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\begin{center}
ECONOMIC ANALYSIS GROUP

DISCUSSION PAPER


\vspace{3cm}


Bias in Reduced-Form Estimates of Pass-through

\vspace{0.25cm} by \vspace{0.25cm}

Alexander MacKay, Nathan H. Miller, Marc Remer, and Gloria Sheu*  \\

EAG 14-1  $\quad$ February 2014

\end{center}

\vspace{2cm}

\small

\noindent EAG Discussion Papers are the primary vehicle used to disseminate research from economists in the Economic Analysis Group (EAG) of the Antitrust Division. These papers are intended to inform interested individuals and institutions of EAG's research program and to stimulate comment and criticism on economic issues related to antitrust policy and regulation. The
Antitrust Division encourages independent research by its economists. The views expressed herein are entirely those of the author and are not purported to reflect those of the United States Department of Justice.
\vspace{0.25cm}

\noindent Information on the EAG research program and discussion paper series may be obtained from Russell Pittman, Director of Economic Research, Economic Analysis Group, Antitrust Division, U.S. Department of Justice, LSB 9004, Washington, DC 20530, or by e-mail at russell.pittman@usdoj.gov. Comments on specific papers may be addressed directly to the authors at the same mailing address or at their e-mail addresses.

\vspace{0.25cm}

\noindent To obtain a complete list of titles or to request single copies of individual papers, please write to Kathy Burt at kathy.burt@usdoj.gov or call (202) 307-5794. In addition, recent papers are now available on the Department of Justice website at http://www.justice.gov/atr/public/eag/discussion-papers.html.

\vspace{0.25cm}

\noindent\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_


\noindent *MacKay: University of Chicago, Department of Economics, mackay@uchicago.edu.  Miller: Georgetown University, McDonough School of Business, nhm27@georgetown.edu.  Remer and Sheu: Economic Analysis Group, U.S. Department of Justice, marc.remer@usdoj.gov, gloria.sheu@usdoj.gov.  The views expressed herein are entirely those of the authors and should not be purported to reflect those of the U.S. Department of Justice.

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\begin{abstract}            % beginning of the abstract
We show that, in general, consistent estimates of cost pass-through are not obtained from reduced-form regressions of price on cost. We derive a formal approximation for the bias that arises even under standard orthogonality conditions. We provide guidance on the conditions under which bias may frustrate inference.
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\section{Introduction}

This paper addresses the conditions under which cost pass-through
can be estimated accurately with reduced-form regressions of price
on costs. Our interest in this subject follows recent articles that
develop the theoretical properties of pass-through (e.g., \citeN{fabinger2009};
\citeN{fabinger2012}). It is now understood that pass-through can
be used to infer structural demand parameters when demand estimation
is infeasible (\citeN{MRS2012}), to make counterfactual predictions
without functional form restrictions (\citeN{jaffeweyl2011}; \citeN{MRRS2013}),
and to evaluate the magnitude of market power (\citeN{scharfstein2013}).
The empirical literature on pass-through, on which our results have
direct bearing, is substantial and spans the fields of industrial
organization (e.g., \citeN{borenstein1997}; \citeN{Fabra2013})
and international trade (e.g., \citeN{atkeson2008}; \citeN{gopinath2011}).

Our main result is that a reduced-form regression of price on costs
-- the standard methodology for pass-through estimation -- only yields
a consistent estimate if the underlying economic environment has specific
properties. The usual assumption that observed cost measures are uncorrelated
with other determinants of cost does not guarantee consistency.

We derive a second-order approximation for the bias under the assumption
that the regressor is uncorrelated with unobserved costs. For all twice differentiable pricing functions, including those that arise
from Bertrand profit-maximizing behavior, bias can be decomposed into
two components. The first component is due to regression misspecification
and occurs if the distribution of costs is skewed and pass-through
varies with costs.  Misspecification bias can be present even if all variables
are observed perfectly.  In isolation, this bias can be accounted for with
standard techniques (such as a polynomial regression or splines).  The second component, which we call ``partial information bias,'' arises if the marginal costs of a firm or its competitors are partially observed, pass-through varies with costs, and the observed and unobserved costs are not independent. Because independence can be a strong assumption, we also provide bounds for partial information bias that can be calculated given information on the underlying demand system and plausible assumptions
on the distribution of costs.

Neither misspecification bias nor partial information bias arise
if the underlying economic environment is characterized by constant pass-through.
Thus, the standard methodology for pass-through estimation can be
motivated by invoking constant pass-through environments, such as those
developed in \citeN{bulow1983}.%
\footnote{Constant pass-through occurs in a class of demand systems that includes
the linear, log-linear, and constant-markup demand systems.%
} Alternatively, because pass-through is not constant generally, the
standard methodology can be motivated by invoking symmetric cost distributions
and independence between observed and unobserved costs. To our knowledge,
prior research has not fully recognized the conditions under which
misspecification bias and partial information bias arise. Thus, our
result is useful to researchers developing new empirical estimates
of pass-through or seeking to interpret the existing literature.

We introduce the two sources of bias by way of numerical example in
Section 2. We then develop the main theoretical result in Section
3 and provide discussion. Bounds to partial information bias are developed
in Section 4.


\section{Numerical Examples}

Consider first the case of a monopolist facing a logit demand schedule.
Let the mean consumer valuation of the monopolist's good equal two
minus the monopolist's price, and let the mean consumer valuation
of the outside option be zero. This gives rise to the demand schedule
\[
s=\frac{\exp(2-p)}{1+\exp(2-p)},
\]
where $s$ is the monopolist's market share and $p$ is its price.
Theoretical pass-through is not constant with logit demand and, in
the case of monopoly, equals $1-s$. Let the monopolist's marginal
costs be drawn from a distribution with expected value 0.5 and variance
0.083. At the expected marginal cost, the profit maximizing price
yields a market share for the monopolist of 0.434 and a pass-through
of 0.566.

Suppose that an econometrician has 100,000 observations of costs and
the associated profit maximizing prices. Will a reduced-form estimate
obtain a meaningful parameter? We simulate results drawing costs from
a uniform distribution and a lognormal distribution.%
\footnote{Specifically we use a uniform distribution on $[0,1]$ and a log-normal
distribution with parameters $\sigma=\sqrt{\ln(4/3)}$ and $\mu=\ln(1/2)-(1/2)\sigma^{2}$.%
} The reduced-form regression estimate nearly matches the theoretical pass-through rate in the case of uniformly distributed costs. The lognormal
distribution, in contrast, results in a pass-through estimate of 0.604,
roughly 6.5\% higher than the actual pass-through rate at the expected
cost. The source of this bias, as we detail below, is misspecification
of the reduced-form regression equation combined with an asymmetric
cost distribution and pass-through that is not constant.

Now suppose instead that the econometrician observes only a fraction
of the monopolist's marginal cost. The observed component, which we
denote $c_{1}$, has a uniform distribution with expected value 0.5
and variance 0.083. The unobserved component takes the form $c_{2}=Rc_{1}$
where $R$ is a Radermacher weight that takes the values of one and
negative one with equal probability. Thus the observed and unobserved
components of cost, while uncorrelated, are not independent because
$c_{1}$ is correlated with the conditional variance of $c_{2}$.
Again the econometrician obtains 100,000 observations of profit-maximizing
prices and observed costs, and uses a univariate reduced-form regression
to estimate pass-through. Our simulations indicate a point estimate
of 0.642, which is 12.4\% higher than the theoretical pass-through
at the expected cost.%
\footnote{The standard errors with 100,000 observations are very small.%
} We refer to this bias as partial information bias because the source
is the the unobserved cost component, which is missing from the regression
equation. We view this as distinct from omitted variable bias, which
typically is derived based on correlation between a regressor and
the error term in linear settings.


\section{Theoretical Result}

We provide a theoretical result that explains the numerical results
developed above, in the form of the following proposition:

\bigbreak \textbf{Proposition 1}: \textit{Let the equilibrium price
of a firm take the form $p=f(c_{1},c_{2})$ where $f$ is a twice differentiable pricing
function and $c_{1}$ and $c_{2}$ are stochastic cost terms with
expected values $a$ and $b$, respectively. Further let $c_{1}$
and $c_{2}$ be mean independent so that $E[c_{2}|c_{1}]=b$. Then,
to a second-order approximation and for any data generating process,
the probability limit of the coefficient obtained from a univariate
regression of $p$ on $c_{1}$ equals }
\[
\mbox{plim}\hat{\rho}=f_{1}(a,b)+\frac{1}{2}f_{11}(a,b)\frac{E[(c_{1}-a)^{3}]}{Var(c_{1})}+\left(f_{12}(a,b)\frac{Cov((c_{1})^{2},c_{2})}{Var(c_{1})}+\frac{1}{2}f_{22}(a,b)\frac{Cov(c_{1},(c_{2})^{2})}{Var(c_{1})}\right),
\]
\textit{where $f_{i}$ is the partial derivative of $f$ with respect
to $c_{i}$ and $f_{ij}(a,b)$ is the second derivative of $f$ with
respect to $c_{i}$ and $c_{j}$.}

\noindent \bigbreak \textbf{Proof}: See the Appendix.

\noindent \bigbreak The pricing function $f$ can be conceptualized
as the equilibrium strategy for a consumer demand schedule and a competitive
game. Thus, it is fully consistent with oligopoly price theory. The
first term on the right hand side of the main equation, $f_{1}(a,b)$,
is the theoretical pass-through that arises at the expected value
of the cost distribution. We take as given that this is the object
of interest, although in some settings the researcher may be interested
in other notions of pass-through.

The second term is the misspecification bias that arises due to the
curvature of the pricing function. There is no misspecification bias
if either (i) pass-through is constant, in which case $f_{11}(a,b)=0$,
or (ii) the distribution of $c_{1}$ is symmetric such that the third
central moment $E[(c_{1}-a)^{3}]$ is zero. Indeed, misspecification
bias explains the result of the lognormal example in Section 2. Given
the parameterization employed, it is possible to calculate that with
log-normally distributed costs $\frac{1}{2}E[(c_{1}-a)^{3}]/Var(c_{1})=0.266$.
This component times the second derivative of the pricing function
gives a value of 0.037, which is close to the empirical bias of 0.038
that arises with the reduced-form regression.

If misspecification is the only source of bias, then the regression
coefficient represents a weighted average of the theoretical pass-through
that arises at the realized cost draws. While this object can be useful
to empirical researchers in some settings, its use also can hinder
inference in studies that treat pass-through as an outcome of interest.
As one example, consider an econometrician that seeks to compare pass-through
across markets or over time as an indicator for changing market power
(e.g., as in \citeN{scharfstein2013}). If the true economic environment
features non-constant pass-through and an asymmetric cost distribution,
and if the realized cost draws vary across markets, then reduced-form
regression results can be unreliable. Misspecification bias in isolation
can be accounted for with standard techniques (e.g., splines or polynomials),
but if partial information bias is also present then these adjustments
can confound rather than improve inference.

The final term, in parentheses, is the partial information bias that
arises from the exclusion of $c_{2}$ from the reduced-form regression.
There is no partial information bias if pass-through is constant.
If pass-through is not constant then the standard assumption that
$c_{1}$ and $c_{2}$ are uncorrelated is insufficient to eliminate
partial information bias.%
\footnote{Intuitively, it may seem that if the variance of $c_{1}$ is small,
the approximation is local, and therefore the impact of higher-order
moments is limited. However, this is not the case, as both $Cov((c_{1})^{2},c_{2})/Var(c_{1})$
and $Cov(c_{1},(c_{2})^{2})/Var(c_{1})$ may be large as $Var(c_{1})$
approaches zero.%
} For example, pass-through estimates are biased upward if $c_{1}$
is positively correlated with the conditional variance of $c_{2}$
and the pricing function is convex, as is the case with the example
in Section 2. Partial information can arise in a variety of settings
because $c_{2}$ can be thought of either as a marginal cost component
or as an unobserved cost of a competitor. If $c_{2}$ is an unobserved
cost term, then it affects pass-through via total marginal costs.
If instead $c_{2}$ is a competitor's costs, then it influences the
competitor's price and affects pass-through indirectly.


\section{Bounds on Partial Information Bias}

The condition that $c_{1}$ and $c_{2}$ are independent is sufficient
to eliminate partial information bias. When independence does not
hold it is possible to bound the magnitude of bias, given an assumption
on the pricing function (which implies a demand system and competitive
game) and restrictions on the distribution of unobserved costs. The
pricing function allows the terms $f_{12}(a,b)$ and $f_{22}(a,b)$
to be obtained. Then, recognizing that $Cov(c_{1},(c_{2})^{2})/Var(c_{1})$
is the slope coefficient from a regression of $(c_{2})^{2}$ on $c_{1}$,
we have that
\[
\frac{Cov(c_{1},(c_{2})^{2})}{Var(c_{1})}<\frac{\max_{c_{1}}E[(c_{2})^{2}|c_{1}]-\min_{c_{1}}E[(c_{2})^{2}|c_{1}]}{c_{1}^{max}-c_{1}^{min}}.
\]
If we further assume that $E[c_{1}|c_{2}]=E[c_{1}]$, then this becomes
\[
\frac{Cov(c_{1},(c_{2})^{2})}{Var(c_{1})}<\frac{\max_{c_{1}}Var(c_{2}|c_{1})-\min_{c_{1}}Var(c_{2}|c_{1})}{c_{1}^{max}-c_{1}^{min}}.
\]
Reasonable guesses for the range of the conditional variance in the
unobserved component of cost will generate bounds for the bias. The
other component of bias can be bounded similarly, by recognizing that
\[
\frac{Cov((c_{1})^{2},c_{2})}{Var(c_{1})}=\frac{Cov((c_{1})^{2},c_{2})}{Var(c_{2})}\frac{Var(c_{2})}{Var(c_{1})}<\frac{\max_{c_{2}}Var(c_{1}|c_{2})-\min_{c_{2}}Var(c_{1}|c_{2})}{c_{2}^{max}-c_{2}^{min}}\frac{Var(c_{2})}{Var(c_{1})}.
\]
Thus, researchers sometimes may be able to assess the reliability
of reduced-form regression even in economic environments with non-constant
pass-through and an unobserved marginal cost term that is not independent
from the observed cost component.

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\appendix
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\section*{Appendix}


\section{Proof of Proposition 1}

The proof is by construction. Consider the second-order Taylor expansion
of the equilibrium price around the expected costs ($a,b$):
\begin{eqnarray*}
p & \approx & f(a,b)+f_{1}(a,b)(c_{1}-a)+f_{2}(a,b)(c_{2}-b)\\
 &  & +\frac{1}{2}\left[f_{11}(a,b)(c_{1}-a)^{2}+2f_{12}(a,b)(c_{1}-a)(c_{2}-b)+f_{22}(a,b)(c_{2}-b)^{2}\right]\\
 & = & [constant]+f_{1}(a,b)c_{1}+\frac{1}{2}f_{11}(a,b)((c_{1})^{2}-2ac_{1})+f_{12}(a,b)(c_{1}c_{2}-ac_{2}-bc_{1})\\
 &  & +f_{2}(a,b)c_{2}+\frac{1}{2}f_{22}(a,b)((c_{2})^{2}-2bc_{2})
\end{eqnarray*}
The plim of $\hat{\rho}$ from a regression of $p$ on $c_{1}$ is
equal to $Cov(p,c_{1})/Var(c_{1})$, where
\begin{eqnarray*}
Cov(p,c_{1}) & \approx & f_{1}(a,b)Var(c_{1})-af_{11}(a,b)Var(c_{1})-bf_{12}(a,b)Var(c_{1})\\
 &  & +\frac{1}{2}f_{11}(a,b)Cov(c_{1},(c_{1})^{2})+f_{12}(a,b)Cov(c_{1},c_{1}c_{2})+\frac{1}{2}f_{22}(a,b)Cov(c_{1},(c_{2})^{2}).
\end{eqnarray*}
Therefore it follows that
\begin{eqnarray*}
\mbox{plim}\hat{\rho} & \approx & f_{1}(a,b)+f_{11}(a,b)\left[\frac{1}{2}\frac{Cov(c_{1},(c_{1})^{2})}{Var(c_{1})}-a\right]\\
 &  & +\left(f_{12}(a,b)\left[\frac{Cov(c_{1},c_{1}c_{2})}{Var(c_{1})}-b\right]+\frac{1}{2}f_{22}(a,b)\frac{Cov(c_{1},(c_{2})^{2})}{Var(c_{1})}\right),
\end{eqnarray*}
and this yields Proposition 1 after minor algebraic manipulations.

\end{document}
