0.022 0.037 42.91 57.09 61.50 38.50 (0.008,
0.032) (–0.005, 0.019) (0.010, 0.035) (0.022, 0.056) The left panel summarizes
point estimates of $TC_{t}(\tau)$ with the corresponding two-sided 95% bias-
corrected confidence intervals in parentheses. Each bank-year is classified as
exhibiting technical progress [$TC_{t}(\tau)>0$] vs. non-progress
[$TC_{t}(\tau)\leq 0$] and technical stasis [$TC_{t}(\tau)=0$] vs. non-stasis
[$TC_{t}(\tau)\neq 0$] using the corresponding one- and two-sided 95% bias-
corrected confidence bounds, respectively. The right panel reports sample
shares for each category and for its corresponding negating alternative.
Percentage points sum up to a hundred within binary groups only.

Our data suggest that, in the period following the financial crisis, only
larger banks in the upper tail of the cost distribution ($\tau\geq 0.75$) have
been benefiting from significant cost-diminishing technological advances:
1.5–2.1% p.a., on average. The share of these banks with statistically
positive technological change estimates is between 54 and 62%, with the rest
of banks in the upper quartile exhibiting no statistically significant cost
diminution. For mid-size banks at the median of the cost distribution,
technical change is statistically positive for modest 37% of banks. Evidence
of significant cost diminution is even weaker among banks in the bottom half
of the cost distribution. Overall, our results suggest that the cost-saving
effects of many recent technological advancements in the banking industry,
such as the growing networks of automated teller machines, growing credit card
networks, electronic payments, internet banking, etc., that were found in the
pre-crisis period by earlier studies (e.g., Wheelock and Wilson, , 1999;
Almanidis, , 2013; Malikov et al., , 2015) have now largely waned, plausibly
because most banks had already capitalized on them to the fullest extent
feasible and/or because they now face new regulatory controls. A significant
technical change among larger banks in the upper tail of the cost distribution
is likely due to their better capability to adapt and innovate.

##  6 Conclusion

Propelled by the recent financial product innovations, banks are becoming more
complex, branching out into many “nontraditional” banking operations beyond
issuance of loans. This broadening of operational scope in a pursuit of
revenue diversification may be beneficial if banks exhibit scope economies.
The existing empirical evidence lends no support for such product-scope-driven
cost economies in banking, but it is greatly outdated and, surprisingly, there
has been little (if any) research on this subject despite the drastic
transformations that the U.S. banking industry has undergone over the past two
decades in the wake of technological advancements and regulatory changes.
Commercial banks have significantly shifted towards nontraditional operations,
and the portfolio of products offered by present-day banks is very different
from that two decades ago. This underscore the importance of taking a fresh
look at scope economies in banks because leveraging operational scope
continues to play a vital role in operations management in banking. It is also
important from a policy evaluation perspective, in the face of new financial
regulations such as the Dodd–Frank Wall Street Reform and the Consumer
Protection Act of 2010 that seek to set restrictions on the scale and scope of
bank operations.

This paper provides new evidence about scope economies in U.S. commercial
banking during the 2009–2018 post-crisis period. We improve upon the prior
literature not only by analyzing the most recent and relevant data and
accounting for bank’s nontraditional off-balance sheet operations, but also in
multiple methodological ways as follows. In a pursuit of robust estimates of
scope economies and statistical inference thereon, we estimate a flexible, yet
parsimonious, time-varying-coefficient panel-data quantile regression model
which accommodates three-way bank heterogeneity: (i) distributional
heterogeneity in the cost structure of banks along the size of their costs,
(ii) temporal variation in cost complementarities and spillovers due to
technological change/innovation, and (iii) unobserved bank confounders such as
latent management quality. Our results provide strong evidence in support of
significantly positive scope economies across banks of virtually all sizes.
Contrary to earlier studies, we find no empirical corroboration for scope
diseconomies.

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## Appendix

##  Appendix A Three-Step Estimation Procedure

This appendix describes the estimation details of a conditional quantile
function in (3). First, for ease of notation, we define
$\boldsymbol{D}_{t}=[D_{2,t},\dots,D_{T,t}]^{\prime}$,
$\boldsymbol{\eta}=[\eta_{2},\dots,\eta_{T}]^{\prime}$ and
$\boldsymbol{\theta}=[\theta_{2},\dots,\theta_{T}]^{\prime}$.

#### Step 1.

We first estimate parameters of the location function. Under the assumption
(ii), from (3.1) it follows that the conditional mean function of the log-cost
$c_{it}$ is

$\mathbb{E}\left[c_{it}|\boldsymbol{v}_{it},\boldsymbol{D}_{t}\right]=\beta_{0}+\sum_{\kappa}\eta_{\kappa}D_{\kappa,t}+\Big{[}\boldsymbol{\beta}_{1}+\boldsymbol{\beta}_{1}^{*}\sum_{\kappa}\eta_{\kappa}D_{\kappa,t}\Big{]}^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\Big{[}\boldsymbol{\beta}_{2}+\boldsymbol{\beta}_{2}^{*}\sum_{\kappa}\eta_{\kappa}D_{\kappa,t}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\lambda_{i},$
(A.1)

which can be consistently estimated in the within-transformed form via
nonlinear least squares after purging additive location fixed effects.888Note
that, although (A.1) is nonlinear, the presence of fixed effects does not give
rise to the incidental parameter problem in this case because
$\\{\lambda_{i}\\}$ enters the model additively and is not inside the
nonlinear mean function.

Given the nonlinearity and high dimensionality of (A.1) in parameters, we
estimate the slope coefficients via concentration by noticing that,
conditional on $\boldsymbol{\eta}$, this mean regression is linear in
$\big{[}\boldsymbol{\beta}_{1}^{\prime},\boldsymbol{\beta}_{2}^{\prime},\boldsymbol{\beta}_{1}^{*}{{}^{\prime}},\boldsymbol{\beta}_{2}^{*}{{}^{\prime}}\big{]}^{\prime}$
yielding the profiled least-squares estimator for
$\big{[}\boldsymbol{\beta}_{1}(\boldsymbol{\eta})^{\prime},\boldsymbol{\beta}_{2}(\boldsymbol{\eta})^{\prime},\boldsymbol{\beta}_{1}^{*}(\boldsymbol{\eta}){{}^{\prime}},\boldsymbol{\beta}_{2}^{*}(\boldsymbol{\eta}){{}^{\prime}}\big{]}^{\prime}$.
Specifically, letting the concentrated sum of (within-transformed) squared
errors be

$\displaystyle M(\boldsymbol{\eta})=\sum_{i}\sum_{t}\Bigg{[}$ $\displaystyle
c_{it}-\overline{c}_{i}-\boldsymbol{\eta}^{\prime}\big{(}\boldsymbol{D}_{t}-\overline{\boldsymbol{D}}\big{)}-\big{(}\boldsymbol{v}_{it}-\overline{\boldsymbol{v}}_{i}\big{)}^{\prime}\boldsymbol{\beta}_{1}\left(\boldsymbol{\eta}\right)\
-$
$\displaystyle\Big{(}\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{v}_{it}-\overline{\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{v}_{i}}\Big{)}^{\prime}\boldsymbol{\beta}_{1}^{*}\left(\boldsymbol{\eta}\right)-\frac{1}{2}\Big{(}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)-\overline{\text{vec}\left(\boldsymbol{v}_{i}\boldsymbol{v}_{i}^{\prime}\right)}\Big{)}\boldsymbol{\beta}_{2}\left(\boldsymbol{\eta}\right)\
-$
$\displaystyle\frac{1}{2}\Big{(}\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}\cdot\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)-\overline{\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}\cdot\text{vec}\left(\boldsymbol{v}_{i}\boldsymbol{v}_{i}^{\prime}\right)}\Big{)}^{\prime}\boldsymbol{\beta}_{2}^{*}\left(\boldsymbol{\eta}\right)\Bigg{]}^{2},$
(A.2)

with the “bar” denoting the cross-time averages of variables that it tops, we
have the profiled estimators
$[\boldsymbol{\beta}_{1}(\boldsymbol{\eta})^{\prime},\boldsymbol{\beta}_{2}(\boldsymbol{\eta})^{\prime},\boldsymbol{\beta}_{1}^{*}(\boldsymbol{\eta}){{}^{\prime}},\boldsymbol{\beta}_{2}^{*}(\boldsymbol{\eta}){{}^{\prime}}]^{\prime}=\big{(}\sum_{i}\sum_{t}\mathbb{X}_{it}\mathbb{X}_{it}^{\prime}\big{)}^{-1}\sum_{i}\sum_{t}\mathbb{X}_{it}\mathbb{Y}^{\dagger}_{it}$,
where
$\mathbb{X}_{it}=\big{[}\left(\boldsymbol{v}_{it}-\overline{\boldsymbol{v}}_{i}\right)^{\prime},$
$\big{(}\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{v}_{it}-\overline{\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{v}_{i}}\big{)}^{\prime}$,
$\frac{1}{2}\big{(}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)-\overline{\text{vec}\left(\boldsymbol{v}_{i}\boldsymbol{v}_{i}^{\prime}\right)}\big{)}$,
$\frac{1}{2}\big{(}\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}\cdot\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)-\overline{\boldsymbol{\eta}^{\prime}\boldsymbol{D}_{t}\cdot\text{vec}\left(\boldsymbol{v}_{i}\boldsymbol{v}_{i}^{\prime}\right)}\big{)}^{\prime}$
and
$\mathbb{Y}_{it}^{\dagger}=c_{it}-\overline{c}_{i}-\boldsymbol{\eta}^{\prime}\big{(}\boldsymbol{D}_{t}-\overline{\boldsymbol{D}}\big{)}$.

Thus, the nonlinear fixed-effects estimators of the slope coefficients
$[\boldsymbol{\eta}^{\prime},\boldsymbol{\beta}_{1}^{\prime},\boldsymbol{\beta}_{2}^{\prime},\boldsymbol{\beta}_{1}^{*}{{}^{\prime}},\boldsymbol{\beta}_{2}^{*}{{}^{\prime}}]^{\prime}$
in the location functions are

$\widehat{\boldsymbol{\eta}}=\arg\min_{\boldsymbol{\eta}}M(\boldsymbol{\eta})\quad\text{and}\quad\widehat{\boldsymbol{\beta}}_{1}=\boldsymbol{\beta}_{1}\left(\widehat{\boldsymbol{\eta}}\right),\
\quad\widehat{\boldsymbol{\beta}}_{2}=\boldsymbol{\beta}_{2}\left(\widehat{\boldsymbol{\eta}}\right),\
\quad\widehat{\boldsymbol{\beta}}_{1}^{*}=\boldsymbol{\beta}_{1}^{*}\left(\widehat{\boldsymbol{\eta}}\right),\
\quad\widehat{\boldsymbol{\beta}}_{2}^{*}=\boldsymbol{\beta}_{2}^{*}\left(\widehat{\boldsymbol{\eta}}\right).$
(A.3)

Under the usual $\sum_{i=1}^{n}\lambda_{i}=0$ normalization, we can then
recover the location-shifting intercept ${\beta}_{0}$ and fixed effects
$\\{\lambda_{i}\\}$ via

$\displaystyle\widehat{\beta}_{0}$
$\displaystyle=\frac{1}{nT}\sum_{i}\sum_{t}\Bigg{(}c_{it}-\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}-\Big{[}\widehat{\boldsymbol{\beta}}_{1}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}-\frac{1}{2}\Big{[}\widehat{\boldsymbol{\beta}}_{2}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)\Bigg{)},$
(A.4) $\displaystyle\widehat{\lambda}_{i}$
$\displaystyle=\frac{1}{T}\sum_{t}\Bigg{(}c_{it}-\widehat{\beta}_{0}-\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}-\Big{[}\widehat{\boldsymbol{\beta}}_{1}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}-\frac{1}{2}\Big{[}\widehat{\boldsymbol{\beta}}_{2}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)\Bigg{)}\
\forall i.$ (A.5)

Hence, the residual estimator is

$\widehat{u}_{it}=c_{it}-\widehat{\beta}_{0}-\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}-\Big{[}\widehat{\boldsymbol{\beta}}_{1}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}-\frac{1}{2}\Big{[}\widehat{\boldsymbol{\beta}}_{2}+\widehat{\boldsymbol{\eta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\beta}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)-\widehat{\lambda}_{i}.$
(A.6)

#### Step 2.

We then estimate parameters of the scale function. Based on the assumptions
(ii)–(iii), we have an auxiliary conditional mean regression:

$\mathbb{E}\left[|u_{it}||\boldsymbol{v}_{it},\boldsymbol{D}_{t}\right]=\gamma_{0}+\sum_{\kappa}\theta_{\kappa}D_{\kappa,t}+\Big{[}\boldsymbol{\gamma}_{1}+\boldsymbol{\gamma}_{1}^{*}\sum_{\kappa}\theta_{\kappa}D_{\kappa,t}\Big{]}^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\Big{[}\boldsymbol{\gamma}_{2}+\boldsymbol{\gamma}_{2}^{*}\sum_{\kappa}\theta_{\kappa}D_{\kappa,t}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\sigma_{i},$
(A.7)

which, just like in the first step, we can estimate via nonlinear least
squares after within-transforming scale fixed effects out. Concretely, with
the concentrated squared residual objective function

$\displaystyle M(\boldsymbol{\theta})=\sum_{i}\sum_{t}\Bigg{[}$
$\displaystyle|\widehat{u}_{it}|-|\overline{\widehat{u}}_{i}|-\boldsymbol{\theta}^{\prime}\big{(}\boldsymbol{D}_{t}-\overline{\boldsymbol{D}}\big{)}-\big{(}\boldsymbol{v}_{it}-\overline{\boldsymbol{v}}_{i}\big{)}^{\prime}\boldsymbol{\gamma}_{1}\left(\boldsymbol{\theta}\right)\
-$
$\displaystyle\Big{(}\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{v}_{it}-\overline{\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{v}_{i}}\Big{)}^{\prime}\boldsymbol{\gamma}_{1}^{*}\left(\boldsymbol{\theta}\right)-\frac{1}{2}\Big{(}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)-\overline{\text{vec}\left(\boldsymbol{v}_{i}\boldsymbol{v}_{i}^{\prime}\right)}\Big{)}\boldsymbol{\gamma}_{2}\left(\boldsymbol{\theta}\right)\
-$
$\displaystyle\frac{1}{2}\Big{(}\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)-\overline{\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\text{vec}\left(\boldsymbol{v}_{i}\boldsymbol{v}_{i}^{\prime}\right)}\Big{)}^{\prime}\boldsymbol{\gamma}_{2}^{*}\left(\boldsymbol{\theta}\right)\Bigg{]}^{2},$
(A.8)

and the corresponding profiled estimators given by
$[\boldsymbol{\gamma}_{1}(\boldsymbol{\theta})^{\prime},\boldsymbol{\gamma}_{2}(\boldsymbol{\theta})^{\prime},\boldsymbol{\gamma}_{1}^{*}(\boldsymbol{\theta}){{}^{\prime}},\boldsymbol{\gamma}_{2}^{*}(\boldsymbol{\theta}){{}^{\prime}}]^{\prime}=\big{(}\sum_{i}\sum_{t}\mathcal{X}_{it}\mathcal{X}_{it}^{\prime}\big{)}^{-1}\times$
$\sum_{i}\sum_{t}\mathcal{X}_{it}\mathcal{Y}^{\dagger}_{it}$, where
$\mathcal{X}_{it}=\big{[}\left(\boldsymbol{v}_{it}-\overline{\boldsymbol{v}}_{i}\right)^{\prime},$
$\big{(}\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{v}_{it}-\overline{\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\boldsymbol{v}_{i}}\big{)}^{\prime}$,
$\frac{1}{2}\big{(}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)-\overline{\text{vec}\left(\boldsymbol{v}_{i}\boldsymbol{v}_{i}^{\prime}\right)}\big{)}$,
$\frac{1}{2}\big{(}\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)-\overline{\boldsymbol{\theta}^{\prime}\boldsymbol{D}_{t}\cdot\text{vec}\left(\boldsymbol{v}_{i}\boldsymbol{v}_{i}^{\prime}\right)}\big{)}^{\prime}$
and
$\mathcal{Y}_{it}^{\dagger}=|\widehat{u}_{it}|-|\overline{\widehat{u}}_{i}|-\boldsymbol{\theta}^{\prime}\big{(}\boldsymbol{D}_{t}-\overline{\boldsymbol{D}}\big{)}$,
the nonlinear fixed-effects estimators of the scale-function slope
coefficients
$[\boldsymbol{\theta}^{\prime},\boldsymbol{\gamma}_{1}^{\prime},\boldsymbol{\gamma}_{2}^{\prime},\boldsymbol{\gamma}_{3}^{\prime},\boldsymbol{\gamma}_{4}^{\prime}]^{\prime}$
are

$\widehat{\boldsymbol{\theta}}=\arg\min_{\boldsymbol{\theta}}M(\boldsymbol{\theta})\quad\text{and}\quad\widehat{\boldsymbol{\gamma}}_{1}=\boldsymbol{\gamma}_{1}\left(\widehat{\boldsymbol{\theta}}\right),\
\quad\widehat{\boldsymbol{\gamma}}_{2}=\boldsymbol{\gamma}_{2}\left(\widehat{\boldsymbol{\theta}}\right),\
\quad\widehat{\boldsymbol{\gamma}}_{1}^{*}=\boldsymbol{\gamma}_{1}^{*}\left(\widehat{\boldsymbol{\theta}}\right),\
\quad\widehat{\boldsymbol{\gamma}}_{2}^{*}=\boldsymbol{\gamma}_{2}^{*}\left(\widehat{\boldsymbol{\theta}}\right).$
(A.9)

To recover the scale-shifting intercept ${\gamma}_{0}$ and fixed effects
$\\{\sigma_{i}\\}$, use $\sum_{i=1}^{n}\sigma_{i}=0$:

$\displaystyle\widehat{\gamma}_{0}$
$\displaystyle=\frac{1}{nT}\sum_{i}\sum_{t}\Bigg{(}|\widehat{u}_{it}|-\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}-\Big{[}\widehat{\boldsymbol{\gamma}}_{1}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}-\frac{1}{2}\Big{[}\widehat{\boldsymbol{\gamma}}_{2}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)\Bigg{)},$
(A.10) $\displaystyle\widehat{\sigma}_{i}$
$\displaystyle=\frac{1}{T}\sum_{t}\Bigg{(}|\widehat{u}_{it}|-\widehat{\gamma}_{0}-\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}-\Big{[}\widehat{\boldsymbol{\gamma}}_{1}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}-\frac{1}{2}\Big{[}\widehat{\boldsymbol{\gamma}}_{2}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)\Bigg{)}\
\forall i.$ (A.11)

#### Step 3.

For any given quantile index $0<\tau<1$ of interest, we next estimate the
unconditional quantile of $\varepsilon_{it}$. From (3.2), we have the
conditional quantile function of $u_{it}$:

$\displaystyle\mathcal{Q}_{u}\left[\tau|\boldsymbol{v}_{it},\boldsymbol{D}_{t}\right]=\Big{(}\gamma_{0}+\sum_{\kappa}\theta_{\kappa}D_{\kappa,t}+\Big{[}\boldsymbol{\gamma}_{1}+\boldsymbol{\gamma}_{1}^{*}\sum_{\kappa}\theta_{\kappa}D_{\kappa,t}\Big{]}^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\Big{[}\boldsymbol{\gamma}_{2}+\boldsymbol{\gamma}_{2}^{*}\sum_{\kappa}\theta_{\kappa}D_{\kappa,t}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\sigma_{i}\Big{)}q_{\tau}.$
(A.12)

We therefore can estimate $q_{\tau}$ via a univariate quantile regression
(with no intercept) via

$\widehat{q}_{\tau}=\arg\min_{q}\sum_{i}\sum_{t}\rho_{\tau}\Bigg{\\{}\widehat{u}_{it}-\Big{(}\widehat{\gamma}_{0}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}+\Big{[}\widehat{\boldsymbol{\gamma}}_{1}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{1}^{*}\Big{]}^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\Big{[}\widehat{\boldsymbol{\gamma}}_{2}+\widehat{\boldsymbol{\theta}}^{\prime}\boldsymbol{D}_{t}\cdot\widehat{\boldsymbol{\gamma}}_{2}^{*}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\widehat{\sigma}_{i}\Big{)}q\Bigg{\\}},$
(A.13)

where
$\rho_{\tau}\\{\xi\\}=\xi\left(\tau-\mathbb{I}\left\\{\xi<0\right\\}\right)$
is the check function, $\widehat{u}_{it}$ is estimated in Step 1, and
$\big{[}\widehat{\boldsymbol{\theta}}^{\prime},\widehat{\gamma}_{0},\widehat{\boldsymbol{\gamma}}_{1}^{\prime},\widehat{\boldsymbol{\gamma}}_{2}^{\prime},\widehat{\boldsymbol{\gamma}}_{1}^{*}{{}^{\prime}},\widehat{\boldsymbol{\gamma}}_{2}^{*}{{}^{\prime}}\big{]}^{\prime}$
and $\\{\widehat{\sigma}_{i}\\}$ are estimated in Step 2.

With all unknown parameters now estimated, we can construct the estimator of
the feasible analogue of the $\tau$th conditional quantile of the log-cost in
(3).

##  Appendix B Bias-Corrected Bootstrap Inference

To correct for finite-sample biases, we employ Efron, ’s (1982) bias-corrected
bootstrap percentile confidence intervals to conduct statistical inference.
Bootstrap also significantly simplifies testing because, owing to a multi-step
nature of our estimator, computation of the asymptotic variance of the
parameter estimators is not trivial. Due to the panel structure of data, we
use wild residual block bootstrap, thereby taking into account the potential
dependence in residuals within each bank over time. The bootstrap algorithm is
as follows.

  1. (i)

Compute the estimator in Step 1. Save the estimated coefficients
$\big{[}\widehat{\boldsymbol{\eta}}^{\prime},\widehat{\beta}_{0},\widehat{\boldsymbol{\beta}_{1}}^{\prime},\widehat{\boldsymbol{\beta}}_{2}^{\prime},\widehat{\boldsymbol{\beta}}_{1}^{*}{{}^{\prime}},\widehat{\boldsymbol{\beta}}_{2}^{*}{{}^{\prime}}\big{]}^{\prime}$,
location fixed effects $\\{\widehat{\lambda}_{i}\\}$ and residuals
$\\{\widehat{u}_{it}\\}$.

  2. (ii)

Generate bootstrap weights $\omega_{i}^{b}$ for each cross-section/bank $i$
from the two-point mass distribution:

$w_{i}^{b}=\begin{cases}\left(1+\sqrt{5}\right)/2&\textrm{with
prob.}\left(\sqrt{5}-1\right)/\left(2\sqrt{5}\right)\\\
\left(1-\sqrt{5}\right)/2&\textrm{with
prob.}\left(\sqrt{5}+1\right)/\left(2\sqrt{5}\right)\end{cases}.$ (B.1)

Next, for each observation $(i,t)$ with $i=1,\dots,n$ and $t=1,\dots,T$,
generate a new bootstrap disturbance as
$u_{it}^{b}=w_{i}^{b}\times\widehat{u}_{it}$.

  3. (iii)

Construct a new bootstrap outcome variable:
$c_{it}^{b}=\widehat{\beta}_{0}+\sum_{\kappa}\widehat{\eta}_{\kappa}D_{\kappa,t}+\Big{[}\widehat{\boldsymbol{\beta}}_{1}+\widehat{\boldsymbol{\beta}}_{1}^{*}\sum_{\kappa}\widehat{\eta}_{\kappa}D_{\kappa,t}\Big{]}^{\prime}\boldsymbol{v}_{it}+\frac{1}{2}\Big{[}\widehat{\boldsymbol{\beta}}_{2}+\widehat{\boldsymbol{\beta}}_{2}^{*}\sum_{\kappa}\widehat{\eta}_{\kappa}D_{\kappa,t}\Big{]}^{\prime}\text{vec}\left(\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right)+\widehat{\lambda}_{i}+u_{it}^{b}$
for all $i=1,\dots,n$ and $t=1,\dots,T$.

  4. (iv)

Recompute the Step 1 estimators in (A.3)–(A.5) using $c_{it}^{b}$ in place of
$c_{it}$ to obtain bootstrap estimates of the location-function coefficients
and fixed effects. Signify these by the superscript “$b$.” Then, compute the
bootstrap estimate of the residual
$\widehat{u}_{it}^{b}=c_{it}-\widehat{\beta}_{0}^{b}-\sum_{\kappa}\widehat{\eta}_{\kappa}^{b}D_{\kappa,t}-\Big{[}\widehat{\boldsymbol{\beta}}_{1}^{b}+\widehat{\boldsymbol{\beta}}_{1}^{*b}\sum_{\kappa}\widehat{\eta}_{\kappa}^{b}D_{\kappa,t}\Big{]}^{\prime}\boldsymbol{v}_{it}-\allowbreak\frac{1}{2}\Big{[}\widehat{\boldsymbol{\beta}}_{2}^{b}+\widehat{\boldsymbol{\beta}}_{2}^{*b}\sum_{\kappa}\widehat{\eta}_{\kappa}^{b}D_{\kappa,t}\Big{]}^{\prime}\text{vec}\left[\boldsymbol{v}_{it}\boldsymbol{v}_{it}^{\prime}\right]-\widehat{\lambda}_{i}^{b}$.

  5. (v)

Reestimate the Step 2 estimator in (A.9) and the Step 3 estimator in (A.13)
using $\widehat{u}_{it}^{b}$ in place of $\widehat{u}_{it}$ to obtain
bootstrap estimates of the scale function coefficients and $q_{\tau}$.

  6. (vi)

Repeat bootstrap steps (ii)–(v) $B$ times ($B=500$ in this study). Use the
empirical distribution of $B$ bootstrap replicas of some estimand of interest
(say, a coefficient or a quantile-specific function thereof such as cost
subadditivity measure $\mathcal{S}_{t}^{*}$) to construct bias-corrected
confidence intervals for this estimand.

To make matters concrete, let the (potentially, observation- and quantile-
specific) estimand of interest be denoted by $\widehat{\mathcal{E}}$. We can
use the empirical distribution of
$\\{\widehat{\mathcal{E}}^{1},\dots,\widehat{\mathcal{E}}^{B}\\}$ to estimate
the bias-corrected two-sided $(1-\alpha)\times 100$% confidence bounds for
$\mathcal{E}$ as an interval between the $[a_{1}\times 100]$th and
$[(1-a_{2})\times 100]$th percentiles of the bootstrap distribution, where
$a_{1}=\Phi\left(2\widehat{z}_{0}+\Phi^{-1}(\alpha/2)\right)$ and
$a_{2}=\Phi\left(2\widehat{z}_{0}+\Phi^{-1}(1-\alpha/2)\right)$ with
$\Phi(\cdot)$ being the standard normal cdf along with its quantile function
$\Phi^{-1}(\cdot)$. Parameter
$\widehat{z}_{0}=\Phi^{-1}\left(\\#\big{\\{}\widehat{\mathcal{E}}^{b}<\widehat{\mathcal{E}}\big{\\}}/{B}\right)$
is a bias-correction factor measuring median bias, with $\\#\\{\mathcal{A}\\}$
being a count function that returns the number of times event $\mathcal{A}$ is
true. Naturally, to estimate the one-sided lower/upper $(1-\alpha)\times 100$%
confidence bound with bias correction, we respectively use the $[o_{1}\times
100]$th or $[(1-o_{2})\times 100]$th percentiles of the bootstrap
distribution, where
$o_{1}=\Phi\left(2\widehat{z}_{0}+\Phi^{-1}(\alpha)\right)$ and
$o_{2}=\Phi\left(2\widehat{z}_{0}+\Phi^{-1}(1-\alpha)\right)$.