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Inference for First-Price Auctions with Guerre, Perrigne, and Vuong's Estimator 2019. This manuscript version is made available under the Creative Commons CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/. First version: March 3, 2016, This version: .
\title{\textbf{Inference for First-Price Auctions with Guerre, Perrigne,
and Vuong's Estimator}\footnote{We thank the editor, Han Hong, the associate editor and two anonymous
referees, whose comments have greatly improved the paper. We also
thank Emmanuel Guerre, Jinyong Hahn, Nianqing Liu, Ryo Okui, Joris
Pinkse, Christoph Rothe, and Quang Vuong for their helpful comments.
Jun Ma's research is supported by the Fundamental Research Funds for
the Central Universities, and the Research Funds of Renmin University
of China, \#15XNF014 and fund for building world-class universities
(disciplines) of Renmin University of China. Vadim Marmer gratefully
acknowledges the financial support of the Social Sciences and Humanities
Research Council of Canada under grants 435-2013-0331 and 435-2017-0329. }
\let\thefootnote\relax\footnotetext{
\textcopyright 2019. This manuscript version is made available under the Creative Commons CC-BY-NC-ND 4.0 license \url{http://creativecommons.org/licenses/by-nc-nd/4.0/}. First version: March 3, 2016, This version: \today.}}
\author{Jun Ma\thanks{School of Economics, Renmin University of China, 59 Zhongguancun Street,
Haidian District, Beijing, China. Email: [email removed]. Tel.:
8610 62511102.}\and Vadim Marmer\thanks{Corresponding author. Vancouver School of Economics, University of
British Columbia, 6000 Iona Drive, Vancouver, BC, V6T 1L4, Canada.
Email: [email removed]. Tel.: 1 (604) 822 8217.} \and Artyom Shneyerov\thanks{Department of Economics, Concordia University, 1455 de Maisonneuve
Blvd. West, Montreal, Quebec, H3G 1M8, Canada. Deceased 24 October
2017.}}
\maketitle
\begin{abstract}
We consider inference on the probability density of valuations in
the first-price sealed-bid auctions model within the independent private
value paradigm. We show the asymptotic normality of the two-step nonparametric
estimator of \citet*{Guerre_Perrigne_Vuong_Auction_Econometrica_2000}
(GPV), and propose an easily implementable and consistent estimator
of the asymptotic variance. We prove the validity of the pointwise
percentile bootstrap confidence intervals based on the GPV estimator.
Lastly, we use the intermediate Gaussian approximation approach to
construct bootstrap-based asymptotically valid uniform confidence
bands for the density of the valuations. \\
\noindent \textit{Keywords}: Asymptotic Normality, Bootstrap, First-Price
Auctions, Gaussian Approximation, Independent Private Values, Two-Step
Nonparametric Estimators, Uniform Confidence Bands
\noindent \emph{JEL classification}: C14, C57
\end{abstract}
\tableofcontents
\newpage
\section{Introduction}
The structural estimation of auctions is an important and rapidly
growing subfield at the junction of econometrics and industrial organization.
Since the seminal work of \citet*[GPV hereafter]{Guerre_Perrigne_Vuong_Auction_Econometrica_2000},
much of theoretical and applied work has focused on nonparametric
estimation of first-price, sealed-bid auctions.\footnote{\citet*{hendricks2007empirical} survey the empirical auction literature,
while \citet*{athey2007nonparametric} survey the nonparametric identification
approaches. \citet{hickman2012structural} provide a recent review.} The object of interest is the probability density function (PDF)
of latent valuations, which can then be used for a variety of policy
counterfactuals such as the optimal reserve price (\citealp{paarsch1997deriving}
and \citealp{li2003semiparametric}).\footnote{In auctions, the valuation of a bidder is simply his or her willingness
to pay for the object.}
The focus on nonparametric estimation is due to several reasons. First,
in empirical applications, large auction datasets are often available.\footnote{E.g., \citet{kawai2014detecting} utilize a dataset of $40,000$ auctions
in their study of collusion in Japan, while \citet{augenblick2015sunk}
employs a dataset $160,000$ penny auctions.} Second, nonparametric methods are flexible since no functional form
assumptions are needed. Third, in auctions, nonparametric estimators
are built directly from the identification arguments, and are often
easy to implement.\footnote{See \citet{athey2002identification} for a number of additional identification
results.}
GPV proposed a two-step nonparametric estimator of the PDF of valuations
in first-price auctions, and showed that it is uniformly consistent
and attains the minimax optimal uniform convergence rate. However,
it has been an open question whether this estimator also converges
in a distributional sense, which would allow empirical researchers
to perform inferences, thereby increasing the scope of applications.
Recently, \citet{Marmer_Shneyerov_Quantile_Auctions} developed an
alternative quantile-based estimator of the PDF of valuations and
showed its asymptotic normality.\footnote{More recently, quantile methods have been used in the context of auctions
in \citet{gimenes2016quantile}, \citet{gimenes2017econometrics},
\citet{liu2017nonparametric}, and \citet{luo2017integrated}.} However, the GPV estimator is well established in the literature
and is used in all the empirical applications we are aware of. Moreover,
our results imply that the GPV estimator has a smaller asymptotic
variance than that of the quantile-based estimator, as long as the
two estimators use the same second-order kernel.
Inference and the closely related problem of nonparametric testing
in structural auction models are important and have been receiving
increasing attention in the literature. Beginning with the fundamental
\citet{haile2003nonparametric}'s test for common values, recent contributions
include testing for the monotonicity of bidding strategies (\citealp{liuvuong2013}),
endogenous entry (\citealp{li2009entry} and \citealp{marmer2013model}),
common versus private values (\citealp{hill2013there}), the affiliation
of bidder valuations (\citealp{jun2010consistent}, \citealp{li2010testing}
and \citealp{de2010testing}), and inferences on bidder risk attitudes
(\citealp{fang2014inference}). In the absence of the asymptotic distribution
framework for the GPV estimator, these papers have adopted problem-specific
approaches in each case.
The first main result we show in this paper is that the GPV estimator
is asymptotically normal. The key difficulty is the presence of the
nonparametric first step, which provides nonparametric estimates of
the valuations in each auction. In the second step, the kernel density
estimator is applied to those estimates, rather than the true valuations.
This creates a unique challenge, to our knowledge not previously addressed
in the econometrics literature. Our main insight is that the leading
term in an asymptotic expansion of the estimator can be viewed as
a V-statistic with a kernel that depends on the bandwidth. A projection
argument shows that the distribution of this V-statistic is asymptotically
normal. Using maximal inequalities for empirical processes and U-processes
developed in recent literature, we show that the remainder term is
uniformly negligible. The proof is rather long due to an intricate
nature of the estimator, and involves some delicate steps.
Note that a working paper version of GPV \citep{GPV1995} also has
an asymptotic normality result for the GPV estimator. However, the
result therein is of a limited nature as it relies on a particular
choice of tuning parameters, which insures that only the second stage
of the estimating procedure contributes to the asymptotic variance.
Thus, in their approach, the uncertainty due to the estimation of
valuations in the first stage can be ignored asymptotically, which
is achieved by applying different rates of smoothing of \emph{auction-specific
covariates} at both stages. The approach is restrictive in two respects:
(i) While GPV's smoothing strategy makes first-stage estimation errors
negligible asymptotically, in finite samples their contribution to
the variance may still be significant. Our approach takes into account
the contribution of both stages and, as a result, is more accurate
in finite samples. (ii) Equally importantly, their approach cannot
be applied in cases with no auction-specific covariates or when covariates
are modeled semi-parametrically as in \citet{haile2003nonparametric}.
Note that treating auction-specific characteristics semi-parametrically
is particularly appealing to practitioners.
One unusual feature of our asymptotic normality result concerns the
form of the asymptotic variance of the GPV estimator. Typically, the
asymptotic variances of kernel density estimators depend on the integral
of the squared kernel, which is a known constant that can be easily
computed analytically or numerically.\footnote{See, e.g., \citet{li2007net}.}
However, in the case of the GPV estimator, this constant is replaced
by a convoluted integral transformation that involves the kernel function,
its derivative, and the derivatives of the bidding strategy. This
is a consequence of the two-step nature of the GPV estimator and happens
due to the impact of the estimation errors from the first stage of
the procedure on the distribution of the estimator. Since the bidding
strategy is unknown, this fact complicates estimation of the asymptotic
variance of the GPV estimator.
Our second contribution is to propose a consistent estimator of the
asymptotic variance that avoids estimation of the bidding strategy
and its derivative. Its uniform rate of convergence is established
using the maximal inequalities. Our third contribution is to show
the validity of the percentile bootstrap for the GPV estimator, which
allows constructing confidence intervals without estimation of the
asymptotic variance.
Our pointwise asymptotic normality results can be used for inference
on the optimal reserve price, as the latter is determined by a nonlinear
equation in the PDF of valuations \citep[see][]{haile2003iim}. In
our fourth contribution, however, we extend the pointwise results
and develop valid uniform confidence bands for the PDF. The uniform
confidence bands can be used, e.g., for specification of valuations'
density. The extension utilizes the uniform rates of convergence of
the remainder terms in our V-statistic approximation of the GPV estimator
and its Hoeffding decomposition; it also relies on Gaussian anti-concentration
inequalities and Gaussian coupling theorems developed in recent literature.
This approach, referred to as the Intermediate Gaussian Approximation
(IGA, hereafter) in the literature, is based on the seminal work of
\citet{chernozhukov2014anti,chernozhukov2014gaussian,chernozhukov2016empirical}.
\citet{chernozhukov2014gaussian} showed that although a random function
based on nonparametric estimation errors does not typically weakly
converge to any tight Gaussian random element, under certain conditions
the supremum of its studentized version can be often approximated
by the supremum of a tight Gaussian random element, the distribution
of which changes with the sample size. \citet{chernozhukov2016empirical,chernozhukov2014anti}
showed that under certain conditions the distribution of the Gaussian
supremum can be approximated by bootstrapping, and the bootstrap consistency
can be shown by applying the coupling theorems and the Gaussian anti-concentration
inequality developed in these papers.\footnote{See, e.g., \citet{Kato_Sasaki_2,Kato_Sasaki_1} for recent applications
of these theorems for constructing confidence bands for different
nonparametric curves. } Our paper is one of the first applications of these results. Our
Monte Carlo simulation results show that the IGA approach produces
confidence bands with excellent finite-sample coverage properties.
Our paper is also related to the recent literature on nonparametrically
generated regressors in nonparametric regression. See, e.g., \citet{rilstone1996nonparametric},
\citet{pinkse2001nonparametric}, and \citet{mammen2012nonparametric}.
Note, however, that while that literature is concerned with nonparametrically
estimated exogenous covariates, we deal with kernel estimation of
the density of a nonparametrically generated ``dependent'' variable,
potentially in presence of observable conditioning variables.
The rest of the paper proceeds as follows. Section \ref{sec:Data-Generating-Process-(DGP)}
introduces the data-generating process (DGP) and describes the GPV
estimator in detail. Due to complexity of the estimator, in Section
\ref{sec:Asymptotic Normality} we show the asymptotic normality of
the GPV estimator in a simplified model that has a constant number
of bidders across auctions and no auction-specific heterogeneity.
Such a simplification allows us to present the main ideas in a more
transparent fashion. In Section \ref{sec:Pointwise-Confidence-Intervals},
we derive an estimator for the asymptotic variance and establish its
uniform rate of convergence. We also show consistency of the percentile
bootstrap confidence intervals. Section \ref{sec:Uniform-Confidence-Bands}
provides results on constructing valid confidence bands within the
same simplified framework. Proofs of the results in Sections \ref{sec:Asymptotic Normality}\textendash \ref{sec:Uniform-Confidence-Bands}
are given in the Appendix. Section \ref{sec:Auction-Specific-Heterogeneity}
provides corresponding theorems in the general model with a random
number of bidders and auction-specific heterogeneity. The proofs of
these results can be found in the Supplement (included). Section \ref{sec:Binding-Reserve-Price}
discusses how our approach can be extended to auctions with binding
reserve prices. We report the results from our Monte Carlo study in
Section \ref{sec:Monte-Carlo-Simulations}. Section \ref{sec:Conclusions}
concludes.
\begin{ntnbold} ``$a\coloneqq b$'' is understood as ``$a$ is
defined by $b$''. ``$a\eqqcolon b$'' is understood as ``$b$
is defined by $a$''. $\mathbbm{1}\left(\cdot\right)$ denotes the
indicator function, and we also denote $\mathbbm{1}_{A}\coloneqq\mathbbm{1}\left(\cdot\in A\right)$.
Let $\mathbb{H}\left(\boldsymbol{c},r\right)$ denote a closed hypercube
centered at some real vector $\boldsymbol{c}$ with edge $r$. Let
$\boldsymbol{c}^{\mathrm{T}}$ denote the transpose of $\boldsymbol{c}$.
``$\overset{d}{=}$'' means ``equal in distribution''. Let $\ell^{\infty}\left(A\right)$
be the class of bounded functions defined on $A$. For any $f\in\ell^{\infty}\left(A\right)$,
let $\left\Vert f\right\Vert _{A}\coloneqq\underset{x\in A}{\mathrm{sup}}\left|f\left(x\right)\right|$
be the sup-norm. \end{ntnbold}
\section{Data-Generating Process (DGP) and the GPV Estimator\label{sec:Data-Generating-Process-(DGP)}}
The econometrician observes data from $L$ auctions. Let $\boldsymbol{X}_{l}$
denote the $d$-dimensional relevant characteristics for the object
in the $l$-th auction. Let $N_{l}$ denote the number of bidders
in the $l$-th auction. Let $B_{il}$ denote the bid submitted by
the $i$-th bidder in the $l$-th auction. The data observed by the
econometrician is given by $\left\{ \left(B_{il},\boldsymbol{X}_{l},N_{l}\right):i=1,...,N_{l},\,l=1,...,L\right\} .$
Unobserved bidders' valuations of the $l$-th auctioned object are
denoted by $\left\{ V_{il}:i=1,...,N_{l},\,l=1,...,L\right\} .$ The
following assumption describes the DGP.\footnote{Assumption \ref{assu:DGP} is similar to Assumptions A1 and A2 of
GPV and \citet[Assumption 1]{Marmer_Shneyerov_Quantile_Auctions}.
Part (f) imposes the condition that the valuations and the random
number of bidders are independent conditionally on the characteristics.
See Footnote 14 of GPV.}
\begin{assumption}[\textbf{DGP}]
\textup{\label{assu:DGP}(a) $\left\{ \left(\boldsymbol{X}_{l},N_{l}\right):l=1,...,L\right\} $
are i.i.d.}
\textup{(b) The marginal PDF of $\boldsymbol{X}_{1}$, denoted by
$\varphi\left(\cdot\right)$, is strictly positive and continuous
on its support $\mathcal{X}\coloneqq\left[\underline{x},\overline{x}\right]^{d}$
for some $\underline{x}<\overline{x}$ assumed to be known and admits
up to $R+1$ ($R\geq2$) continuous partial derivatives.}
\textup{(c) The conditional probability mass function of $N_{1}$
given $\boldsymbol{X}_{1}=\boldsymbol{x}$, denoted by $\pi\left(\cdot|\boldsymbol{x}\right)$,
has a known support $\mathcal{N}\coloneqq\left\{ \underline{n},...,\overline{n}\right\} $
for all $\boldsymbol{x}\in\mathcal{X}$, $\underline{n}\geq2$.}
\textup{(d) For all $n\in\mathcal{N}$, $\pi\left(n|\cdot\right)$
is strictly positive and admits up to $R+1$ continuous partial derivatives.}
\textup{(e) $\left(n,\boldsymbol{x}\right)\mapsto\pi\left(n|\boldsymbol{x}\right)\varphi\left(\boldsymbol{x}\right)$
is bounded above and away from zero on its support $\mathcal{N}\times\mathcal{X}$. }
\textup{(f) For each $l=1,...,L$, given $\boldsymbol{X}_{l}=\boldsymbol{x}$
and $N_{l}=n$, $\left\{ V_{il}:i=1,...,n\right\} $ are i.i.d. with
conditional PDF $f\left(\cdot|\boldsymbol{x}\right)$ and conditional
cumulative distributional function (CDF) $F\left(\cdot|\boldsymbol{x}\right)$.}
\textup{(g) For each $n\in\mathcal{N}$, the support of $\left(V_{11},\boldsymbol{X}_{1}\right)$
is $\mathcal{S}_{V,\boldsymbol{X}}\coloneqq\left\{ \left(v,\boldsymbol{x}\right):\boldsymbol{x}\in\mathcal{X},\,v\in\left[\underline{v}\left(\boldsymbol{x}\right),\overline{v}\left(\boldsymbol{x}\right)\right]\right\} ,$
with some positive boundary functions $\underline{v}\left(\cdot\right)$
and $\overline{v}\left(\cdot\right)$.}
\textup{(h) $f\left(\cdot|\cdot\right)$ is strictly positive and
bounded away from zero and admits up to $R$ continuous partial derivatives
on $\mathcal{S}_{V,\boldsymbol{X}}$.}
\end{assumption}
Bidders' valuations are not directly observable. Following GPV, we
assume that $B_{il}$ is the equilibrium bid of risk-neutral bidder
$i$ submitted in the $l$-th auction. Therefore the valuations are
linked to the observed bids through the Bayesian Nash equilibrium
(BNE) bidding strategy:
\begin{equation}
B_{il}=s\left(V_{il},\boldsymbol{X}_{l},N_{l}\right)\coloneqq V_{il}-\frac{1}{\left(F\left(V_{il}|\boldsymbol{X}_{l}\right)\right)^{N_{l}-1}}\int_{\underline{v}\left(\boldsymbol{X}_{l}\right)}^{V_{il}}\left(F\left(u|\boldsymbol{X}_{l}\right)\right)^{N_{l}-1}\mathrm{d}u.\footnotemark\label{eq:BNE strategy}
\end{equation}
\footnotetext{See Equations (1) and (8) of GPV.}Under Assumptions
\ref{assu:DGP}(a) and \ref{assu:DGP}(f), $\left\{ B_{il}:i=1,...,N_{l}\right\} $
are conditionally i.i.d. draws given $\boldsymbol{X}_{l}$ and $N_{l}$.
Let $\overline{b}\left(\boldsymbol{x},n\right)\coloneqq s\left(\overline{v}\left(\boldsymbol{x}\right),\boldsymbol{x},n\right)$
and $\underline{b}\left(\boldsymbol{x}\right)\coloneqq\underline{v}\left(\boldsymbol{x}\right)$.
Proposition 1(i) of GPV shows that the support of $\left(B_{il},\boldsymbol{X}_{l},N_{l}\right)$
is $\left\{ \left(b,\boldsymbol{x},n\right):n\in\mathcal{N},\,\left(b,\boldsymbol{x}\right)\in\mathcal{S}_{B,\boldsymbol{X}}^{n}\right\} ,$
where $\mathcal{S}_{B,\boldsymbol{X}}^{n}\coloneqq\left\{ \left(b,\boldsymbol{x}\right):\boldsymbol{x}\in\mathcal{X},\,b\in\left[\underline{b}\left(\boldsymbol{x}\right),\overline{b}\left(\boldsymbol{x},n\right)\right]\right\} .$
Let $G\left(\cdot|\boldsymbol{x},n\right)$ denote the conditional
CDF of $B_{il}$ given $\boldsymbol{X}_{l}=\boldsymbol{x}$ and $N_{l}=n$.
Let $g\left(\cdot|\boldsymbol{x},n\right)$ be the corresponding conditional
PDF. GPV established identification of the inverse bidding strategy:
\begin{equation}
V_{il}=\xi\left(B_{il},\boldsymbol{X}_{l},N_{l}\right)\coloneqq B_{il}+\frac{1}{N_{l}-1}\frac{G\left(B_{il}|\boldsymbol{X}_{l},N_{l}\right)}{g\left(B_{il}|\boldsymbol{X}_{l},N_{l}\right)}.\label{eq:inverse bidding strategy}
\end{equation}
By replacing $G\left(\cdot|\cdot,\cdot\right)$ and $g\left(\cdot|\cdot,\cdot\right)$
in (\ref{eq:inverse bidding strategy}) with their nonparametric estimators,
GPV proposed an estimator of $\xi\left(\cdot,\cdot,\cdot\right)$,
denoted by $\widehat{\xi}\left(\cdot,\cdot,\cdot\right)$. The GPV
estimator of $f\left(v|\boldsymbol{x}\right)$ is the kernel density
estimator that, in place of the true valuations, uses the so-called
pseudo valuations $\left\{ \widehat{V}_{il}\coloneqq\widehat{\xi}\left(B_{il},\boldsymbol{X}_{l},N_{l}\right):i=1,...,N_{l},\,l=1,...,L\right\} $.
Below we provide the details of GPV's estimation procedure. Let $K_{0}$
and $K_{1}$ be univariate kernel functions of different orders satisfying
the following assumption:
\begin{assumption}[\textbf{Kernel}]
\label{assu:Assumption 2 Kernel}\textup{(a) $K_{0}$ and $K_{1}$
are symmetric, compactly supported on $[-1,1]$ and twice continuously
differentiable on $\mathbb{R}$ with Lipschitz derivatives.}
\textup{(b) $K_{0}$ is of order $R$, and $K_{1}$ is of order $1+R$:
$\int K_{0}(u)\mathrm{d}u=1$ and $\int u^{k}K_{0}(u)\mathrm{d}u=0$
for $k=1,...,R-1$; $\int K_{1}(u)\mathrm{d}u=1$ and $\int u^{k}K_{1}(u)\mathrm{d}u=0$
for $k=1,...,R$.}
\end{assumption}
With $K_{g}\coloneqq K_{1}$ and the multi-dimensional product kernels
\[
K_{f}\left(v,\boldsymbol{x}\right)\coloneqq K_{0}\left(v\right)\cdot\prod_{k=1}^{d}K_{0}\left(x_{k}\right)\textrm{ and }K_{\boldsymbol{X}}\left(\boldsymbol{x}\right)\coloneqq\prod_{k=1}^{d}K_{1}\left(x_{k}\right),\textrm{ for }v\in\mathbb{R},\,\boldsymbol{x}=\left(x_{1},...,x_{d}\right)\in\mathbb{R}^{d},
\]
define the following nonparametric estimators:
\[
\widehat{\varphi}\left(\boldsymbol{x}\right)\coloneqq\frac{1}{L}\sum_{l=1}^{L}\frac{1}{h^{d}}K_{\boldsymbol{X}}\left(\frac{\boldsymbol{X}_{l}-\boldsymbol{x}}{h}\right)\textrm{ and }\widehat{\pi}\left(n|\boldsymbol{x}\right)\coloneqq\frac{1}{\widehat{\varphi}\left(\boldsymbol{x}\right)L}\sum_{l=1}^{L}\mathbbm{1}\left(N_{l}=n\right)\frac{1}{h^{d}}K_{\boldsymbol{X}}\left(\frac{\boldsymbol{X}_{l}-\boldsymbol{x}}{h}\right),
\]
where $\widehat{\varphi}\left(\cdot\right)$ is the kernel density
estimator of $\varphi$ and $\widehat{\pi}\left(\cdot|\cdot\right)$
is the Nadaraya-Watson estimator of the conditional probability mass
function $\pi\left(\cdot|\cdot\right)$. Based on these, we define
below the nonparametric estimators of the conditional CDF and PDF
of the bids:
\begin{gather*}
\widehat{G}\left(b|\boldsymbol{x},n\right)\coloneqq\frac{1}{\widehat{\pi}\left(n|\boldsymbol{x}\right)\widehat{\varphi}\left(\boldsymbol{x}\right)L}\sum_{l=1}^{L}\mathbbm{1}\left(N_{l}=n\right)\frac{1}{N_{l}}\sum_{i=1}^{N_{l}}\mathbbm{1}\left(B_{il}\leq b\right)\frac{1}{h^{d}}K_{\boldsymbol{X}}\left(\frac{\boldsymbol{X}_{l}-\boldsymbol{x}}{h}\right),\\
\widehat{g}\left(b|\boldsymbol{x},n\right)\coloneqq\frac{1}{\widehat{\pi}\left(n|\boldsymbol{x}\right)\widehat{\varphi}\left(\boldsymbol{x}\right)L}\sum_{l=1}^{L}\mathbbm{1}\left(N_{l}=n\right)\frac{1}{N_{l}}\sum_{i=1}^{N_{l}}\frac{1}{h^{1+d}}K_{g}\left(\frac{B_{il}-b}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{X}_{l}-\boldsymbol{x}}{h}\right).
\end{gather*}
Consider a partition of $\mathbb{R}^{d}$ with generic half-open hypercubes
of side $h_{\partial}>0$:
\[
\Pi_{k_{1},...,k_{d}}\coloneqq\left[k_{1}h_{\partial},\left(k_{1}+1\right)h_{\partial}\right)\times\cdots\times\left[k_{d}h_{\partial},\left(k_{d}+1\right)h_{\partial}\right),
\]
where $\left(k_{1},...,k_{d}\right)$ runs over $\mathbb{Z}^{d}$.
Let $\Pi_{h_{\partial}}\left(\boldsymbol{x}\right)$ denote the hypercube
that contains $\boldsymbol{x}$ in this partition. Define
\begin{gather}
\widehat{\overline{b}}\left(\boldsymbol{x},n\right)\coloneqq\mathrm{\mathrm{max}}\left\{ B_{pl}:p=1,...,N_{l},\,\boldsymbol{X}_{l}\in\Pi_{h_{\partial}}\left(\boldsymbol{x}\right),\,N_{l}=n,\,l=1,...,L\right\} ,\nonumber \\
\widehat{\underline{b}}\left(\boldsymbol{x}\right)\coloneqq\mathrm{min}\left\{ B_{pl}:p=1,...,N_{l},\,\boldsymbol{X}_{l}\in\Pi_{h_{\partial}}\left(\boldsymbol{x}\right),\,l=1,...,L\right\} \label{eq:bids support estimator}
\end{gather}
to be the estimators of the boundaries of the support. Note that the
estimators of the boundaries are super consistent. Let $\mathcal{\widehat{S}}_{B,\boldsymbol{X}}^{n}\coloneqq\left\{ \left(b,\boldsymbol{x}\right):\boldsymbol{x}\in\mathcal{X},\,b\in\left[\widehat{\underline{b}}\left(\boldsymbol{x}\right),\widehat{\overline{b}}\left(\boldsymbol{x},n\right)\right]\right\} .$
The support of $\left(B_{il},\boldsymbol{X}_{l},N_{l}\right)$ then
can be estimated by $\left\{ \left(b,\boldsymbol{x},n\right):n\in\mathcal{N},\,\left(b,\boldsymbol{x}\right)\in\mathcal{\widehat{S}}_{B,\boldsymbol{X}}^{n}\right\} .$
The kernel density estimator $\widehat{g}\left(b|\boldsymbol{x},n\right)$
is asymptotically biased when $\left(b,\boldsymbol{x}\right)$ is
near the boundaries of the support. GPV suggested that trimming should
be applied to the observations near the estimated boundaries using
the trimming factor $\mathbb{T}_{il}\coloneqq\mathbbm{1}\left(\mathbb{H}\left(\left(B_{il},\boldsymbol{X}_{l}\right),2h\right)\subseteq\mathcal{\widehat{S}}_{B,\boldsymbol{X}}^{N_{l}}\right).$
The two-step nonparametric estimator of $f\left(v|\boldsymbol{x}\right)$
developed by GPV is
\begin{equation}
\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)\coloneqq\frac{1}{\widehat{\varphi}\left(\boldsymbol{x}\right)L}\sum_{l=1}^{L}\frac{1}{N_{l}}\sum_{i=1}^{N_{l}}\mathbb{T}_{il}\frac{1}{h^{1+d}}K_{f}\left(\frac{\widehat{V}_{il}-v}{h},\frac{\boldsymbol{X}_{l}-\boldsymbol{x}}{h}\right).\label{eq:f_hat_GPV definition}
\end{equation}
For deriving the asymptotic properties of the GPV estimator, we make
the following assumption on the bandwidths $h$ and $h_{\partial}$.\footnote{Assumption \ref{assu: rate of bandwidth}(a) is the same as the assumption
on the rate of bandwidth for \citet{Marmer_Shneyerov_Quantile_Auctions}'s
quantile-based estimator. See Assumption 3 therein. }
\begin{assumption}[\textbf{Bandwidth}]
\label{assu: rate of bandwidth}\textup{ (a) The bandwidth $h$ is
of the form $h=\lambda_{1}L^{-\gamma_{1}}$, for some strictly positive
constants $\lambda_{1}$ and $\gamma_{1}$ satisfying $\nicefrac{1}{\left(2R+3+d\right)}\leq\gamma_{1}<\nicefrac{1}{\left(3+d\right)}$.}
\textup{(b) When $d>0$, the ``boundary'' bandwidth is of the form
$h_{\partial}=\lambda_{\partial}\left(\nicefrac{\mathrm{log}\left(L\right)}{L}\right)^{\nicefrac{1}{\left(1+d\right)}}$,
where $\lambda_{\partial}$ is a strictly positive constant.}
\end{assumption}
GPV showed that the optimal uniform convergence rate of their estimator
is attained when the bandwidth $h$ is of order $O\left(\left(\nicefrac{\mathrm{log}\left(L\right)}{L}\right)^{\nicefrac{1}{\left(2R+3+d\right)}}\right)$.
Note that the bandwidth in Assumption \ref{assu: rate of bandwidth}
is of smaller order. Under-smoothing imposed in Assumption \ref{assu: rate of bandwidth}
is needed to control the asymptotic bias of the GPV estimator, which
is important for the validity of inference.
\section{Asymptotic Normality of the GPV Estimator\label{sec:Asymptotic Normality}}
For clarity of the presentation of the main ideas and results, in
this section we first establish pointwise asymptotic normality of
the GPV estimator in a simplified version of the model that has a
fixed number of bidders and no auction-specific heterogeneity. When
there are covariates capturing auction-specific heterogeneity present,
these results can be used by treating the covariates additively semi-parametrically
as in \citet[Section 6]{haile2003nonparametric}. In that case, there
is no kernel smoothing over the covariates, as the GPV procedure would
be applied to the ``homogenized'' bids, which are constructed as
residuals from the parametric regression of the bids against the covariates.
In the simplified model, the econometrician observes data on bids
in $L$ identical auctions, with a fixed number of bidders $N$ in
each auction: $\left\{ B_{il}:i=1,\ldots,N,\,l=1,\ldots,L\right\} .$
Under Assumption \ref{assu:DGP}, the valuations $\left\{ V_{il}:i=1,\ldots,N,\,l=1,\ldots,L\right\} $
are i.i.d. with a compact support $\left[\underline{v},\overline{v}\right]\subseteq\mathbb{R}_{+}$,
PDF $f$ and CDF $F$. The object of interest is the PDF of the valuation
at interior points of $\left[\underline{v},\overline{v}\right]$.
Suppose that $v_{l}>\underline{v}$, $v_{u}<\overline{v}$ and $I\coloneqq\left[v_{l},v_{u}\right]$
is an inner closed sub-interval of $\left[\underline{v},\overline{v}\right]$.
Fix
\[
\overline{\delta}\coloneqq\mathrm{min}\left\{ \nicefrac{\left(\overline{v}-v_{u}\right)}{2},\nicefrac{\left(v_{l}-\underline{v}\right)}{2}\right\} .
\]
Under Assumption \ref{assu:DGP}, $f$ is strictly positive and bounded
away from zero on its support and admits at least $R$ continuous
derivatives. Lemma A1 of GPV showed that under Assumption \ref{assu:DGP},
the BNE bidding strategy is strictly increasing and $R+1$ times continuously
differentiable. In this simplified framework, the inverse of the BNE
bidding strategy is
\begin{equation}
\xi\left(b\right)\coloneqq b+\frac{1}{N-1}\frac{G\left(b\right)}{g\left(b\right)},\label{eq: inverse bid strat}
\end{equation}
where $G$ and $g$ are the CDF and PDF of bids respectively. Denote
$\overline{b}\coloneqq s\left(\overline{v}\right)$ and $\underline{b}\coloneqq s\left(\underline{v}\right)$.
Proposition 1(ii) of GPV shows that under Assumption \ref{assu:DGP},
$g$ is also bounded away from zero on its support $\left[\underline{b},\overline{b}\right]$:
\begin{equation}
\underline{C}_{g}\coloneqq\underset{b\in\left[\underline{b},\overline{b}\right]}{\mathrm{inf}}g\left(b\right)>0.\label{eq:density of bids bounded from 0}
\end{equation}
The inverse bidding strategy (\ref{eq: inverse bid strat}) can be
estimated by
\[
\widehat{\xi}\left(b\right)\coloneqq b+\frac{1}{N-1}\frac{\widehat{G}\left(b\right)}{\widehat{g}\left(b\right)},
\]
where we use the usual nonparametric estimators of $G$ and $g$:
\[
\widehat{G}\left(b\right)\coloneqq\frac{1}{N\cdot L}\sum_{i,l}\mathbbm{1}\left(B_{il}\leq b\right)\textrm{ and }\widehat{g}\left(b\right)\coloneqq\frac{1}{N\cdot L}\sum_{i,l}\frac{1}{h}K_{g}\left(\frac{B_{il}-b}{h}\right),
\]
where $\sum_{i,l}$ is understood as $\sum_{l=1}^{L}\sum_{i=1}^{N}$.
Let $\widehat{\overline{b}}\coloneqq\mathrm{max}\left\{ B_{il}:i=1,\ldots,N,\,l=1,...,L\right\} $,
and $\widehat{\underline{b}}\coloneqq\mathrm{min}\left\{ B_{il}:i=1,\ldots,N,\,l=1,...,L\right\} $.
The trimming factor is now simply $\mathbb{T}_{il}\coloneqq\mathbbm{1}\left(\widehat{\underline{b}}+h\leq B_{il}\leq\widehat{\overline{b}}-h\right)$.
The GPV estimator of $f\left(v\right)$ is now given by
\[
\widehat{f}_{GPV}\left(v\right)=\frac{1}{N\cdot L}\sum_{i,l}\mathbb{T}_{il}\frac{1}{h}K_{f}\left(\frac{\widehat{V}_{il}-v}{h}\right),
\]
where $K_{f}=K_{0}$ in this simplified framework.
We derive the following stochastic expansion of $\widehat{f}_{GPV}\left(v\right)$
around $f\left(v\right)$:
\begin{equation}
\widehat{f}_{GPV}\left(v\right)-f\left(v\right)=\frac{1}{\left(N\cdot L\right)^{2}}\sum_{i,l}\widetilde{\mathbb{T}}_{il}\frac{1}{h^{2}}K_{f}'\left(\frac{V_{il}-v}{h}\right)\left(\widehat{V}_{il}-V_{il}\right)+o_{p}\left(\left(Lh^{3}\right)^{-\nicefrac{1}{2}}\right),\label{eq:the first stochastic approximation result}
\end{equation}
where $\widetilde{\mathbb{T}}_{il}\coloneqq\mathbbm{1}\left(\left|V_{il}-v\right|\leq\overline{\delta}\right)$
is an infeasible trimming factor and the remainder term is uniform
in $v\in I$. In the above expression, the derivative $K_{f}'$ of
the kernel function appears due to the linearization of $K_{f}\left(\nicefrac{\left(\widehat{V}_{il}-v\right)}{h}\right)$
around $K_{f}\left(\nicefrac{\left(V_{il}-v\right)}{h}\right)$. The
result in (\ref{eq:the first stochastic approximation result}) shows
that the distribution of the GPV estimator depends not only on the
variation in $V_{il}$'s, but also on the estimation errors of pseudo
valuations. In other words, the errors from estimation of the inverse
bidding strategy affect the asymptotic distribution of the GPV estimator.
Since $\widehat{G}$ has a faster rate of convergence than $\widehat{g}$,
the discrepancy between $V_{il}$ and $\widehat{V}_{il}$ depends
on that between the true PDF $g\left(B_{il}\right)$ and the estimated
PDF $\widehat{g}\left(B_{il}\right)$, which in turn depends on the
averaged discrepancy between $K_{g}\left(\nicefrac{\left(B_{jm}-B_{il}\right)}{h}\right)$
and $g\left(B_{il}\right)$, where the averaging is across $B_{jm}$'s.
Lemma \ref{lem:lemma 2} establishes a further asymptotic expansion
for the GPV estimator:
\begin{equation}
\widehat{f}_{GPV}\left(v\right)-f\left(v\right)=\frac{1}{\left(N-1\right)}\frac{1}{\left(N\cdot L\right)^{2}}\sum_{i,l}\sum_{j,k}\mathcal{M}\left(B_{il},B_{jk};v\right)+o_{p}\left(\left(Lh^{3}\right)^{-\nicefrac{1}{2}}\right),\label{eq:f_GPV - f leading term}
\end{equation}
where the remainder term is uniform in $v\in I$, and
\begin{equation}
\mathcal{M}\left(b,b';v\right)\coloneqq-\frac{1}{h^{2}}K_{f}'\left(\frac{\xi\left(b\right)-v}{h}\right)\frac{G\left(b\right)}{g\left(b\right)^{2}}\left(\frac{1}{h}K_{g}\left(\frac{b'-b}{h}\right)-g\left(b\right)\right).\label{eq:V_stat kernel definition}
\end{equation}
For any fixed $v$, the leading term in (\ref{eq:f_GPV - f leading term})
is a V-statistic with a kernel that depends on the bandwidth $h$.
We now apply Hoeffding decomposition to this leading term. Define
\begin{eqnarray}
\mathcal{M}_{1}\left(b;v\right) & \coloneqq & \int\mathcal{M}\left(b,b';v\right)\mathrm{d}G\left(b'\right)\nonumber \\
& = & -\frac{1}{h^{2}}K_{f}'\left(\frac{\xi\left(b\right)-v}{h}\right)\frac{G\left(b\right)}{g\left(b\right)^{2}}\left(\mathrm{E}\left[\frac{1}{h}K_{g}\left(\frac{B_{11}-b}{h}\right)\right]-g(b)\right),\label{eq: M1 explained}
\end{eqnarray}
and further,
\[
\mathcal{M}_{2}\left(b;v\right)\coloneqq\int\mathcal{M}\left(b',b;v\right)\mathrm{d}G\left(b'\right),\textrm{ and }\mu_{\mathcal{M}}\left(v\right)\coloneqq\int\int\mathcal{M}\left(b,b';v\right)\mathrm{d}G\left(b\right)\mathrm{d}G\left(b'\right).
\]
Note that $\mu_{\mathcal{M}}\left(v\right)=\mathrm{E}\left[\mathcal{M}_{1}\left(B_{11};v\right)\right]=\mathrm{E}\left[\mathcal{M}_{2}\left(B_{11};v\right)\right]$.
The Hoeffding decomposition yields
\begin{align}
& \frac{1}{\left(N\cdot L\right)^{2}}\sum_{i,l}\sum_{j,k}\mathcal{M}\left(B_{il},B_{jk};v\right)\nonumber \\
= & \mu_{\mathcal{M}}\left(v\right)+\left\{ \frac{1}{N\cdot L}\sum_{i,l}\mathcal{M}_{1}\left(B_{il};v\right)-\mu_{\mathcal{M}}\left(v\right)\right\} +\left\{ \frac{1}{N\cdot L}\sum_{i,l}\mathcal{M}_{2}\left(B_{il};v\right)-\mu_{\mathcal{M}}\left(v\right)\right\} \nonumber \\
& +\frac{1}{\left(N\cdot L\right)\left(N\cdot L-1\right)}\sum_{\left(i,l\right)\neq\left(j,k\right)}\left\{ \mathcal{M}\left(B_{il},B_{jk};v\right)-\mathcal{M}_{1}\left(B_{il};v\right)-\mathcal{M}_{2}\left(B_{jk};v\right)+\mu_{\mathcal{M}}\left(v\right)\right\} \nonumber \\
& +\frac{1}{\left(N\cdot L\right)^{2}}\sum_{i,l}\mathcal{M}\left(B_{il},B_{il};v\right)-\frac{1}{\left(N\cdot L\right)^{2}\left(N\cdot L-1\right)}\sum_{\left(i,l\right)\neq\left(j,k\right)}\mathcal{M}\left(B_{il},B_{jk};v\right).\label{eq:M Hoeffding decomposition}
\end{align}
In the proof of Theorem \ref{thm:Asymptotic Normality GPV} below,
we use results for empirical processes and U-processes to show that
the terms in the third and fourth lines of (\ref{eq:M Hoeffding decomposition})
and $\left(N\cdot L\right)^{-1}\sum_{i,l}\mathcal{M}_{1}\left(B_{il};v\right)$
are asymptotically negligible uniformly in $v\in I$. As is apparent
from the definition of $\mathcal{M}_{1}$ in (\ref{eq: M1 explained}),
the contribution of the $\mathcal{M}_{1}\left(b;v\right)$ terms is
negligible because they depend on the difference between the expectation
$\mathrm{E}\left[h^{-1}K_{g}\left(\nicefrac{\left(B_{il}-b\right)}{h}\right)\right]$
and $g\left(b\right)$, i.e., the bias of the kernel density estimator,
which is of order $O\left(h^{1+R}\right)$. Thus, the asymptotic distribution
of the GPV estimator is driven solely by
\begin{equation}
\frac{1}{(N-1)}\frac{1}{N\cdot L}\sum_{i,l}\left(\mathcal{M}_{2}\left(B_{il};v\right)-\mu_{\mathcal{M}}\left(v\right)\right),\label{eq:leading term of the stochastic expansion identical auctions}
\end{equation}
where $\mathcal{M}_{2}\left(B_{il};v\right)-\mu_{\mathcal{M}}\left(v\right)$,
$i=1,...,N$, $l=1,...,L$ are independent, zero-mean and depend on
the bandwidth.
We also show in the proof of Theorem \ref{thm:Asymptotic Normality GPV}
that the rescaled variance of (\ref{eq:leading term of the stochastic expansion identical auctions})
satisfies
\begin{align}
& \mathrm{E}\left[\frac{Lh^{3}}{\left(N-1\right)^{2}}\left(\frac{1}{N\cdot L}\sum_{i,l}\left(\mathcal{M}_{2}\left(B_{il};v\right)-\mu_{\mathcal{M}}\left(v\right)\right)\right)^{2}\right]\nonumber \\
= & \frac{1}{N\left(N-1\right)^{2}h^{3}}\int\left\{ \int K_{f}'\left(\frac{\xi\left(b'\right)-v}{h}\right)\frac{G\left(b'\right)}{g\left(b'\right)^{2}}K_{g}\left(\frac{b-b'}{h}\right)\mathrm{d}G\left(b'\right)\right\} ^{2}\mathrm{d}G\left(b\right)+O\left(h^{3}\right)\nonumber \\
\eqqcolon & \mathrm{V}_{\mathcal{M}}\left(v\right)+O\left(h^{3}\right),\label{eq:E_m2^2 expansion}
\end{align}
where the remainder term is uniform in $v\in I$. Thus, the asymptotic
variance of the GPV estimator is the limit of the leading term in
(\ref{eq:E_m2^2 expansion}) as $h\downarrow0$. Note that
\begin{multline}
\lim_{h\downarrow0}\frac{1}{h^{3}}\int\left\{ \int K_{f}'\left(\frac{\xi\left(b'\right)-v}{h}\right)\frac{G\left(b'\right)}{g\left(b'\right)^{2}}K_{g}\left(\frac{b-b'}{h}\right)\mathrm{d}G\left(b'\right)\right\} ^{2}\mathrm{d}G\left(b\right)\\
=\frac{G(s(v))^{2}(s'(v))^{2}}{g(s(v))}\int\left\{ \int K_{f}'\left(u\right)K_{g}\left(w-s'\left(v\right)u\right)\mathrm{d}u\right\} ^{2}\mathrm{d}w.\label{eq:variance limit simple}
\end{multline}
We have the following result.
\begin{thm}[\textbf{Asymptotic Normality}]
\label{thm:Asymptotic Normality GPV}Suppose Assumptions \ref{assu:DGP}
- \ref{assu: rate of bandwidth} hold. Then for any interior point
$v\in\left(\underline{v},\overline{v}\right)$, $\left(Lh^{3}\right)^{\nicefrac{1}{2}}\left(\widehat{f}_{GPV}\left(v\right)-f\left(v\right)\right)\rightarrow_{d}\mathrm{N}\left(0,\mathrm{V}_{GPV}\left(v\right)\right),$
where
\begin{equation}
\mathrm{V}_{GPV}\left(v\right)\coloneqq\frac{1}{N\left(N-1\right)^{2}}\frac{F\left(v\right)^{2}f\left(v\right)^{2}}{g\left(s\left(v\right)\right)^{3}}\int\left\{ \int K_{f}'\left(u\right)K_{g}\left(w-s'\left(v\right)u\right)\mathrm{d}u\right\} ^{2}\mathrm{d}w.\label{eq:asymptotic covariance}
\end{equation}
\end{thm}
\begin{rembold}The asymptotic variance $\mathrm{V}_{GPV}\left(v\right)$
has an interesting and non-standard feature. Typically, the asymptotic
variance of a kernel estimator involves a constant which is a simple
functional of the kernel and does not involve any elements of the
DGP. However, in the case of the GPV estimator, the constant has a
convolution form. Moreover, this constant also involves the derivative
of the unknown bidding function. As can be seen from (\ref{eq:the first stochastic approximation result}),
the convolution form is due to the fact that the variance of the GPV
estimator is determined not only by the variation of $V_{il}$, but
also by the estimation errors $\widehat{V}_{il}-V_{il}$. The $s'(v)$
term appears due to averaging of $\xi(b)$'s in a small neighborhood
of $v$, as one can see from (\ref{eq:E_m2^2 expansion}).
The presence of the $s'(v)$ term inside the integral in (\ref{eq:asymptotic covariance})
can cause additional complications in estimation of the asymptotic
variance $\mathrm{V}_{GPV}(v)$, as the econometrician now also has
to estimate the functional of the kernel function. Potentially, one
could estimate $s'(v)$ and then estimate the convolution by the plug-in
approach. However, we show in the next section that the asymptotic
variance can be estimated directly without separate estimation of
$s'(v)$ and the convolution by using the sample analogue of the expression
in the second line of (\ref{eq:E_m2^2 expansion}). \end{rembold}
\begin{rembold}[Extensions]Theorem \ref{thm:Asymptotic Normality GPV}
also implies asymptotic normality of some modified GPV estimators.
\citet{henderson_2012_JOE} proposed to modify the standard GPV approach
by estimating $\xi$ under a monotonicity constraint implied by the
structural model. However, if the auction model is correctly specified,
the unconstrained estimator of $\xi$ will be monotone with probability
approaching one. Hence, we expect the standard GPV estimator and the
estimator proposed in \citet{henderson_2012_JOE} to be first-order
asymptotically equivalent.
\citet{hickman_hubbard_2014_JAE} proposed another modified GPV estimator
by replacing sample trimming used in the standard GPV procedure with
a boundary correction. Their method uses a boundary-bias-corrected
kernel estimator in estimation of the density of bids. The corrected
estimator is uniformly consistent over the entire support of the distribution
of bids. When estimating the PDF of valuations away from the boundaries,
our proof of (\ref{eq:f_GPV - f leading term}) can be adapted to
show that the same V-statistic approximation holds for the estimator
in \citet{hickman_hubbard_2014_JAE}. Hence, their estimator is first-order
asymptotically equivalent to the standard GPV estimator, when $v$
is chosen away from the boundaries. \end{rembold}
\begin{rembold}[Asymptotic Bias]In the proof of Theorem \ref{thm:Asymptotic Normality GPV},
we incorporate the bias term:
\begin{multline}
\left(Lh^{3}\right)^{\nicefrac{1}{2}}\left(\widehat{f}_{GPV}\left(v\right)-f\left(v\right)-\frac{1}{R!}f^{\left(R\right)}\left(v\right)\left(\int K_{f}\left(u\right)u^{R}\mathrm{d}u\right)h^{R}+o\left(h^{R}\right)\right)\\
\rightarrow_{d}\mathrm{N}\left(0,\mathrm{V}_{GPV}\left(v\right)\right).\label{eq:asymptotic normality with bias term}
\end{multline}
The leading bias term of the GPV estimator is the same as that of
the infeasible estimator constructed using the unobserved true valuations.
This is due to the following feature of the first-price auction model:
at interior points, the bid density has $1+R$ continuous derivatives
rather than $R$. It is natural to incorporate this structural feature
in the estimation procedure by using a higher-order kernel in the
first stage. \end{rembold}
\begin{rembold}[Comparison with the Quantile-Based Estimator]\label{rem:comparison QB}The
quantile-based (QB) estimator of \citet{Marmer_Shneyerov_Quantile_Auctions}
does not require estimation of latent valuations, and instead relies
on a direct representation of the PDF of valuations using the distribution
functions of observable bids. While the two estimators have the same
rate of convergence, the limiting distribution of the GPV estimator
shows that it indeed improves on the QB estimator in the following
sense. One can show that the GPV estimator has a smaller asymptotic
variance than that of the quantile-based estimator, as long as the
two estimators use the same second-order kernel functions. Suppose
now $K_{f}=K_{g}=K$, for some second-order kernel function $K$.
Let $\mathrm{V}_{QB}\left(v\right)$ be the asymptotic variance of
the QB estimator in the simplest setting without auction-specific
heterogeneity.\footnote{See \citet[Theorem 2]{Marmer_Shneyerov_Quantile_Auctions} for the
expression of the asymptotic variance of the quantile-based estimator.} By Jensen's inequality and standard calculus techniques, it can be
easily shown that $\nicefrac{\mathrm{V}_{QB}\left(v\right)}{\mathrm{V}_{GPV}\left(v\right)}\geq1$.
The details of the proof can be found in the Supplement.
Consider the following family of PDF's: $f_{\theta}(v)=\theta v^{\theta-1}\cdot\mathbbm{1}\left(0\leq v\leq1\right)$
for some $\theta>0$. The corresponding BNE bidding strategy is $s(v)=\left(1-\left(\theta(N-1)+1\right)^{-1}\right)v$.
Note that in this case $s'(v)$ is constant, and we can compute the
ratio $\nicefrac{\mathrm{V}_{QB}\left(v\right)}{\mathrm{V}_{GPV}\left(v\right)}$
analytically. In the case of the triweight kernel $K$, ratio is found
to be, e.g., 1.3259 when $\left(\theta,N\right)=\left(1,2\right)$
and 2.3038 when $\left(\theta,N\right)=\left(2,7\right)$. Thus, depending
on the model, the GPV estimator could be substantially more precise
than the QB estimator.\footnote{We believe that this finding is of interest in a more general context
outside of the empirical auctions literature. It illustrates that
two-step nonparametric estimators can outperform more direct estimators
that avoid first-stage estimation of latent variables.}\end{rembold}
\section{Pointwise Confidence Intervals\label{sec:Pointwise-Confidence-Intervals}}
If the asymptotic variance $\mathrm{V}_{GPV}(v)$ can be consistently
estimated by some estimator $\widehat{\mathrm{V}}_{GPV}\left(v\right)$,
one can construct an asymptotically valid pointwise confidence interval
for $f\left(v\right)$ as
\begin{equation}
CI^{\dagger}\left(v\right)\coloneqq\left[\widehat{f}_{GPV}\left(v\right)-z_{1-\nicefrac{\alpha}{2}}\sqrt{\frac{\widehat{\mathrm{V}}_{GPV}\left(v\right)}{Lh^{3}}},\widehat{f}_{GPV}\left(v\right)+z_{1-\nicefrac{\alpha}{2}}\sqrt{\frac{\widehat{\mathrm{V}}_{GPV}\left(v\right)}{Lh^{3}}}\right],\label{eq:CI_n definitions}
\end{equation}
where $z_{1-\nicefrac{\alpha}{2}}$ denotes the $1-\nicefrac{\alpha}{2}$
quantile of the standard normal distribution.
While the formula for the asymptotic variance in (\ref{eq:asymptotic covariance})
can be used for plug-in estimation of $\mathrm{V}_{GPV}(v)$, such
an estimator would be difficult to implement in practice. Firstly,
it would require estimating the bidding strategy and its derivative.
Secondly, even with an estimate of $s'\left(v\right)$, it is not
always easy to compute analytically the double integral in the definition
of $\mathrm{V}_{GPV}\left(v\right)$. This issue becomes even more
severe when there is auction-specific heterogeneity, as we discuss
in Section \ref{sec:Auction-Specific-Heterogeneity}. In that case,
one would need to evaluate a multidimensional integral.
To avoid those issues, we propose an alternative approach to estimation
of the asymptotic variance. As we discuss in the previous section,
the asymptotic variance of the GPV estimator is the limit of the expression
in (\ref{eq:E_m2^2 expansion}). The leading term on the right-hand
side of (\ref{eq:E_m2^2 expansion}) can be estimated using a U-type-statistic,
while replacing the unknown $G$, $g$, and $\xi$ with $\widehat{G}$,
$\widehat{g}$, and $\widehat{\xi}$ respectively. The resulting estimator
is given by
\begin{eqnarray}
\widehat{\mathrm{V}}_{GPV}\left(v\right) & \coloneqq & \frac{1}{N\left(N-1\right)^{2}h^{3}}\frac{1}{\left(N\cdot L\right)\left(N\cdot L-1\right)\left(N\cdot L-2\right)}\nonumber \\
& & \times\sum_{i,l}\sum_{\left(j,k\right)\neq\left(i,l\right)}\sum_{\left(j',k'\right)\neq\left(i,l\right),\,\left(j',k'\right)\neq\left(j,k\right)}\eta_{il,jk}(v)\eta_{il,j'k'}(v),\text{ where}\nonumber \\
\eta_{il,jk}(v) & \coloneqq & \mathbb{T}_{jk}K_{f}'\left(\frac{\widehat{V}_{jk}-v}{h}\right)\frac{\widehat{G}\left(B_{jk}\right)}{\widehat{g}\left(B_{jk}\right)^{2}}K_{g}\left(\frac{B_{il}-B_{jk}}{h}\right).\label{eq:definition estimator of asymptotic variance}
\end{eqnarray}
The estimator avoids estimation of the bidding strategy and its derivative
and evaluation of multidimensional integrals. It is very easily implementable
in practice since it depends only on the bids, $\widehat{G}$, $\widehat{g}$,
and the pseudo valuations. The next theorem shows consistency of the
proposed estimator and provides an estimate of its uniform convergence
rate.
\begin{thm}[\textbf{Variance Estimation}]
\label{thm:variance estimator}Suppose Assumptions \ref{assu:DGP}
- \ref{assu: rate of bandwidth} hold. Then,
\[
\underset{v\in I}{\mathrm{sup}}\left|\widehat{\mathrm{V}}_{GPV}\left(v\right)-\mathrm{V}_{\mathcal{M}}\left(v\right)\right|=O_{p}\left(\left(\frac{\mathrm{log}\left(L\right)}{Lh^{3}}\right)^{\nicefrac{1}{2}}+h^{R}\right).
\]
\end{thm}
An alternative to the confidence interval (\ref{eq:CI_n definitions})
is the bootstrap. We show below that the bootstrap approximation to
the distribution of $S\left(v\right)\coloneqq\left(Lh^{3}\right)^{\nicefrac{1}{2}}\left(\widehat{f}_{GPV}\left(v\right)-f\left(v\right)\right)$
is asymptotically valid. Our focus is on the percentile bootstrap
as it does not require estimation of the asymptotic variance, which
makes it fairly popular among practitioners.
Let $\left\{ B_{il}^{*}:i=1,\ldots,N,l=1,\ldots,L\right\} $ denote
the bootstrap sample, i.e., a set of independent random variables
drawn from the distribution $\widehat{G}$ conditionally on the original
sample of bids. Let $\widehat{G}^{*}$ and $\widehat{g}^{*}$ denote
the bootstrap analogues of $\widehat{G}$ and $\widehat{g}$ respectively:
they are constructed by following exactly the same procedure as that
for constructing $\widehat{G}$ and $\widehat{g}$, however using
the (empirical) bootstrap sample instead of the original sample. Let
$\widehat{\xi}^{*}$ be the bootstrap analogue of $\widehat{\xi}$
defined using $\widehat{G}^{*}$ and $\widehat{g}^{*}$ in place of
$\widehat{G}$ and $\widehat{g}$. We generate bootstrap samples of
pseudo values as $\widehat{V}_{il}^{*}\coloneqq\widehat{\xi}^{*}\left(B_{il}^{*}\right)$.
Lastly, we construct a bootstrap analogue of $\widehat{f}_{GPV}\left(v\right)$:
\[
\widehat{f}_{GPV}^{*}\left(v\right)\coloneqq\frac{1}{N\cdot L}\sum_{i,l}\mathbb{T}_{il}^{*}\frac{1}{h}K_{f}\left(\frac{\widehat{V}_{il}^{*}-v}{h}\right),
\]
where $\mathbb{T}_{il}^{*}\coloneqq\mathbbm{1}\left(\widehat{\underline{b}}+h\leq B_{il}^{*}\leq\widehat{\overline{b}}-h\right)$.
Let $q_{\tau}^{*}(v)$ be the $\tau$-th quantile of the conditional
distribution of $\widehat{f}_{GPV}^{*}\left(v\right)$ given the original
sample. The percentile bootstrap confidence interval is
\[
CI^{*}\left(v\right)\coloneqq\left[q_{\nicefrac{\alpha}{2}}^{*}(v),\,q_{1-\nicefrac{\alpha}{2}}^{*}(v)\right]=\left[\widehat{f}_{GPV}\left(v\right)+\frac{s_{\nicefrac{\alpha}{2}}^{*}(v)}{\sqrt{Lh^{3}}},\,\widehat{f}_{GPV}\left(v\right)+\frac{s_{\nicefrac{1-\alpha}{2}}^{*}(v)}{\sqrt{Lh^{3}}}\right],
\]
where $s_{\tau}^{*}(v)$ is the $\tau-$th quantile of the conditional
distribution of
\[
S^{*}\left(v\right)\coloneqq\left(Lh^{3}\right)^{\nicefrac{1}{2}}\left(\widehat{f}_{GPV}^{*}\left(v\right)-\widehat{f}_{GPV}\left(v\right)\right)
\]
given the original sample. The conditional distributions of the bootstrap
statistics $\widehat{f}_{GPV}^{*}\left(v\right)$ and $S^{*}\left(v\right)$
given the original sample can be easily approximated by Monte Carlo
methods. We show below that the bootstrap estimator of the finite-sample
distribution of $S\left(v\right)$ is consistent. Let $\mathrm{P}^{*}\left[\cdot\right]$
denote the conditional probability given the original sample of bids.
\begin{thm}[\textbf{Bootstrap Consistency}]
\label{thm: bootstrap consistency}Suppose Assumptions \ref{assu:DGP}
- \ref{assu: rate of bandwidth} hold. Then for any interior point
$v\in\left(\underline{v},\overline{v}\right)$,
\[
\underset{z\in\mathbb{R}}{\mathrm{sup}}\left|\mathrm{P}^{*}\left[S^{*}\left(v\right)\leq z\right]-\mathrm{P}\left[S\left(v\right)\leq z\right]\right|\rightarrow_{p}0,\,\textrm{as \ensuremath{L\uparrow\infty}}.
\]
\end{thm}
\begin{rembold}\label{rem: bootstrap validity pointwise confidence interval}Theorems
\ref{thm:Asymptotic Normality GPV}, \ref{thm: bootstrap consistency}
and P\'{o}lya's theorem yield
\[
\underset{z\in\mathbb{R}}{\mathrm{sup}}\left|\mathrm{P}^{*}\left[S^{*}\left(v\right)\leq z\right]-\mathrm{P}\left[\mathrm{N}\left(0,\mathrm{V}_{GPV}\left(v\right)\right)\leq z\right]\right|\rightarrow_{p}0,\,\textrm{as \ensuremath{L\uparrow\infty}},
\]
for each $v\in\left(\underline{v},\overline{v}\right)$. The above
result and standard arguments (see, e.g., \citealp[Lemma 23.3]{VanDerVaart_Asymptotic_Statistics_Book})
yield the asymptotic validity (consistency) of the percentile bootstrap
confidence interval $CI^{*}\left(v\right)$, i.e., $\mathrm{P}\left[f\left(v\right)\in CI^{*}\left(v\right)\right]\rightarrow1-\alpha$
as $\textrm{\ensuremath{L\uparrow\infty}}.$\end{rembold}
\begin{rembold}One can also studentize $S\left(v\right)$ using our
estimator of the asymptotic variance:
\begin{equation}
Z\left(v\right)\coloneqq\frac{\widehat{f}_{GPV}\left(v\right)-f\left(v\right)}{\left(Lh^{3}\right)^{-\nicefrac{1}{2}}\widehat{\mathrm{V}}_{GPV}\left(v\right)^{\nicefrac{1}{2}}}.\label{eq:Z definition}
\end{equation}
Since $\widehat{\mathrm{V}}_{GPV}\left(v\right)$ is consistent for
the asymptotic variance, $Z\left(v\right)$ is asymptotically distributed
as a standard normal random variable. Let $\widehat{\mathrm{V}}_{GPV}^{*}\left(v\right)$
be the bootstrap analogue of $\widehat{\mathrm{V}}_{GPV}\left(v\right)$.
The bootstrap analogue of $Z\left(v\right)$ is $Z^{**}\left(v\right)\coloneqq\nicefrac{S^{*}\left(v\right)}{\sqrt{\widehat{\mathrm{V}}_{GPV}^{*}\left(v\right)}}$.
Let $z_{\tau}^{**}(v)$ be the $\tau-$th quantile of the conditional
distribution of $Z^{**}(v)$ given the original sample. The ``bootstrap-$t$''
(or studentized bootstrap) confidence intervals can be obtained by
replacing the critical value $z_{1-\nicefrac{\alpha}{2}}$ in $CI^{\dagger}\left(v\right)$
with their bootstrap counterpart $z_{\tau}^{**}(v)$. The asymptotic
validity of this alternative bootstrap confidence interval easily
follows as a corollary to Theorem \ref{thm: bootstrap consistency}.\end{rembold}
\section{Uniform Confidence Bands\label{sec:Uniform-Confidence-Bands}}
Consider the stochastic process
\begin{equation}
\mathit{\Gamma}\left(v\right)\coloneqq\frac{1}{N^{\nicefrac{1}{2}}\left(N-1\right)}\frac{1}{\left(N\cdot L\right)^{\nicefrac{1}{2}}}\sum_{i,l}\frac{\mathcal{M}_{2}\left(B_{il};v\right)-\mu_{\mathcal{M}}\left(v\right)}{\mathrm{Var}\left[N^{-\nicefrac{1}{2}}\left(N-1\right)^{-1}\mathcal{M}_{2}\left(B_{11};v\right)\right]^{\nicefrac{1}{2}}},\,v\in I.\label{eq:GAMMA defnition}
\end{equation}
Note that $\mathrm{E}\left[\mathit{\Gamma}\left(v\right)\right]=0$
and $\mathrm{E}\left[\mathit{\Gamma}\left(v\right)^{2}\right]=1$
for all $v\in I$.
The following theorem shows that (a version of) the centered Gaussian
process with index set $I$ and covariance function $\mathrm{E}\left[\mathit{\Gamma}\left(v\right)\mathit{\Gamma}\left(v'\right)\right]$,
for $\left(v,v'\right)\in I^{2}$, is a tight random element in $\ell^{\infty}\left(I\right)$.
This Gaussian process, denoted by $\left\{ \mathit{\Gamma}_{G}\left(v\right):v\in I\right\} $,
is the \emph{intermediate} Gaussian process. The tightness of $\varGamma_{G}$
as a random element in $\ell^{\infty}\left(I\right)$ can be established
using standard results (see, e.g., \citealp[Lemma 2.1]{chernozhukov2014gaussian}).
The following theorem also shows that one can approximate the distribution
of the sup-norm $\left\Vert Z\right\Vert _{I}=\underset{v\in I}{\mathrm{sup}}\left|Z\left(v\right)\right|$
with that of $\mathit{\Gamma}_{G}$. The result follows from uniform
approximations of (\ref{eq:leading term of the stochastic expansion identical auctions})
and $\widehat{\mathrm{V}}_{GPV}\left(v\right)$ and uses the coupling
theorem for suprema of empirical processes of \citet{chernozhukov2014gaussian}
and the Gaussian anti-concentration inequality of \citet{chernozhukov2014anti}.
\begin{thm}
\label{thm:Gaussian coupling}Suppose Assumptions \ref{assu:DGP}
- \ref{assu: rate of bandwidth} hold. Then there exists a tight Gaussian
random element $\varGamma_{G}$ in $\ell^{\infty}\left(I\right)$
that has mean zero and the same covariance structure as that of $\mathit{\Gamma}$.
Moreover,
\[
\underset{z\in\mathbb{R}}{\mathrm{sup}}\left|\mathrm{P}\left[\left\Vert Z\right\Vert _{I}\leq z\right]-\mathrm{P}\left[\left\Vert \mathit{\Gamma}_{G}\right\Vert _{I}\leq z\right]\right|\rightarrow0,\,\textrm{as \ensuremath{L\uparrow\infty}}.
\]
\end{thm}
The next result shows that the distribution of the sup-norm of the
(empirical) bootstrap process
\begin{equation}
Z^{*}\left(v\right)\coloneqq\frac{\widehat{f}_{GPV}^{*}\left(v\right)-\widehat{f}_{GPV}\left(v\right)}{\left(Lh^{3}\right)^{-\nicefrac{1}{2}}\widehat{\mathrm{V}}_{GPV}\left(v\right)^{\nicefrac{1}{2}}},\,v\in I\label{eq:Z_star process}
\end{equation}
can be similarly approximated by that of $\mathit{\Gamma}_{G}$.
\begin{thm}
\label{thm:Gaussian coupling for empirical bootstrap process}Suppose
Assumptions \ref{assu:DGP} - \ref{assu: rate of bandwidth} hold.
Then,
\[
\underset{z\in\mathbb{R}}{\mathrm{sup}}\left|\mathrm{P}^{*}\left[\left\Vert Z^{*}\right\Vert _{I}\leq z\right]-\mathrm{P}\left[\left\Vert \mathit{\Gamma}_{G}\right\Vert _{I}\leq z\right]\right|\rightarrow_{p}0,\,\textrm{as \ensuremath{L\uparrow\infty}}.
\]
\end{thm}
Since the distributions of the suprema of (the absolute values of)
$\left\{ Z\left(v\right):v\in I\right\} $ and that of $\left\{ Z^{*}\left(v\right):v\in I\right\} $
are both well approximated by that of $\left\{ \varGamma_{G}\left(v\right):v\in I\right\} $,
one can use the bootstrap critical values based on $\left\Vert Z^{*}\right\Vert _{I}$
for construction of uniform confidence bands. Let
\begin{equation}
\zeta_{L,\alpha}^{*}\coloneqq\mathrm{inf}\left\{ z\in\mathbb{R}:\mathrm{P}^{*}\left[\left\Vert Z^{*}\right\Vert _{I}\leq z\right]\geq1-\alpha\right\} \label{eq:zeta_L^* definition}
\end{equation}
be the $\left(1-\alpha\right)$-quantile of the conditional distribution
of $\left\Vert Z^{*}\right\Vert _{I}$ given the original sample.
The uniform confidence band is given by
\[
CB^{*}\left(v\right)\coloneqq\left[\widehat{f}_{GPV}\left(v\right)-\zeta_{L,\alpha}^{*}\sqrt{\frac{\widehat{\mathrm{V}}_{GPV}\left(v\right)}{Lh^{3}}},\,\widehat{f}_{GPV}\left(v\right)+\zeta_{L,\alpha}^{*}\sqrt{\frac{\widehat{\mathrm{V}}_{GPV}\left(v\right)}{Lh^{3}}}\right],\,\textrm{for \ensuremath{v\in I}}.
\]
The following corollary establishes its asymptotic validity and provides
an estimate of the order of the bootstrap critical value $\zeta_{L,\alpha}^{*}$.\footnote{It also implies that the (supremum) width of the band $CB^{*}$ is
of order $O_{p}\left(\mathrm{log}\left(h^{-1}\right)^{\nicefrac{1}{2}}\left(Lh^{3}\right)^{-\nicefrac{1}{2}}\right)$.}
\begin{cor}[\textbf{Validity of Bootstrap Confidence Band}]
\label{cor:consistency of IGA bootstrap}Suppose Assumptions \ref{assu:DGP}
- \ref{assu: rate of bandwidth} hold. Then, $\mathrm{P}\left[f\left(v\right)\in CB^{*}\left(v\right),\,\textrm{for all \ensuremath{v\in I}}\right]\rightarrow1-\alpha$
as $L\uparrow\infty$. Moreover, $\zeta_{L,\alpha}^{*}=O_{p}\left(\mathrm{log}\left(h^{-1}\right)^{\nicefrac{1}{2}}\right)$.
\end{cor}
\begin{rembold}[Limiting Distribution of Uniform Error]Since the
seminal work of \citet{bickel1973some}, it has been found that in
many cases a suitable normalization of the supremum of a studentized
absolute difference between a nonparametric curve and its kernel-based
estimator converges in distribution to standard Gumbel distribution.
Asymptotically valid uniform confidence bands can be based on the
Gumbel approximation. We do not pursue such an approach in this paper,
since it is known that the accuracy of such approximation is poor.
See \citet[Section 2.7]{gine2015mathematical} and \citet{chernozhukov2014anti}
for discussion. On the other hand, for the auction model one can show
that (a suitable normalization of) $\left\Vert Z\right\Vert _{I}$
converges in distribution to standard Gumbel distribution when the
true bidding strategy is linear. Derivation of the limiting distribution
for the general case is interesting but beyond the scope of this paper.
See the Supplement for more discussion.\end{rembold}
\begin{rembold}Taking the IGA approach, we establish consistency
of bootstrap uniform confidence bands by showing that the distributions
of both $\left\Vert Z\right\Vert _{I}$ and its empirical bootstrap
counterpart can be approximated by the distribution of $\left\Vert \mathit{\Gamma}_{G}\right\Vert _{I}$.
The proof hinges on using the coupling theorems of \citet{chernozhukov2014gaussian,chernozhukov2016empirical}
and the Gaussian anti-concentration inequality of \citet{chernozhukov2014anti}.
In the literature, the multiplier bootstrap is used when the IGA approach
is taken to construct confidence bands for nonparametric curves. See,
e.g., \citet{chernozhukov2014anti} and \citet{Kato_Sasaki_2,Kato_Sasaki_1}.
Here, we chose the empirical bootstrap since it is practically convenient
given the two-step nature of the GPV estimator.\end{rembold}
\section{Auction-Specific Heterogeneity\label{sec:Auction-Specific-Heterogeneity}}
\subsection{Asymptotic Normality and Estimation of the Asymptotic Variance}
We now turn to the general model with auction-specific heterogeneity
and a random number of bidders. Firstly, we establish the asymptotic
normality of the GPV estimator by following the same approach and
steps as in the case of the simplified model in Section \ref{sec:Asymptotic Normality}.
While handling the general case is complicated by much heavier notations,
all the results provided in this section can be viewed as straightforward
generalizations of the results in Sections \ref{sec:Asymptotic Normality}-\ref{sec:Uniform-Confidence-Bands}.
The proofs of the results for the general case can be found in the
Supplement.
In comparison with the simplified model, one of the main differences
is in the form of the asymptotic variance. Recall that in the simplified
case, the asymptotic variance of the GPV estimator depends on the
derivative of the bidding strategy. As we show below, when there is
auction-specific heterogeneity, the asymptotic variance also involves
the partial derivatives of the bidding strategy with respect to the
auction-specific characteristics. This is in addition to the partial
derivative with respect to the valuation.
For some fixed $\boldsymbol{x}$ which is an interior point of $\mathcal{X}$,
let $I\left(\boldsymbol{x}\right)\coloneqq\left[v_{l}\left(\boldsymbol{x}\right),v_{u}\left(\boldsymbol{x}\right)\right]$
be an inner closed sub-interval of $\left[\underline{v}\left(\boldsymbol{x}\right),\overline{v}\left(\boldsymbol{x}\right)\right]$.
The fact that the conditional density of the valuations given $\boldsymbol{X}=\boldsymbol{x}$
and $N=n$ is $f\left(\cdot|\boldsymbol{x}\right)$ under Assumption
\ref{assu:DGP} motivates the following two-step estimator of $f\left(v|\boldsymbol{x}\right)$:
\[
\widehat{f}_{GPV}\left(v|\boldsymbol{x},n\right)\coloneqq\frac{1}{\widehat{\pi}\left(n|\boldsymbol{x}\right)\widehat{\varphi}\left(\boldsymbol{x}\right)L}\sum_{l=1}^{L}\mathbbm{1}\left(N_{l}=n\right)\frac{1}{N_{l}}\sum_{i=1}^{N_{l}}\mathbb{T}_{il}\frac{1}{h^{1+d}}K_{f}\left(\frac{\widehat{V}_{il}-v}{h},\frac{\boldsymbol{X}_{l}-\boldsymbol{x}}{h}\right).
\]
Note that the above estimator only uses data from auctions with $N_{l}=n$.
Since the PDF of valuations does not depend on the number of bidders,
an estimator for $f(v|\boldsymbol{x})$ can be constructed as a weighted
average of $\left\{ \widehat{f}_{GPV}\left(v|\boldsymbol{x},n\right):n\in\mathcal{N}\right\} $.
E.g., GPV suggested using estimates of the conditional probabilities
of drawing $N_{l}=n$ as the weights:
\begin{equation}
\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)=\sum_{n\in\mathcal{N}}\widehat{\pi}\left(n|\boldsymbol{x}\right)\widehat{f}_{GPV}\left(v|\boldsymbol{x},n\right).\label{eq:f_hat_GPV heterogeneity}
\end{equation}
Note that this gives an expression that is the same as the right hand
side of (\ref{eq:f_hat_GPV definition}).
By repeating the steps from Section \ref{sec:Asymptotic Normality},
one can show that the following analogue of the linearization results
in (\ref{eq:f_GPV - f leading term}) holds for the general model:
\begin{multline}
\widehat{f}_{GPV}\left(v|\boldsymbol{x},n\right)-f\left(v|\boldsymbol{x}\right)\\
=\frac{1}{\widehat{\pi}\left(n|\boldsymbol{x}\right)\widehat{\varphi}\left(\boldsymbol{x}\right)L^{2}}\sum_{l=1}^{L}\sum_{k=1}^{L}\mathcal{M}^{n}\left(\left(\boldsymbol{B}_{\cdot l},\boldsymbol{X}_{l},N_{l}\right),\left(\boldsymbol{B}_{\cdot k},\boldsymbol{X}_{k},N_{k}\right);v\right)+o_{p}\left(\left(Lh^{3+d}\right)^{-\nicefrac{1}{2}}\right),\label{eq:auction heterogeneity V statistic approximation}
\end{multline}
where $\boldsymbol{B}_{\cdot l}\coloneqq\left(B_{1l},...,B_{N_{l}l}\right)$,
and the remainder term is uniform in $v\in I\left(\boldsymbol{x}\right)$.
For $\boldsymbol{b}_{\cdot}\coloneqq(b_{1},\ldots,b_{m})$ , the kernel
function $\mathcal{M}^{n}$ is given by:
\begin{eqnarray}
& & \mathcal{M}^{n}\left(\left(\boldsymbol{b}_{\cdot},\boldsymbol{z},m\right),\left(\boldsymbol{b}_{\cdot}',\boldsymbol{z}',m'\right);v\right)\nonumber \\
& \coloneqq & -\mathbbm{1}\left(m=n\right)\frac{1}{m}\sum_{i=1}^{m}\frac{1}{h^{2+d}}K_{f}'\left(\frac{\xi\left(b_{i},\boldsymbol{z},m\right)-v}{h},\frac{\boldsymbol{z}-\boldsymbol{x}}{h}\right)\frac{G\left(b_{i},\boldsymbol{z},m\right)}{\left(m-1\right)g\left(b_{i},\boldsymbol{z},m\right)^{2}}\nonumber \\
& & \times\left(\mathbbm{1}\left(m'=m\right)\frac{1}{m'}\sum_{j=1}^{m'}\frac{1}{h^{1+d}}K_{g}\left(\frac{b_{j}'-b_{i}}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{z}'-\boldsymbol{z}}{h}\right)-g\left(b_{i},\boldsymbol{z},m\right)\right),\label{eq:auction heterogeneity V statistic kernel}
\end{eqnarray}
where $K_{f}'\left(\cdot,\cdot\right)$ denotes the partial derivative
function of $K_{f}$ with respect to its first argument,
\[
G\left(b,\boldsymbol{z},m\right)\coloneqq G\left(b|\boldsymbol{z},m\right)\pi\left(m|\boldsymbol{z}\right)\varphi\left(\boldsymbol{z}\right)\text{ and }g\left(b,\boldsymbol{z},m\right)\coloneqq g\left(b|\boldsymbol{z},m\right)\pi\left(m|\boldsymbol{z}\right)\varphi\left(\boldsymbol{z}\right).
\]
Note that the leading term on the right-hand side of (\ref{eq:auction heterogeneity V statistic approximation})
involves a V-statistic (with a kernel that depends on the bandwidth)
and, therefore, can be analyzed using the Hoeffding decomposition.
Thus, (\ref{eq:M Hoeffding decomposition}) can be generalized as
\begin{align*}
& \frac{1}{L^{2}}\sum_{l=1}^{L}\sum_{m=1}^{L}\mathcal{M}^{n}\left(\left(\boldsymbol{B}_{\cdot l},\boldsymbol{X}_{l},N_{l}\right),\left(\boldsymbol{B}_{\cdot m},\boldsymbol{X}_{m},N_{m}\right);v\right)\\
= & \mu_{\mathcal{M}^{n}}\left(v\right)+\left\{ \frac{1}{L}\sum_{l=1}^{L}\left(\mathcal{M}_{1}^{n}\left(\boldsymbol{B}_{\cdot l},\boldsymbol{X}_{l},N_{l};v\right)-\mu_{\mathcal{M}^{n}}\left(v\right)\right)\right\} \\
& +\left\{ \frac{1}{L}\sum_{l=1}^{L}\left(\mathcal{M}_{2}^{n}\left(\boldsymbol{B}_{\cdot l},\boldsymbol{X}_{l},N_{l};v\right)-\mu_{\mathcal{M}^{n}}\left(v\right)\right)\right\} +o_{p}\left(\left(Lh^{3+d}\right)^{-\nicefrac{1}{2}}\right),
\end{align*}
where the remainder term is uniform in $v\in I\left(\boldsymbol{x}\right)$,
\begin{align*}
\mathcal{M}_{1}^{n}\left(\boldsymbol{b}.,\boldsymbol{z},m;v\right) & \coloneqq\mathrm{E}\left[\mathcal{M}^{n}\left(\left(\boldsymbol{b}.,\boldsymbol{z},m\right),\left(\boldsymbol{B}_{\cdot1},\boldsymbol{X}_{1},N_{1}\right);v\right)\right],\\
\mathcal{M}_{2}^{n}\left(\boldsymbol{b}.,\boldsymbol{z},m;v\right) & \coloneqq\mathrm{E}\left[\mathcal{M}^{n}\left(\left(\boldsymbol{B}_{\cdot1},\boldsymbol{X}_{1},N_{1}\right),\left(\boldsymbol{b}.,\boldsymbol{z},m\right);v\right)\right],
\end{align*}
and $\mu_{\mathcal{M}^{n}}\left(v\right)\coloneqq\mathrm{E}\left[\mathcal{M}^{n}\left(\left(\boldsymbol{B}_{\cdot1},\boldsymbol{X}_{1},N_{1}\right),\left(\boldsymbol{B}_{\cdot2},\boldsymbol{X}_{2},N_{2}\right);v\right)\right]$.
The projection term $\mathcal{M}_{1}^{n}$ is the expectation of the
kernel $\mathcal{M}^{n}\left(\left(\boldsymbol{b}.,\boldsymbol{z},m\right),\left(\boldsymbol{B}_{\cdot1},\boldsymbol{X}_{1},N_{1}\right);v\right)$
with the first argument fixed at $\left(\boldsymbol{b}.,\boldsymbol{z},m\right)$.
The expression for $\mathcal{M}_{1}^{n}$ is:
\begin{eqnarray*}
& & \mathcal{M}_{1}^{n}\left(\boldsymbol{b}.,\boldsymbol{z},m;v\right)\\
& = & -\mathbbm{1}\left(m=n\right)\frac{1}{m}\sum_{i=1}^{m}\frac{1}{h^{2+d}}K_{f}'\left(\frac{\xi\left(b_{i},\boldsymbol{z},m\right)-v}{h},\frac{\boldsymbol{z}-\boldsymbol{x}}{h}\right)\frac{G\left(b_{i},\boldsymbol{z},m\right)}{g\left(b_{i},\boldsymbol{z},m\right)^{2}}\\
& & \times\int\sum_{m'\in\mathcal{N}}\int\cdots\int\left(\mathbbm{1}\left(m'=n\right)\frac{1}{m'}\sum_{j=1}^{m'}\frac{1}{h^{1+d}}K_{g}\left(\frac{b_{j}'-b_{i}}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{z}'-\boldsymbol{z}}{h}\right)-g\left(b_{i},\boldsymbol{z},m\right)\right)\\
& & \times\left(\prod_{j=1}^{m'}g\left(b_{j}'|\boldsymbol{z}',m'\right)\right)\pi\left(m'|\boldsymbol{z}'\right)\varphi\left(\boldsymbol{z}'\right)\mathrm{d}b_{1}'\cdots\mathrm{d}b_{m'}'\mathrm{d}\boldsymbol{z}'.
\end{eqnarray*}
As in the case of the simplified model, the contribution of $\mathcal{M}_{1}^{n}$
is asymptotically negligible. This happens for the same reason as
in Section \ref{sec:Asymptotic Normality}: $\mathcal{M}_{1}^{n}$
depends on the difference between the expectation of the kernel function
and the true density. Hence, the asymptotic distribution of the GPV
estimator is driven solely by the $\mathcal{M}_{2}^{n}$ term, which
is the expectation of the kernel $\mathcal{M}^{n}\left(\left(\boldsymbol{B}_{\cdot1},\boldsymbol{X}_{1},N_{1}\right),\left(\boldsymbol{b}.,\boldsymbol{z},m\right);v\right)$
with the second argument fixed at $\left(\boldsymbol{b}.,\boldsymbol{z},m\right)$.
We show in the supplement that a generalized version of (\ref{eq:E_m2^2 expansion})
holds:
\begin{align}
& \mathrm{E}\left[Lh^{3+d}\left\{ \frac{1}{L}\sum_{l=1}^{L}\left(\mathcal{M}_{2}^{n}\left(\boldsymbol{B}_{\cdot l},\boldsymbol{X}_{l},N_{l};v\right)-\mu_{\mathcal{M}^{n}}\left(v\right)\right)\right\} ^{2}\right]\nonumber \\
= & \frac{1}{n\left(n-1\right)^{2}}\frac{1}{h^{3\left(1+d\right)}}\int\int\left(\int_{\mathcal{X}}\int_{\underline{b}\left(\boldsymbol{z}'\right)}^{\overline{b}\left(\boldsymbol{z}',n\right)}K_{f}'\left(\frac{\xi\left(b',\boldsymbol{z}',n\right)-v}{h},\frac{\boldsymbol{z}'-\boldsymbol{x}}{h}\right)\frac{G\left(b',\boldsymbol{z}',n\right)}{g\left(b',\boldsymbol{z}',n\right)}\right.\nonumber \\
& \left.\times K_{g}\left(\frac{b-b'}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{z}-\boldsymbol{z}'}{h}\right)\mathrm{d}b'\mathrm{d}\boldsymbol{z}'\right)^{2}g\left(b,\boldsymbol{z},n\right)\mathrm{d}b\mathrm{d}\boldsymbol{z}+O\left(h^{3}\right)\nonumber \\
\eqqcolon & \mathrm{V}_{\mathcal{M}}\left(v|\boldsymbol{x},n\right)+O\left(h^{3}\right),\label{eq:EM_2 heterogeneity}
\end{align}
where the remainder term is uniform in $v\in I\left(\boldsymbol{x}\right)$.
Moreover, similarly to (\ref{eq:variance limit simple}),
\begin{align}
& \underset{h\downarrow0}{\mathrm{lim}}\frac{1}{h^{3\left(1+d\right)}}\int\int\left(\int_{\mathcal{X}}\int_{\underline{b}\left(\boldsymbol{z}'\right)}^{\overline{b}\left(\boldsymbol{z}',n\right)}K_{f}'\left(\frac{\xi\left(b',\boldsymbol{z}',n\right)-v}{h},\frac{\boldsymbol{z}'-\boldsymbol{x}}{h}\right)\frac{G\left(b',\boldsymbol{z}',n\right)}{g\left(b',\boldsymbol{z}',n\right)}\right.\nonumber \\
& \left.\times K_{g}\left(\frac{b-b'}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{z}-\boldsymbol{z}'}{h}\right)\mathrm{d}b'\mathrm{d}\boldsymbol{z}'\right)^{2}g\left(b,\boldsymbol{z},n\right)\mathrm{d}b\mathrm{d}\boldsymbol{z}\nonumber \\
= & \frac{G(s(v,\boldsymbol{x},n),\boldsymbol{x},n)^{2}s_{v}(v,\boldsymbol{x},n)^{2}}{g(s(v,\boldsymbol{x},n),\boldsymbol{x},n)}\nonumber \\
& \times\int\int\left\{ \int\int K_{f}'\left(w,\boldsymbol{y}\right)K_{\boldsymbol{X}}\left(\boldsymbol{y}-\boldsymbol{z}\right)K_{g}\left(u-s_{v}w-s_{\boldsymbol{x}}^{\mathrm{T}}\boldsymbol{y}\right)\mathrm{d}w\mathrm{d}\boldsymbol{y}\right\} ^{2}\mathrm{d}u\mathrm{d}\boldsymbol{z},\label{eq:variance limit general}
\end{align}
where $s_{v}$ and $s_{\boldsymbol{x}}$ denote the partial derivatives
of the bidding function:
\begin{equation}
s_{v}\coloneqq\left.\frac{\partial s\left(u,\boldsymbol{z},n\right)}{\partial u}\right|_{\left(u,\boldsymbol{z}\right)=\left(v,\boldsymbol{x}\right)}\textrm{ and }s_{\boldsymbol{x}}\coloneqq\left.\frac{\partial s\left(u,\boldsymbol{z},n\right)}{\partial\boldsymbol{z}}\right|_{\left(u,\boldsymbol{z}\right)=\left(v,\boldsymbol{x}\right)}.\label{eq:derivatives}
\end{equation}
The asymptotic variance of the GPV estimator is the limit of $\left(\pi\left(n|\boldsymbol{x}\right)\varphi\left(\boldsymbol{x}\right)\right)^{-2}\mathrm{V}_{\mathcal{M}}\left(v|\boldsymbol{x},n\right)$.
After using a change of variable argument and (\ref{eq:variance limit general}),
the variance is shown to be
\begin{multline}
\mathrm{V}_{GPV}\left(v|\boldsymbol{x},n\right)\coloneqq\frac{1}{n\left(n-1\right)^{2}}\frac{F\left(v|\boldsymbol{x}\right)^{2}f\left(v|\boldsymbol{x}\right)^{2}}{\pi\left(n|\boldsymbol{x}\right)\varphi\left(\boldsymbol{x}\right)g\left(s\left(v,\boldsymbol{x},n\right)|\boldsymbol{x},n\right)^{3}}\\
\times\int\int\left\{ \int\int K_{f}'\left(w,\boldsymbol{y}\right)K_{\boldsymbol{X}}\left(\boldsymbol{y}-\boldsymbol{z}\right)K_{g}\left(u-s_{v}w-s_{\boldsymbol{x}}^{\mathrm{T}}\boldsymbol{y}\right)\mathrm{d}w\mathrm{d}\boldsymbol{y}\right\} ^{2}\mathrm{d}u\mathrm{d}\boldsymbol{z}.\label{eq:V_GPV heterogeneity definition}
\end{multline}
The following theorem is a generalization of Theorem \ref{thm:Asymptotic Normality GPV}.
The proof of the theorem as well as the proofs of all other results
provided in Section \ref{sec:Auction-Specific-Heterogeneity} are
in the Supplement.
\begin{thm}
\label{thm:heterogeneity} Suppose Assumptions \ref{assu:DGP} - \ref{assu: rate of bandwidth}
hold. Then, for any interior point $\left(v,\boldsymbol{x}\right)\in\mathcal{S}_{V,\boldsymbol{X}},$
\[
\left(Lh^{3+d}\right)^{\nicefrac{1}{2}}\left(\widehat{f}_{GPV}\left(v|\boldsymbol{x},n\right)-f\left(v|\boldsymbol{x}\right)\right)\rightarrow_{d}\mathrm{N}\left(0,\mathrm{V}_{GPV}\left(v|\boldsymbol{x},n\right)\right).
\]
Moreover, $\left\{ \widehat{f}_{GPV}\left(v|\boldsymbol{x},n\right):n\in\mathcal{N}\right\} $
are asymptotically independent.
\end{thm}
\begin{rembold}As in the simplified model, the asymptotic variance
of the GPV estimator depends on a convoluted integral transformation
involving the kernel function, its derivative and the derivatives
of the bidding strategy. Auction-specific heterogeneity complicates
the expression in two ways. Firstly, the dimension of the integral
depends on the number of auction characteristics (covariates), and
analytical calculation of the integral becomes cumbersome when there
are many covariates. Secondly, the expression now contains the derivatives
of the bidding strategy with respect to the covariates. The reason
for that is apparent from the expression for the second moment of
$\mathcal{M}_{2}^{n}\left(\boldsymbol{B}_{\cdot1},\boldsymbol{X}_{1},N_{1};v\right)$
in (\ref{eq:EM_2 heterogeneity}) as it involves averaging of the
inverse bidding function $\xi(u,\boldsymbol{z}',n)$ over the auction
characteristics $\boldsymbol{z}'$ in a shrinking neighborhood of
$\boldsymbol{x}$.
While the GPV estimator and the quantile-based estimator have the
same rate of convergence (see \citealp[Theorem 2]{Marmer_Shneyerov_Quantile_Auctions}
for the rate of convergence and the expression of the asymptotic variance
$\mathrm{V}_{QB}\left(v|\boldsymbol{x},n\right)$ of the quantile-based
estimator in the general model), one can show that the GPV estimator
has a smaller asymptotic variance than that of the quantile-based
estimator, as long as the two estimators use the same second-order
kernel function. It can be shown that the ratio $\nicefrac{\mathrm{V}_{GPV}\left(v|\boldsymbol{x},n\right)}{\mathrm{V}_{QB}\left(v|\boldsymbol{x},n\right)}\leq1$
by applying Jensen's inequality and standard techniques for multi-dimensional
integration. A detailed proof can be found in the Supplement.\end{rembold}
\begin{rembold} [Optimal Weights]Since the distribution of valuations
does not depend on the number of bidders, i.e., $f\left(v|\boldsymbol{x},n\right)=f\left(v|\boldsymbol{x}\right)$
for all $\left(v,\boldsymbol{x},n\right)\in\mathcal{S}_{V,\boldsymbol{X}}\times\mathcal{N}$,
one can average the estimators $\left\{ \widehat{f}_{GPV}\left(v|\boldsymbol{x},n\right):n\in\mathcal{N}\right\} $
to obtain a weighted estimator of the density $f\left(v|\boldsymbol{x}\right)$:
\[
\widehat{f}_{GPV}^{w}\left(v|\boldsymbol{x}\right)\coloneqq\sum_{n\in\mathcal{N}}\widehat{w}\left(n,v,\boldsymbol{x}\right)\widehat{f}_{GPV}\left(v|\boldsymbol{x},n\right),
\]
where the weights $\widehat{w}\left(n,v,\boldsymbol{x}\right)$, $n\in\mathcal{N}$
should satisfy $\widehat{w}\left(n,v,\boldsymbol{x}\right)\rightarrow_{p}w\left(n,v,\boldsymbol{x}\right)$,
$n\in\mathcal{N}$ and $\sum_{n\in\mathcal{N}}w\left(n,v,\boldsymbol{x}\right)=1$.
As in \citet{Marmer_Shneyerov_Quantile_Auctions}, the optimal weights
that minimize the asymptotic variance of of the resulting weighted
estimator are inversely related to $\mathrm{V}_{GPV}\left(v|\boldsymbol{x},n\right)$
and given by
\[
w^{\mathrm{opt}}\left(n,v,\boldsymbol{x}\right)\coloneqq\frac{\frac{1}{\mathrm{V}_{GPV}\left(v|\boldsymbol{x},n\right)}}{\sum_{n\in\mathcal{N}}\frac{1}{\mathrm{V}_{GPV}\left(v|\boldsymbol{x},n\right)}}.
\]
These weights can be consistently estimated by the plug-in principle
using an estimator of $\mathrm{V}_{GPV}\left(v|\boldsymbol{x},n\right)$.
The original GPV estimator uses the weights $\widehat{w}\left(v,\boldsymbol{x},n\right)=\widehat{\pi}(n|\boldsymbol{x})$,
$n\in\mathcal{N}$. See (\ref{eq:f_hat_GPV definition}) and (\ref{eq:f_hat_GPV heterogeneity}).
Note, however, that such weights would be sub-optimal from the point
of view of minimizing the asymptotic variance. We provide the asymptotic
normality of $\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)$ in
the corollary below.\end{rembold}
\begin{cor}
\label{cor:asymptotic normality heterogeneity}Suppose Assumptions
\ref{assu:DGP} - \ref{assu: rate of bandwidth} hold. Then, for any
interior point $\left(v,\boldsymbol{x}\right)\in\mathcal{S}_{V,\boldsymbol{X}}$,
\[
\left(Lh^{3+d}\right)^{\nicefrac{1}{2}}\left(\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)-f\left(v|\boldsymbol{x}\right)\right)\rightarrow_{d}\mathrm{N}\left(0,\mathrm{V}_{GPV}\left(v|\boldsymbol{x}\right)\right),
\]
where $\mathrm{V}_{GPV}\left(v|\boldsymbol{x}\right)\coloneqq\sum_{n\in\mathcal{N}}\pi\left(n|\boldsymbol{x}\right)^{2}\mathrm{V}_{GPV}\left(v|\boldsymbol{x},n\right)$.
\end{cor}
For practical purposes, it is important to have a consistent estimator
of the asymptotic variance $\mathrm{V}_{GPV}\left(v|\boldsymbol{x}\right)$
that avoids estimation of the bidding strategy $s\left(\cdot,\cdot,n\right)$
and its derivatives. It is also highly desirable to avoid analytical
or numerical evaluation of a multidimensional integral in the definition
of the asymptotic variance. Following the same approach we used in
the case of the simplified model (see (\ref{eq:definition estimator of asymptotic variance})),
we rely on the sample analogue of (\ref{eq:EM_2 heterogeneity}):
\begin{multline}
\widehat{\mathrm{V}}_{GPV}\left(v|\boldsymbol{x},n\right)\coloneqq\frac{1}{n(n-1)^{2}}\frac{1}{\widehat{\pi}\left(n|\boldsymbol{x}\right)^{2}\widehat{\varphi}\left(\boldsymbol{x}\right)^{2}h^{3\left(1+d\right)}}\frac{1}{L\left(L-1\right)\left(L-2\right)}\\
\times\sum_{l=1}^{L}\sum_{k\neq l}\sum_{k'\neq k,k'\neq l}\mathbbm{1}\left(N_{l}=n,N_{k}=n,N_{k'}=n\right)\frac{1}{N_{l}}\sum_{i=1}^{N_{l}}\eta_{il,k}(v,\boldsymbol{x})\eta_{il,k'}(v,\boldsymbol{x}),\label{eq:V_GPV estimator heterogeneity}
\end{multline}
where
\begin{multline*}
\eta_{il,k}(v,\boldsymbol{x})\coloneqq\frac{1}{N_{k}}\\
\times\sum_{j=1}^{N_{k}}\mathbb{T}_{jk}K_{f}'\left(\frac{\widehat{V}_{jk}-v}{h},\frac{\boldsymbol{X}_{k}-\boldsymbol{x}}{h}\right)\frac{\widehat{G}\left(B_{jk},\boldsymbol{X}_{k},N_{k}\right)}{\widehat{g}\left(B_{jk},\boldsymbol{X}_{k},N_{k}\right)^{2}}K_{g}\left(\frac{B_{il}-B_{jk}}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{X}_{l}-\boldsymbol{X}_{k}}{h}\right).
\end{multline*}
The following result is a generalization of Theorem \ref{thm:variance estimator}.
\begin{thm}
\label{thm:variance estimator convergence rate heterogeneity}Suppose
Assumptions \ref{assu:DGP} - \ref{assu: rate of bandwidth} hold.
Then, for any interior point $\boldsymbol{x}$,
\[
\underset{v\in I\left(\boldsymbol{x}\right)}{\mathrm{sup}}\left|\widehat{\mathrm{V}}_{GPV}\left(v|\boldsymbol{x},n\right)-\left(\pi\left(n|\boldsymbol{x}\right)\varphi\left(\boldsymbol{x}\right)\right)^{-2}\mathrm{V}_{\mathcal{M}}\left(v|\boldsymbol{x},n\right)\right|=O_{p}\left(\left(\frac{\mathrm{log}\left(L\right)}{Lh^{3+d}}\right)^{\nicefrac{1}{2}}+h^{R}\right).
\]
\end{thm}
\begin{rembold} An estimator for the asymptotic variance $\mathrm{V}_{GPV}(v|\boldsymbol{x})$
of the estimator $\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)$
in Corollary \ref{cor:asymptotic normality heterogeneity} can be
constructed using the plug-in approach:
\[
\mathrm{\widehat{V}}_{GPV}\left(v|\boldsymbol{x}\right)\coloneqq\sum_{n\in\mathcal{N}}\widehat{\pi}\left(n|\boldsymbol{x}\right)^{2}\widehat{\mathrm{V}}_{GPV}\left(v|\boldsymbol{x},n\right).
\]
Its rate of convergence is the same as in Theorem \ref{thm:variance estimator convergence rate heterogeneity}.
\end{rembold}
\subsection{Bootstrap-based Inference and Uniform Confidence Bands}
To generate bootstrap samples, we apply the same resampling procedure
as that proposed in \citet[Section 4]{Marmer_Shneyerov_Quantile_Auctions}.
First, we randomly draw $L$ observations from $\left\{ \left(\boldsymbol{X}_{l},N_{l}\right):l=1,...,L\right\} $
(i.e., the auction-specific characteristics) with replacement. Next,
we randomly draw bids with replacement from the bids corresponding
to each selected auction. Given $\left(\boldsymbol{X}_{l}^{*},N_{l}^{*}\right)=\left(\boldsymbol{X}_{l'},N_{l'}\right)$
in the first step, in the second step $\{B_{il}^{*}:i=1,...,N_{l}^{*}\}$
is generated as an empirical bootstrap sample drawn from $\{B_{il'}:i=1,...,N_{l'}\}$.
Let $\widehat{\xi}^{*}\left(\cdot,\cdot,\cdot\right)$ and $\widehat{\varphi}^{*}\left(\cdot\right)$
be the bootstrap analogues of $\widehat{\xi}\left(\cdot,\cdot,\cdot\right)$
and $\widehat{\varphi}\left(\cdot\right)$ respectively. Let $\widehat{f}_{GPV}^{*}\left(v|\boldsymbol{x}\right)$
denote the bootstrap version of the GPV estimator:
\[
\widehat{f}_{GPV}^{*}\left(v|\boldsymbol{x}\right)\coloneqq\frac{1}{\widehat{\varphi}^{*}\left(\boldsymbol{x}\right)L}\sum_{l=1}^{L}\frac{1}{N_{l}^{*}}\sum_{i=1}^{N_{l}^{*}}\mathbb{T}_{il}^{*}\frac{1}{h^{1+d}}K_{f}\left(\frac{\widehat{V}_{il}^{*}-v}{h},\frac{\boldsymbol{X}_{l}^{*}-\boldsymbol{x}}{h}\right),
\]
where $\widehat{V}_{il}^{*}\coloneqq\widehat{\xi}^{*}\left(B_{il}^{*},\boldsymbol{X}_{l}^{*},N_{l}^{*}\right)$
and the bootstrap version of the trimming factor is given by
\[
\mathbb{T}_{il}^{*}\coloneqq\mathbbm{1}\left(\mathbb{H}\left(\left(B_{il}^{*},\boldsymbol{X}_{l}^{*}\right),2h\right)\subseteq\mathcal{\widehat{S}}_{B,\boldsymbol{X}}^{N_{l}}\right).
\]
Consider the scaled deviation of the GPV estimator from the true PDF,
and its bootstrap analogue:
\begin{gather*}
S\left(v|\boldsymbol{x}\right)\coloneqq\left(Lh^{3+d}\right)^{\nicefrac{1}{2}}\left(\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)-f\left(v|\boldsymbol{x}\right)\right)\textrm{ and}\\
S^{*}\left(v|\boldsymbol{x}\right)\coloneqq\left(Lh^{3+d}\right)^{\nicefrac{1}{2}}\left(\widehat{f}_{GPV}^{*}\left(v|\boldsymbol{x}\right)-\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)\right).
\end{gather*}
The following result, which is a generalization of Theorem \ref{thm: bootstrap consistency},
establishes the validity of the percentile bootstrap for $f\left(v|\boldsymbol{x}\right)$.
\begin{thm}
Suppose Assumptions \ref{assu:DGP} - \ref{assu: rate of bandwidth}
hold. Then, for any interior point $\left(v,\boldsymbol{x}\right)\in\mathcal{S}_{V,\boldsymbol{X}}$,
\[
\underset{z\in\mathbb{R}}{\mathrm{sup}}\left|\mathrm{P}^{*}\left[S^{*}\left(v|\boldsymbol{x}\right)\leq z\right]-\mathrm{P}\left[S\left(v|\boldsymbol{x}\right)\leq z\right]\right|\rightarrow_{p}0,\,\textrm{as \ensuremath{L\uparrow\infty}}.
\]
\end{thm}
We now turn to construction of uniform confidence bands for $\left\{ f\left(v|\boldsymbol{x}\right):v\in I\left(\boldsymbol{x}\right)\right\} $
given a fixed interior point $\boldsymbol{x}$. Consider the following
processes:
\[
Z\left(v|\boldsymbol{x}\right)\coloneqq\frac{\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)-f\left(v|\boldsymbol{x}\right)}{\left(Lh^{3+d}\right)^{-\nicefrac{1}{2}}\widehat{\mathrm{V}}_{GPV}\left(v|\boldsymbol{x}\right)^{\nicefrac{1}{2}}}\textrm{ and }Z^{*}\left(v|\boldsymbol{x}\right)\coloneqq\frac{\widehat{f}_{GPV}^{*}\left(v|\boldsymbol{x}\right)-\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)}{\left(Lh^{3+d}\right)^{-\nicefrac{1}{2}}\widehat{\mathrm{V}}_{GPV}\left(v|\boldsymbol{x}\right)^{\nicefrac{1}{2}}},\textrm{ \ensuremath{v\in I\left(\boldsymbol{x}\right)}}.
\]
Similarly to the simplified model, the distribution of $\left\Vert Z\left(\cdot|\boldsymbol{x}\right)\right\Vert _{I\left(\boldsymbol{x}\right)}$
can be approximated by the conditional distribution of $\left\Vert Z^{*}\left(\cdot|\boldsymbol{x}\right)\right\Vert _{I\left(\boldsymbol{x}\right)}$.
Let $\zeta_{L,\alpha}^{*}$ be the $\left(1-\alpha\right)$-quantile
of the conditional distribution of $\left\Vert Z^{*}\left(\cdot|\boldsymbol{x}\right)\right\Vert _{I\left(\boldsymbol{x}\right)}$
given the original sample. Consider the following confidence band:
for $\ensuremath{v\in I\left(\boldsymbol{x}\right)}$,
\[
CB^{*}\left(v|\boldsymbol{x}\right)\coloneqq\left[\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)-\zeta_{L,\alpha}^{*}\sqrt{\frac{\widehat{\mathrm{V}}_{GPV}\left(v|\boldsymbol{x}\right)}{Lh^{3+d}}},\,\widehat{f}_{GPV}\left(v|\boldsymbol{x}\right)+\zeta_{L,\alpha}^{*}\sqrt{\frac{\widehat{\mathrm{V}}_{GPV}\left(v|\boldsymbol{x}\right)}{Lh^{3+d}}}\right].
\]
The following result, which is a generalization of Corollary \ref{cor:consistency of IGA bootstrap},
establishes the validity of $CB^{*}\left(\cdot|\boldsymbol{x}\right)$.
\begin{thm}
Suppose Assumptions \ref{assu:DGP} - \ref{assu: rate of bandwidth}
hold. Then,
\[
\mathrm{P}\left[f\left(v|\boldsymbol{x}\right)\in CB^{*}\left(v|\boldsymbol{x}\right),\,\textrm{for all \ensuremath{v\in I\left(\boldsymbol{x}\right)}}\right]\rightarrow1-\alpha,\,\textrm{as \ensuremath{L\uparrow\infty}}.
\]
\end{thm}
\section{Binding Reserve Price\label{sec:Binding-Reserve-Price}}
Section 4 of GPV shows how to modify their identification and estimation
strategy when there is a binding reserve price. Here, we discuss how
our approach can be applied in that case.
When there is a binding reserve price, it is assumed that only bidders
with valuations exceeding the reserve price submit bids. Thus, one
has to distinguish between the numbers of potential and actual (active)
bidders. Let $N$ denote the number of potential bidders, which is
assumed to be known to players. The bidding strategy depends on the
number of potential bidders instead of the number of active bidders.
As discussed in GPV, $N$ can be estimated by taking the maximum of
the observed numbers of actual bidders across the auctions: $\widehat{N}\coloneqq\mathrm{max}\left\{ N_{l}:l=1,\ldots,L\right\} ,$
where $N_{l}$ is the number of actual bidders in auction $l$.
GPV assume that the reserve price in auction $l$, denoted $P_{0l}$,
is some unknown deterministic function of the auction characteristics
$\boldsymbol{X}_{l}$: $P_{0l}=p_{0}\left(\boldsymbol{X}_{l}\right)$.
The probability of drawing a valuation below the reserve price is
given by $\Phi\left(\boldsymbol{X}_{l}\right)\coloneqq F\left(P_{0l}|\boldsymbol{X}_{l}\right)$.
The conditional CDF and PDF of the distribution of valuations given
participation (submitting a bid) are
\begin{equation}
F^{\star}\left(v|\boldsymbol{x}\right)\coloneqq\frac{F\left(v|\boldsymbol{x}\right)-\Phi\left(\boldsymbol{x}\right)}{1-\Phi\left(\boldsymbol{x}\right)}\text{ and }f^{\star}(v|\boldsymbol{x})\coloneqq\frac{f\left(v|\boldsymbol{x}\right)}{1-\Phi\left(\boldsymbol{x}\right)}\label{eq:binding CDF PDF}
\end{equation}
respectively. The third displayed equation on page 550 of GPV shows
that $\Phi(\cdot)$ can be estimated using a nonparametric regression
of the number of actual bidders:
\[
\widehat{\Phi}\left(\boldsymbol{x}\right)\coloneqq1-\frac{1}{\widehat{N}\widehat{\varphi}\left(\boldsymbol{x}\right)L}\sum_{l=1}^{L}\frac{1}{h^{d}}N_{l}K_{\boldsymbol{X}}\left(\frac{\boldsymbol{X}_{l}-\boldsymbol{x}}{h}\right).
\]
GPV point out that the density of bids is unbounded at the reserve
price $p_{0}\left(\boldsymbol{x}\right)$ and, in its neighborhood,
behaves as $\nicefrac{1}{\sqrt{b-p_{0}(\boldsymbol{x})}}$ . To avoid
technical problems due to the unbounded density, they propose to transform
the bids as
\[
B_{\dagger il}=\left(B_{il}-P_{0l}\right)^{\nicefrac{1}{2}}.
\]
The support of $\left(B_{\dagger11},\boldsymbol{X}_{1}\right)$ is
given by $\mathcal{S}_{B_{\dagger},\boldsymbol{X}}\coloneqq\left\{ \left(b,\boldsymbol{x}\right):\boldsymbol{x}\in\mathcal{X},\,b\in\left[0,\overline{b}_{\dagger}\left(\boldsymbol{x}\right)\right]\right\} $,
where $\overline{b}_{\dagger}\left(\boldsymbol{z}\right)\coloneqq\left(\overline{b}_{\dagger}\left(\boldsymbol{z}\right)-p_{0}\left(\boldsymbol{z}\right)\right)^{\nicefrac{1}{2}}$.
The support can be estimated by $\mathcal{\widehat{S}}_{B_{\dagger},\boldsymbol{X}}\coloneqq\left\{ \left(b,\boldsymbol{x}\right):\boldsymbol{z}\in\mathcal{X},\,b\in\left[0,\widehat{\overline{b}}_{\dagger}\left(\boldsymbol{x}\right)\right]\right\} $,
where $\widehat{\overline{b}}_{\dagger}\left(\boldsymbol{x}\right)\coloneqq\mathrm{\mathrm{max}}\left\{ B_{\dagger pl}:p=1,...,N_{l},\,\boldsymbol{X}_{l}\in\Pi_{h_{\partial}}\left(\boldsymbol{x}\right),\,l=1,...,L\right\} $,
see page 550 in GPV.
Let $G_{\dagger}\left(\cdot|\cdot\right)$ and $g_{\dagger}\left(\cdot|\cdot\right)$
denote respectively the conditional CDF and PDF of the transformed
bids $B_{\dagger11}$ given $\boldsymbol{X}_{1}$. Let $G_{\dagger}\left(b_{\dagger},\boldsymbol{z}\right)\coloneqq G_{\dagger}\left(b_{\dagger}|\boldsymbol{z}\right)\varphi\left(\boldsymbol{z}\right)$
and $g_{\dagger}\left(b_{\dagger},\boldsymbol{z}\right)\coloneqq g_{\dagger}\left(b_{\dagger}|\boldsymbol{z}\right)\varphi\left(\boldsymbol{z}\right)$.
GPV show that $g_{\dagger}\left(\cdot,\cdot\right)$ is bounded on
its support, and that latent valuations can be recovered using
\[
V_{il}=\xi_{\dagger}\left(B_{\dagger il},\boldsymbol{X}_{l}\right)\coloneqq P_{0l}+B_{\dagger il}^{2}+\frac{2B_{\dagger il}}{N-1}\frac{\left(1-\Phi\left(\boldsymbol{X}_{l}\right)\right)G_{\dagger}\left(B_{\dagger il},\boldsymbol{X}_{l}\right)+\Phi\left(\boldsymbol{X}_{l}\right)\varphi\left(\boldsymbol{X}_{l}\right)}{\left(1-\Phi\left(\boldsymbol{X}_{l}\right)\right)g_{\dagger}\left(B_{\dagger il},\boldsymbol{X}_{l}\right)}.
\]
In the modified GPV procedure, one first estimates $\xi_{\dagger}\left(\cdot,\cdot\right)$
by replacing $N$, $\Phi\left(\cdot\right)$, $G_{\dagger}\left(\cdot,\cdot\right)$,
$g_{\dagger}\left(\cdot,\cdot\right)$ and $\varphi\left(\cdot\right)$
with their estimators. $G_{\dagger}\left(\cdot,\cdot\right)$ and
$g_{\dagger}\left(\cdot,\cdot\right)$ can be estimated using the
transformed bids $B_{\dagger il}$:
\begin{gather*}
\widehat{G}_{\dagger}(b_{\dagger},\boldsymbol{x})\coloneqq\frac{1}{L}\sum_{l=1}^{L}\frac{1}{N_{l}}\sum_{i=1}^{N_{l}}\mathbbm{1}\left(B_{\dagger il}\leq b_{\dagger}\right)\frac{1}{h^{d}}K_{\boldsymbol{X}}\left(\frac{\boldsymbol{X}_{l}-\boldsymbol{x}}{h}\right),\\
\widehat{g}_{\dagger}(b_{\dagger},\boldsymbol{x})\coloneqq\frac{1}{L}\sum_{l=1}^{L}\frac{1}{N_{l}}\sum_{i=1}^{N_{l}}\frac{1}{h^{1+d}}K_{g}\left(\frac{B_{\dagger il}-b_{\dagger}}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{X}_{l}-\boldsymbol{x}}{h}\right).
\end{gather*}
In the second step of the modified GPV procedure, one uses the pseudo
valuations
\[
\left\{ \widehat{V}_{il}\coloneqq\widehat{\xi}_{\dagger}\left(B_{\dagger il},\boldsymbol{X}_{l}\right):i=1,\dots,N_{l},l=1,\ldots,L\right\} ,
\]
where $\widehat{\xi}_{\dagger}\left(\cdot,\cdot\right)$ is the estimated
version of $\xi_{\dagger}(\cdot,\cdot)$, in place of latent valuations
to construct a kernel density estimator of $f^{\star}\left(v|\boldsymbol{x}\right)$:
\[
\widehat{f}_{GPV}^{\star}(v|\boldsymbol{x})\coloneqq\frac{1}{\widehat{\varphi}(\boldsymbol{x})L}\sum_{l=1}^{L}\frac{1}{N_{l}}\sum_{i=1}^{N_{l}}\mathbb{T}_{il}\frac{1}{h^{1+d}}K_{f}\left(\frac{\widehat{V}_{il}-v}{h},\frac{\boldsymbol{X}_{l}-\boldsymbol{x}}{h}\right),
\]
where the trimming factors $\mathbb{T}_{il}$ can be defined analogously
to the case with no binding reserve price: $\mathbb{T}_{il}\coloneqq\mathbbm{1}\left(\mathbb{H}\left(\left(B_{\dagger il},\boldsymbol{X}_{l}\right),2h\right)\subseteq\mathcal{\widehat{S}}_{B_{\dagger},\boldsymbol{X}}\right).$
Our approach can be used to obtain the asymptotic distribution of
the modified GPV estimator as follows. In view of the definitions
of $\xi_{\dagger}\left(\cdot.\cdot\right)$ and its estimator, the
$\widehat{V}_{il}-V_{il}$ term in the analogue of (\ref{eq:the first stochastic approximation result})
can be expanded as
\[
\widehat{V}_{il}-V_{il}=\frac{2B_{\dagger il}}{N-1}\frac{\left(1-\Phi\left(\boldsymbol{X}_{l}\right)\right)G_{\dagger}\left(B_{\dagger il},\boldsymbol{X}_{l}\right)+\Phi\left(\boldsymbol{X}_{l}\right)\varphi\left(\boldsymbol{X}_{l}\right)}{\left(1-\Phi\left(\boldsymbol{X}_{l}\right)\right)g_{\dagger}\left(B_{\dagger il},\boldsymbol{X}_{l}\right)^{2}}\left(\widehat{g}_{\dagger}\left(B_{\dagger il},\boldsymbol{X}_{l}\right)-g_{\dagger}\left(B_{\dagger il},\boldsymbol{X}_{l}\right)\right)+s.o.,
\]
where ``$s.o.$'' stands for smaller order terms. Hence, the GPV
estimator of $f^{\star}\left(v|\boldsymbol{x}\right)$ still has a
representation of the same form as in (\ref{eq:auction heterogeneity V statistic approximation}):
\begin{multline*}
\widehat{f}_{GPV}^{\star}\left(v|\boldsymbol{x}\right)-f^{\star}\left(v|\boldsymbol{x}\right)\\
=\frac{1}{\widehat{\varphi}\left(\boldsymbol{x}\right)L^{2}}\sum_{l=1}^{L}\sum_{k=1}^{L}\mathcal{M}_{\dagger}\left(\left(\boldsymbol{B}_{\dagger\cdot l},\boldsymbol{X}_{l},N_{l}\right),\left(\boldsymbol{B}_{\dagger\cdot k},\boldsymbol{X}_{k},N_{k}\right);v\right)+o_{p}\left(\left(Lh^{3+d}\right)^{-\nicefrac{1}{2}}\right),
\end{multline*}
where $\boldsymbol{B}_{\dagger\cdot l}\coloneqq\left(B_{\dagger1l},...,B_{\dagger N_{l}l}\right)$,
and
\begin{eqnarray*}
& & \mathcal{M}_{\dagger}\left(\left(\boldsymbol{b}_{\cdot},\boldsymbol{z},m\right),\left(\boldsymbol{b}_{\cdot}',\boldsymbol{z}',m'\right);v\right)\\
& \coloneqq & -\frac{1}{m}\sum_{i=1}^{m}\frac{1}{h^{2+d}}K_{f}'\left(\frac{\xi_{\dagger}\left(b_{i},\boldsymbol{z}\right)-v}{h},\frac{\boldsymbol{z}-\boldsymbol{x}}{h}\right)\frac{2b_{i}\left(\left(1-\Phi\left(\boldsymbol{z}\right)\right)G_{\dagger}\left(b_{i},\boldsymbol{z}\right)+\Phi\left(\boldsymbol{z}\right)\varphi\left(\boldsymbol{z}\right)\right)}{\left(N-1\right)\left(1-\Phi\left(\boldsymbol{z}\right)\right)g_{\dagger}\left(b_{i},\boldsymbol{z}\right)^{2}}\\
& & \times\left(\frac{1}{m'}\sum_{j=1}^{m'}\frac{1}{h^{1+d}}K_{g}\left(\frac{b_{j}'-b_{i}}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{z}'-\boldsymbol{z}}{h}\right)-g_{\dagger}\left(b_{i},\boldsymbol{z}\right)\right).
\end{eqnarray*}
Similarly to the case with no reserve price, one can apply the Hoeffding
decomposition with only $\mathcal{M}_{\dagger2}\left(\boldsymbol{b}.,\boldsymbol{z},m;v\right)\coloneqq\mathrm{E}\left[\mathcal{M}_{\dagger}\left(\left(\boldsymbol{B}_{\dagger\cdot1},\boldsymbol{X}_{1},N_{1}\right),\left(\boldsymbol{b}.,\boldsymbol{z},m\right);v\right)\right]$
contributing to the asymptotic variance:
\begin{eqnarray*}
& & \mathcal{M}_{\dagger2}\left(\boldsymbol{b}.,\boldsymbol{z},m;v\right)\\
& = & -\frac{1}{h^{2+d}}\int\int K_{f}'\left(\frac{\xi_{\dagger}\left(b',\boldsymbol{z}'\right)-v}{h},\frac{\boldsymbol{z}'-\boldsymbol{x}}{h}\right)\frac{2b'\left((1-\Phi\left(\boldsymbol{z}'\right))G_{\dagger}\left(b',\boldsymbol{z}'\right)+\Phi\left(\boldsymbol{z}'\right)\varphi\left(\boldsymbol{z}'\right)\right)}{\left(N-1\right)\left(1-\Phi\left(\boldsymbol{z}'\right)\right)g_{\dagger}\left(b',\boldsymbol{z}'\right)^{2}}\\
& & \times\left(\frac{1}{m}\sum_{j=1}^{m}\frac{1}{h^{1+d}}K_{g}\left(\frac{b_{j}-b'}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{z}-\boldsymbol{z}'}{h}\right)-g_{\dagger}\left(b',\boldsymbol{z}'\right)\right)g_{\dagger}\left(b',\boldsymbol{z}'\right)\mathrm{d}b'\mathrm{d}\boldsymbol{z}'.
\end{eqnarray*}
Similarly to (\ref{eq:EM_2 heterogeneity}), the asymptotic variance
of the GPV estimator is now given by the limit of
\begin{align}
& \frac{\overline{\pi}\left(\boldsymbol{x}\right)}{\varphi\left(\boldsymbol{x}\right)^{2}\left(N-1\right)^{2}h^{3(1+d)}}\int\int\left\{ \int_{\mathcal{X}}\int_{0}^{\overline{b}_{\dagger}\left(\boldsymbol{z}'\right)}K_{f}'\left(\frac{\xi_{\dagger}\left(b',\boldsymbol{z}'\right)-v}{h},\frac{\boldsymbol{z}'-\boldsymbol{x}}{h}\right)\right.\nonumber \\
& \times\frac{2b'\left((1-\Phi\left(\boldsymbol{z}'\right))G_{\dagger}\left(b',\boldsymbol{z}'\right)+\Phi\left(\boldsymbol{z}'\right)\varphi\left(\boldsymbol{z}'\right)\right)}{1-\Phi\left(\boldsymbol{z}'\right)g_{\dagger}\left(b',\boldsymbol{z}'\right)}\nonumber \\
& \left.\times K_{g}\left(\frac{b-b'}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{z}-\boldsymbol{z}'}{h}\right)\mathrm{d}b'\mathrm{d}\boldsymbol{z}'\right\} ^{2}g_{\dagger}\left(b,\boldsymbol{z}\right)\mathrm{d}b\mathrm{d}\boldsymbol{z}\label{eq:Em_2^2 reserve price}
\end{align}
as $h\downarrow0$, where $\overline{\pi}\left(\boldsymbol{x}\right)\coloneqq\mathrm{E}\left[N_{1}^{-1}\mid\boldsymbol{X}_{1}=\boldsymbol{x}\right]$.
Note that conditionally on $\boldsymbol{X}_{1}$, the number of active
bidders $N_{1}$ has a binomial distribution with parameters $N$
and $1-\Phi\left(\boldsymbol{X}_{1}\right)$. Lastly, similarly to
Theorem \ref{thm:heterogeneity}, the expression in (\ref{eq:variance limit general}),
and for any interior point $v\in\left(p_{0}\left(\boldsymbol{x}\right),\bar{v}\left(\boldsymbol{x}\right)\right)$,
the GPV estimator of $f^{\star}\left(v\mid\boldsymbol{x}\right)$
is asymptotically normal with the asymptotic variance given by
\begin{eqnarray*}
& & \mathrm{V}_{\dagger GPV}\left(v,\boldsymbol{x}\right)\\
& \coloneqq & \frac{\overline{\pi}\left(\boldsymbol{x}\right)}{\varphi\left(\boldsymbol{x}\right)^{2}\left(N-1\right)^{2}}\frac{\left(2s_{\dagger}\left(v,\boldsymbol{x}\right)s_{\dagger v}\left(v,\boldsymbol{x}\right)\left(\left(1-\Phi\left(\boldsymbol{x}\right)\right)G_{\dagger}\left(s_{\dagger}\left(v,\boldsymbol{x}\right),\boldsymbol{x}\right)+\Phi\left(\boldsymbol{x}\right)\varphi\left(\boldsymbol{x}\right)\right)\right)^{2}}{\left(1-\Phi\left(\boldsymbol{x}\right)\right)^{2}g_{\dagger}\left(s_{\dagger}\left(v,\boldsymbol{x}\right),\boldsymbol{x}\right)}\\
& & \times\int\int\left\{ \int\int K_{f}'\left(w,\boldsymbol{y}\right)K_{\boldsymbol{X}}\left(\boldsymbol{y}-\boldsymbol{z}\right)K_{g}\left(u-s_{\dagger v}w-s_{\dagger\boldsymbol{x}}^{\mathrm{T}}\boldsymbol{y}\right)\mathrm{d}w\mathrm{d}\boldsymbol{y}\right\} ^{2}\mathrm{d}u\mathrm{d}\boldsymbol{z},
\end{eqnarray*}
where $s_{\dagger}(\cdot,\boldsymbol{x})\coloneqq\xi_{\dagger}^{-1}(\cdot,\boldsymbol{x})$,
and the partial derivatives $s_{\dagger v}$ and $s_{\dagger\boldsymbol{x}}$
are defined similarly to $s_{v}$ and $s_{\boldsymbol{x}}$ in (\ref{eq:derivatives}).
Similarly to Corollary \ref{cor:asymptotic normality heterogeneity},
one can show:
\[
\left(Lh^{3+d}\right)^{\nicefrac{1}{2}}\left(\widehat{f}^{\star}\left(v\mid\boldsymbol{x}\right)-f^{\star}\left(v\mid\boldsymbol{x}\right)\right)\rightarrow_{d}\mathrm{N}\left(0,\mathrm{V}_{\dagger GPV}\left(v,\boldsymbol{x}\right)\right).
\]
To estimate the asymptotic variance $\mathrm{V}_{\dagger GPV}(v,\boldsymbol{x})$,
one can use the sample analogue of (\ref{eq:Em_2^2 reserve price})
in the same way as that used to construct the estimator of $\mathrm{V}_{GPV}(v,\boldsymbol{x})$
defined by (\ref{eq:V_GPV estimator heterogeneity}) from (\ref{eq:EM_2 heterogeneity}).
As before, the approach does not require estimation of the bidding
strategy $s_{\dagger}(v,\boldsymbol{x})$ or its derivatives. The
analogue estimator is given by
\begin{multline*}
\mathrm{\widehat{V}}_{\dagger GPV}\left(v,\boldsymbol{x}\right)\coloneqq\frac{\widehat{\overline{\pi}}\left(\boldsymbol{x}\right)}{\widehat{\varphi}\left(\boldsymbol{x}\right)^{2}\left(\widehat{N}-1\right)^{2}h^{3(1+d)}}\frac{1}{L\left(L-1\right)\left(L-2\right)}\\
\times\sum_{l=1}^{L}\sum_{k\neq l}\sum_{k'\neq k,k'\neq l}\frac{1}{N_{l}}\sum_{i=1}^{N_{l}}\eta_{\dagger il,k}(v,\boldsymbol{x})\eta_{\dagger il,k'}(v,\boldsymbol{x}),
\end{multline*}
where $\widehat{\overline{\pi}}\left(\boldsymbol{x}\right)$ is the
Nadaraya-Watson estimator of $\overline{\pi}\left(\boldsymbol{x}\right)$,
and
\begin{multline*}
\eta_{\dagger il,k}\left(v,\boldsymbol{x}\right)\coloneqq\frac{1}{N_{k}}\sum_{j=1}^{N_{k}}\mathbb{T}_{jk}K_{f}'\left(\frac{\widehat{V}_{jk}-v}{h},\frac{\boldsymbol{X}_{k}-\boldsymbol{x}}{h}\right)K_{g}\left(\frac{B_{\dagger il}-B_{\dagger jk}}{h}\right)K_{\boldsymbol{X}}\left(\frac{\boldsymbol{X}_{l}-\boldsymbol{X}_{k}}{h}\right)\\
\times\frac{2B_{\dagger jk}\left((1-\widehat{\Phi}\left(\boldsymbol{X}_{k}\right))\widehat{G}_{\dagger}\left(B_{\dagger jk},\boldsymbol{X}_{k}\right)+\widehat{\Phi}\left(\boldsymbol{X}_{k}\right)\widehat{\varphi}\left(\boldsymbol{X}_{k}\right)\right)}{1-\widehat{\Phi}\left(\boldsymbol{X}_{k}\right)\widehat{g}_{\dagger}\left(B_{\dagger jk},\boldsymbol{X}_{k}\right)^{2}}.
\end{multline*}
Suppose $I\left(\boldsymbol{x}\right)$ is an inner closed sub-interval
of $\left[p_{0}\left(\boldsymbol{x}\right),\bar{v}\left(\boldsymbol{x}\right)\right]$.
The uniform convergence rate of $\mathrm{\widehat{V}}_{\dagger GPV}\left(v,\boldsymbol{x}\right)$
to (\ref{eq:Em_2^2 reserve price}) can be shown to be the same as
that in the statement of Theorem \ref{thm:variance estimator convergence rate heterogeneity}.
In view of the definitions in (\ref{eq:binding CDF PDF}), the nonparametric
estimator for $f\left(v|\boldsymbol{x}\right)$ is $\left(1-\widehat{\Phi}\left(\boldsymbol{x}\right)\right)\widehat{f}^{\star}\left(v|\boldsymbol{x}\right)$.
Since $\widehat{\Phi}\left(\boldsymbol{x}\right)$ converges at a
faster rate than the PDF estimator $\widehat{f}^{\star}\left(v\mid\boldsymbol{x}\right)$,
one can see that
\[
\left(Lh^{3+d}\right)^{\nicefrac{1}{2}}\left(\left(1-\widehat{\Phi}\left(\boldsymbol{x}\right)\right)\widehat{f}^{\star}\left(v\mid\boldsymbol{x}\right)-f\left(v\mid\boldsymbol{x}\right)\right)\rightarrow_{d}\mathrm{N}\left(0,\left(1-\Phi\left(\boldsymbol{x}\right)\right)^{2}\mathrm{V}_{\dagger GPV}\left(v,\boldsymbol{x}\right)\right).
\]
A valid uniform confidence band of $\left\{ f\left(v|\boldsymbol{x}\right):v\in I\left(\boldsymbol{x}\right)\right\} $
can be constructed by adapting the methods described in Section \ref{sec:Auction-Specific-Heterogeneity}.
\section{Monte Carlo Simulations\label{sec:Monte-Carlo-Simulations}}
In this section, we assess the finite-sample coverage accuracy of
the uniform confidence bands. Our simulation design follows \citet*{Marmer_Shneyerov_Quantile_Auctions},
and the DGP is described in Remark \ref{rem:comparison QB}. We consider
$\theta\in\left\{ 1,2\right\} $ and draw valuations from $f_{\theta}$.
We choose the triweight kernel when implementing the two-step estimator.
We used the second-order triweight kernel in the second step and used
the fourth-order triweight kernel in the first step.
We need to choose the bandwidths in the first step when we construct
the pseudo valuations and the second step when we implement kernel
density estimation using the pseudo valuations. We follow GPV (see
Section 2.4) and use $h_{g}=3.72\cdot\widehat{\sigma}_{b}\cdot\left(N\cdot L\right)^{-\nicefrac{1}{5}}$as
the first-step bandwidth, where $\widehat{\sigma}_{b}$ is the estimated
standard deviation of the observed bids. We use $h_{f}=3.15\cdot\widehat{\sigma}_{v}\cdot\left(\left(N\cdot L\right)_{\mathbb{T}}\right)^{-\nicefrac{1}{5}}$as
the second-step bandwidth, where $\widehat{\sigma}_{v}$ is the estimated
standard deviation of the trimmed pseudo valuations and $\left(N\cdot L\right)_{\mathbb{T}}$
is the number of bids remaining after the trimming. The constants
$3.72$ and $3.15$ are Silverman's rule-of-thumb constants corresponding
to fourth-order and second-order triweight kernels.\footnote{See \citet{li2007net} for a description of the Silverman approach;
see also \citet[Section 3.2]{li2003semiparametric}.} We consider different numbers of bidders $N\in\left\{ 3,5,7\right\} $,
and also the density function over different ranges: $v\in\left[0.2,0.8\right]$
and $v\in\left[0.3,0.7\right]$.\footnote{We use grid maximization, where the grid is chosen as $[v_{l}:0.001:v_{u}]$.
We have also tried a finer grid $[v_{l}:0.0001:v_{u}],$ which produced
similar results.} The number of auctions $L$ is chosen so the total number of observations
is fixed as $N\cdot L=2100$.
\begin{table}[t]
\caption{Coverage probabilities of the uniform confidence band $CB^{*}$ for
the number of bidders $N=3,5,7$, the distribution parameter $\theta=1,2$,
different ranges of valuations $v$, and the nominal coverage probability
$=0.90,0.95,0.99$. The number of auctions $L$ is determined by $N\cdot L=2100$\label{tab:Coverage-probabilities}\medskip{}
}
$\,$\thispagestyle{empty}
\centering{}
\begin{tabular}{c>{\centering}p{2cm}>{\centering}p{2cm}>{\centering}p{2cm}>{\centering}p{2cm}>{\centering}p{2cm}>{\centering}p{2cm}}
\toprule
$N$ & 0.90 & 0.95 & 0.99 & 0.90 & 0.95 & 0.99\tabularnewline
\midrule
& \multicolumn{3}{c}{\uline{\mbox{$\theta=1,\quad v\in[0.3,0.7]$}}} & \multicolumn{3}{c}{\uline{\mbox{$\theta=1,\quad v\in[0.2,0.8]$}}}\tabularnewline
& & & & & & \tabularnewline
$3$ & 0.880 & 0.924 & 0.988 & 0.880 & 0.930 & 0.982\tabularnewline
$5$ & 0.922 & 0.962 & 0.998 & 0.918 & 0.962 & 0.994\tabularnewline
$7$ & 0.888 & 0.946 & 0.990 & 0.898 & 0.948 & 0.984\tabularnewline
& & & & & & \tabularnewline
& \multicolumn{3}{c}{\uline{\mbox{$\theta=2,\quad v\in[0.3,0.7]$}}} & \multicolumn{3}{c}{\uline{\mbox{$\theta=2,\quad v\in[0.2,0.8]$}}}\tabularnewline
& & & & & & \tabularnewline
$3$ & 0.912 & 0.954 & 0.994 & 0.900 & 0.944 & 0.988\tabularnewline
$5$ & 0.882 & 0.938 & 0.990 & 0.872 & 0.932 & 0.990\tabularnewline
$7$ & 0.916 & 0.950 & 0.986 & 0.914 & 0.964 & 0.994\tabularnewline
& & & & & & \tabularnewline
\bottomrule
\end{tabular}
\end{table}
In Table \ref{tab:Coverage-probabilities}, we report our simulation
results for the bootstrap-based IGA uniform confidence band $CB^{*}$.
We find that the IGA bootstrap approach provides accurate coverage
probabilities. Additional simulation results are reported in the Supplement.
\section{Conclusion\label{sec:Conclusions}}
The GPV estimator has proven to be the essential input in virtually
all nonparametric structural auction models. By proving the asymptotic
normality and the first-order validity of the bootstrap uniform confidence
bands, this paper completes the econometric theory of the GPV estimator
and opens way to new applications.
Our pointwise asymptotic normality results can be used for inference
on an important policy variable: the optimal reserve price. As discussed,
e.g., in \citet{haile2003iim}, the optimal reserve price $r(\boldsymbol{x})$
in auctions with $\boldsymbol{X}_{l}=\boldsymbol{x}$ satisfies the
following equation:
\[
r(\boldsymbol{x})-\frac{1-F(r(\boldsymbol{x})|\boldsymbol{x})}{f(r(\boldsymbol{x})|\boldsymbol{x})}=c(\boldsymbol{x}),
\]
where $c(\boldsymbol{x})$ is the seller's own valuation. Suppose
that the estimator $\widehat{r}(\boldsymbol{x})$ is constructed by
solving an estimated version of the above equation with $f(\cdot|\boldsymbol{x})$
replaced by its GPV estimator $\widehat{f}_{GPV}\left(\cdot|\boldsymbol{x}\right)$.
In that case, our pointwise normality results imply the asymptotic
normality of the estimated optimal reserve price:
\begin{multline*}
\left(Lh^{3+d}\right)^{\nicefrac{1}{2}}\left(\widehat{r}\left(\boldsymbol{x}\right)-r(\boldsymbol{x})\right)\\
\rightarrow_{d}\mathrm{N}\left(0,\left(\frac{1-F(r(\boldsymbol{x})|\boldsymbol{x})}{2f(r(\boldsymbol{x})|\boldsymbol{x})^{2}+f'(r(\boldsymbol{x})|\boldsymbol{x})\left(1-F(r(\boldsymbol{x})|\boldsymbol{x})\right)}\right)^{2}\mathrm{V}_{GPV}\left(r(\boldsymbol{x})|\boldsymbol{x}\right)\right).
\end{multline*}
Our results for the validity of the percentile bootstrap of the GPV
estimator naturally carry over to the above estimator of the optimal
reserve price. Thus in practice, one can use the percentile bootstrap
for inference on the optimal reserve price.
Our uniform confidence bands can be used for specification of the
density of valuations.
In future research, our results could be extended in several directions.
Below, we briefly describe some of potentially interesting extensions.
These extensions would address the limitations of the independent
private values model that underlies the GPV estimator.
First, in GPV the bidders are treated symmetrically. Empirically this
is not always the case. See, e.g., \citet{flambard2006asymmetry}
for an application to snow removal contracts. Second, we abstract
from the empirically relevant issue of unobserved heterogeneity, as
in \citet{krasnokutskaya2011identification}, \citet{hu2013identification}
and \citet{roberts2013unobserved}. Third, correlated values may also
be important empirically. \citet{li2002structural} and \citet{hubbard2012semiparametric}
extend the GPV estimator to the affiliated value environment. Fourth,
\citet{guerre2009nonparametric} and \citet{zincenko2018nonparametric}
provide extensions to an environment with risk-averse bidders. Fifth,
the literature on endogenous entry in auctions has developed rapidly.
See, e.g., \citet{li2009entry}, \citet{krasnokutskaya2011bid}, \citet{marmer2013model},
\citet{roberts2013should} and \citet{gentry2014identification}.
In the above models, the basic GPV estimator is often adapted to suit
the needs of a particular application. But most of these estimators
share the underlying two-step structure of the GPV estimator. We conjecture
that our main results will also prove useful for establishing the
asymptotic normality and validity of certain uniform confidence bands
for these GPV-like estimators.
A remaining unresolved important practical issue is bandwidth selection
for the GPV estimator. It is possible that the recent advances in
that area (e.g., \citealp{calonico2014robust} and \citealp{armstrong2016simple})
can be adapted to the framework of GPV.
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