Monotonicity-Constrained Nonparametric Estimation and Inference for First-Price Auctions

Monotonicity-Constrained Nonparametric Estimation and Inference for First-Price Auctions

Introduction

Shape restrictions on infinite-dimensional parameters have received much attention in econometric research. The roles of shape restrictions in the literature include facilitating identification, providing testable implications and improving estimation and inference. See chetverikov2018econometrics for a recent review.

This paper focuses on the first-price sealed-bid auction model with symmetric bidders, which is the same as that studied in vuong2000one. GPV's estimation strategy uses a nonparametrically estimated inverse bidding function to generate pseudo valuations. However, the true bidding strategy must be strictly increasing and the plug-in nonparametric estimator of GPV ignores this shape restriction. Imposing such a constraint at the estimation stage nonparametrically is interesting. For this purpose, we may use the methods in the monotone nonparametric regression literature.\footnote{See henderson2009imposing for a comprehensive survey of the literature on monotone nonparametric regression.} Some of these methods can be easily adapted to producing constrained estimators for the inverse bidding strategy in the auction setting. E.g., henderson_2012_JOE take the constrained re-weighting approach pioneered by hall2001nonparametric, which was originally used to build a monotone estimator for the conditional expectation function. luo2017integrated impose monotonicity by using the greatest convex minorant of the integrated quantile function of values.

In this paper, we pursue a different approach. We investigate the asymptotic properties of a new monotonicity-constrained estimator based on the smooth rearrangement approach of dette2006simple and propose a uniform confidence band around this monotonicity-constrained nonparametric estimator. We show that this rearrangement-based monotonicity-constrained estimator is asymptotically normal with an asymptotic variance smaller than the unconstrained estimator as in GPV. Since the asymptotic variance plays an important role in determining the width of a uniform confidence band in large samples, the fact that the rearrangement-based estimator has a smaller asymptotic variance will result in sharper inference. Our estimator also has substantial computational advantage over henderson_2012_JOE.\footnote{See dette2006comparative for simulation studies that compare the rearrangement and reweighting approaches to monotone regression. dette2006comparative notice that the rearrangement approach has computational advantage since the reweighting approach requires solving constrained optimization.}

As a by-product, our method also produces a simple estimator for the true bidding function. Note that GPV's procedure is based on the inverse-bidding strategy, which has a simple form. On the other hand, the bidding function has an integral expression that depends on the unknown distribution of latent valuations and constructing its direct plug-in type estimator would be cumbersome. Our simple estimator of the bidding function can be of interest on its own, as it can be used in practical applications for computing counterfactual bids.

We compare the finite sample performances of the confidence bands based on the rearrangement-based and the unconstrained estimators in Monte Carlo experiments. We find that the confidence band based on the rearrangement-based estimator tends to be narrower without sacrificing the coverage accuracy.

The literature on structural econometrics of auctions is vast. See gentry2018structural for a recent review; see also the reviews of the literature in athey2007nonparametric and hendricks2007empirical. In their seminal paper, GPV demonstrate nonparametric identification of the first-price auction model with independent private values, and propose two-step nonparametric estimation of the density of latent valuations. This paper improves on the GPV estimator by incorporating the monotonicity constraint in the nonparametric estimation procedure.

In a recent paper, ma2016inference describe the asymptotic distribution of the GPV estimator and propose a valid bootstrap procedure based on the GPV estimator. They also propose a procedure for constructing uniform confidence bands for the density of latent valuations. This paper builds on their results.

The GPV estimator has been widely used in the empirical literature. For examples of applications, see the literature review in athey2007nonparametric and hendricks2007empirical. The GPV approach has been also utilized in models with risk aversion (guerre2009nonparametric, zincenko2018nonparametric), unobserved heterogeneity (krasnokutskaya2011identification), bidder asymmetry and affiliated values (li2002sea), common values (haile2003nonparametric, hendricks2003empirical), and entry (li2009entry, marmer2013model, gentry2014identification).

The recent related literature includes liuvuong2013, who propose a test for the monotonicity of the bidding function, liu2017nonparametric, who propose a procedure for comparing valuation distributions, Marmer_Shneyerov_Quantile_Auctions, luo2017integrated, gimenes2017econometrics, who propose quantile based methods in the context of auctions. Our paper is also related to the econometrics literature on two-step nonparametric estimation. See, e.g., mammen2012nonparametric.

The rest of the paper is organized as follows. Section (ref) introduces the empirical auction model studied in this paper and its estimation technique, including the GPV estimator and a new monotonicity-constrained estimator. In Section (ref), we show that the new estimator is asymptotically normal with a smaller asymptotic variance, compared to the unconstrained estimator. Section (ref) provides an estimator for the asymptotic variance and a uniform confidence band around the new monotonicity-constrained estimator. We extend the proposed estimation and inference method to an auction model with observed auction heterogeneity in Section (ref). Section (ref) reports Monte Carlo simulation results. Proofs are collected in the appendix.

The Auction Model and Estimation

In this section, we consider an auction model for homogeneous goods and with a fixed number of bidders. A model with observed covariates capturing auction-specific heterogeneity will be considered in Section (ref). The econometrician observes bids from $L$ auctions, with a fixed number of bidders in each auction:

Bidders' valuations \[ \left\{ V_{il}:i=1,\ldots,N,l=1,\ldots,L\right\} \] are not unobservable to the econometrician. We assume the distribution of the valuations satisfy the following assumption.

Assumption (ref)(a) assumes that the bidders are symmetric and the auctions are identical. Assumption (ref) is similar to Assumptions A1 and A2 of GPV and Assumption 1 of MMS. The object of interest is the PDF of the valuations 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 $\left[\underline{v},\overline{v}\right]$.

We assume that the observed bids are generated from the valuations and by the Bayesian Nash equilibrium (BNE) bidding strategy:

The BNE requires that the bidding strategy $s$ is strictly increasing. Moreover, GPV show that $s$ is at least three times continuously differentiable on $\left(\underline{v},\overline{v}\right)$. The inverse of the BNE bidding strategy can be written as

where $G$ and $g$ are CDF and PDF of the bids, respectively. Let $\overline{b}\coloneqq s\left(\overline{v}\right)$ and $\underline{b}\coloneqq s\left(\underline{v}\right)$ denote the boundaries of support for the observed i.i.d. bids. GPV show that $g$ is three-times continuously differentiable and also bounded away from zero on its support $\left[\underline{b},\overline{b}\right]$:

Let $\widehat{G}$ be the empirical CDF of the bids: \[ \widehat{G}\left(b\right)\coloneqq\frac{1}{N\cdot L}\sum_{i,l}\mathbbm{1}\left(B_{il}\leq b\right) \] and $\widehat{g}$ be the kernel density estimator of $g$:

with some bandwidth $h_{g}>0$ and kernel $K_{g}$. Therefore, the plug-in nonparametric estimator of the inverse bidding strategy $\xi$ is

For each $B_{il}$, we construct a pseudo valuation by $\widehat{V}_{il}\coloneqq\widehat{\xi}\left(B_{il}\right)$. There is a boundary bias issue when ordinary kernel density estimator as in ((ref)) is used. Thus, the pseudo valuations corresponding to bids in boundary regions are contaminated. GPV propose to trim off bids that lie in $\left[\widehat{\underline{b}},\widehat{\underline{b}}+h_{g}\right)\cup\left(\widehat{\overline{b}}-h_{g},\widehat{\overline{b}}\right]$, where

We modify the kernel density estimator $\widehat{g}\left(b\right)$ in the boundary region $b\in\left[\underline{b},\underline{b}+h_{g}\right)\cup\left(\overline{b}-h_{g},\overline{b}\right]$ to avoid boundary bias and trimming.\footnote{See hickman_hubbard_2014_JAE for more discussion on why we should avoid trimming when estimating the auction model using GPV approach.} Another concern is that, in order to remove more bias, we need to make use of the fact that the bid density $g$ is smoother than the valuation density $f$ and use a higher-order kernel when estimating the inverse bidding strategy.\footnote{See Remark 2.3 of MMS.} A suitable choice is the local quadratic minimum contrast estimator (MCE). See bickel2015mathematical.\footnote{See jales2017optimal and Ma_EL_MC for more recent applications of MCE in econometrics.} Similar to the local polynomial regression, the local quadratic MCE automatically adapts to the boundary so that the rate of the bias is the same in the boundary region and the interior.\footnote{Note that the MCE requires knowledge of the locations of the endpoints $\underline{b}$ and $\overline{b}$. Since the estimators $\widehat{\underline{b}}$ and $\widehat{\overline{b}}$ are super-consistent: $\widehat{\underline{b}}=\underline{b}+O_{p}\left(\mathrm{log}\left(L\right)/L\right)$ and $\widehat{\overline{b}}=\overline{b}+O_{p}\left(\mathrm{log}\left(L\right)/L\right)$, we can replace the unknown endpoints in the MCE with these estimators without affecting the validity of the asymptotic results.} The local quadratic MCE coincides with the kernel density estimator ((ref)) with a fourth-order kernel, and therefore we achieve desired bias removal in the interior region $\left[\underline{b}+h_{g},\overline{b}-h_{g}\right]$. The GPV estimator is \[ \widehat{f}_{GPV}\left(v\right)\coloneqq\frac{1}{N\cdot L}\sum_{i,l}\frac{1}{h_{f}}K_{f}\left(\frac{\widehat{V}_{il}-v}{h_{f}}\right), \] for some bandwidth $h_{f}>0$ and kernel $K_{f}$. We note again that trimming can be avoided.

Another important observation is that the plug-in estimator ((ref)) may not be monotone in finite samples, although its population counterpart $\xi$ is strictly increasing under the assumption that the empirical auction model is correctly-specified. We apply smooth rearrangement to build a new monotonicity-constrained estimator of $\xi$ on the plug-in estimator $\widehat{\xi}$. Define

for some bandwidth $h_{r}>0$ and (second-order) kernel function $K_{r}$. Denote $\widetilde{K}_{r}\left(u\right)\coloneqq\int_{-\infty}^{u}K_{r}\left(t\right)\mathrm{d}t$. Alternatively we can write

((ref)) contains less challenge in computation as the expression of $\widetilde{K}_{r}$ is always available for standard kernel functions.

It is clear that $\widehat{s}$ is increasing on $\mathbb{R}$ and \[ \widehat{s}\left(t\right)=

\] It is easy to see that $\widehat{s}$ can be viewed as an estimator of the bidding function $s$ (see Lemma (ref)). Note that this estimator is of interest on its own, as the bidding function $s$ is an important structural object. Moreover, constructing an estimator of $s$ directly from its expression in ((ref)) can be cumbersome.

Let $\widehat{s}^{-1}$ denote the pseudo inverse of $\widehat{s}$: \[ \widehat{s}^{-1}\left(b\right)\coloneqq\mathrm{inf}\left\{ u\in\mathbb{R}:\widehat{s}\left(u\right)\geq b\right\} . \] $\widehat{s}^{-1}$ is a rearrangement-based estimator of $\xi$ with the monotonicity constraint imposed. A new modified GPV estimation procedure now can be proposed: First, we construct $\widehat{\xi}$, the plug-in nonparametric estimator of the inverse bidding strategy $\xi$. To avoid trimming, we use the local quadratic MCE instead of the ordinary kernel density estimator. Then, we construct the monotonicity-imposed estimator of the inverse bidding strategy: $\widehat{s}^{-1}$ and generate monotonicity-constrained pseudo valuations $\widehat{V}_{il}^{\dagger}\coloneqq\widehat{s}^{-1}\left(B_{il}\right)$ for $i=1,...,N$, $l=1,...,L$. A new estimator, the rearrangement-based GPV (RGPV), is \[ \widehat{f}_{RGPV}\left(v\right)\coloneqq\frac{1}{N\cdot L}\sum_{i,l}\frac{1}{h_{f}}K_{f}\left(\frac{\widehat{V}_{il}^{\dagger}-v}{h_{f}}\right). \]

The RGPV estimation procedure is computationally more involved than the standard GPV procedure. When implementing it in practice, the integral in ((ref)) can be approximated by an upper Riemann sum. Let $M\in\mathbb{N}$ be a very large number. Let $d\coloneqq\left(\widehat{\overline{b}}-\widehat{\underline{b}}\right)/M$. We can approximate $\int_{\widehat{\underline{b}}}^{\widehat{\overline{b}}}\widetilde{K}_{r}\left(\left(t-\widehat{\xi}\left(b\right)\right)/h_{r}\right)\mathrm{d}b$ by \[ \sum_{i=1}^{M}\widetilde{K}_{r}\left(\frac{t-\widehat{\xi}\left(\widehat{\underline{b}}+i\cdot d\right)}{h_{r}}\right)d. \] However, unlike the estimator in henderson_2012_JOE, it is not required to solve a constrained optimization problem. Thus the RGPV estimation procedure is computationally less demanding than henderson_2012_JOE's estimation procedure, which is based on constrained reweighting.

An alternative, and closely related, approach is to impose the monotonicity restriction through the “non-smooth” rearrangement. Instead of using ((ref)), we can define \[ \widehat{s}_{0}\left(t\right)\coloneqq\int_{\widehat{\underline{b}}}^{\widehat{\overline{b}}}\mathbbm{1}\left(\widehat{\xi}\left(b\right)\leq t\right)\mathrm{d}b+\widehat{\underline{b}},\;t\in\mathbb{R}, \] and use $\widehat{s}_{0}^{-1}$ as a monotonicity-constrained estimator of $\xi$ to generated pseudo valuations. If the empirical auction model is correctly specified so that $\xi$ is strictly increasing, $\widehat{s}_{0}^{-1}$ is always an improvement over the plug-in estimator $\widehat{\xi}$, in the sense that $\widehat{s}_{0}^{-1}$ has a strictly smaller (finite-sample) integrated mean square error whenever $\widehat{\xi}$ is not monotonic. See chernozhukov2009improving. However, for the structural auction model, the parameter of interest is the density $f$. It is unclear whether the density estimator based on pseudo valuations generated by $\widehat{s}_{0}^{-1}$ is an improvement over the unconstrained GPV estimator theoretically. In this paper, we focus on smooth rearrangement and in the next section, we show that based on the RGPV estimator we could potentially achieve sharper inference in large samples, which can be regarded as theoretical advantage over the unconstrained GPV estimator.

Asymptotic Properties

The following assumptions are imposed on the kernel functions and the bandwidths, respectively.

Assumption (ref) implies that the kernel functions are of second order. When the kernel used in the local quadratic MCE of the bid density is second-order, at the interior points, the MCE is the same as the ordinary kernel density estimator ((ref)) with $K_{g}$ being fourth-order. Assumption (ref) is similar to Assumption 3 in MMS.

It is shown in MMS that under assumptions (ref)-(ref),

where

MMS show that the asymptotic variance ((ref)) can be consistently estimated by some estimator $\widehat{\mathrm{V}}_{GPV}\left(v\right)$ and an asymptotically valid confidence interval for $f\left(v\right)$ can be constructed:

where $z_{1-\alpha/2}$ denotes the $1-\alpha/2$ quantile of the standard normal distribution. Note that $\widehat{\mathrm{V}}_{GPV}\left(v\right)$ and its probabilistic limit $\mathrm{V}_{GPV}\left(v\right)$ play important roles in determining the length of the confidence interval. Furthermore, a bootstrap uniform confidence band is given by \[ CB_{GPV}\left(v\right)\coloneqq\left[\widehat{f}_{GPV}\left(v\right)-\zeta_{GPV,\alpha}\sqrt{\frac{\widehat{\mathrm{V}}_{GPV}\left(v\right)}{Lh_{f}^{2}h_{g}}},\widehat{f}_{GPV}\left(v\right)+\zeta_{GPV,\alpha}\sqrt{\frac{\widehat{\mathrm{V}}_{GPV}\left(v\right)}{Lh_{f}^{2}h_{g}}}\right],\textrm{ for \ensuremath{v\in I}}, \] where $\zeta_{GPV,\alpha}$ is some bootstrap critical value. It can be shown that \[ \mathrm{P}\left[f\left(v\right)\in CB_{GPV}\left(v\right),\textrm{ for all \ensuremath{v\in I}}\right]\rightarrow1-\alpha,\textrm{ as \ensuremath{L\uparrow\infty}}. \]

Next, we show that a similar asymptotic normality result holds for the RGPV estimator, but with a smaller asymptotic variance. The proof uses the same arguments as in the proof of Theorem 2.1 in MMS. First, we derive the following asymptotic representation:

where the remainder term is uniform in $v\in I$ . Moreover, \[ \mathcal{M}\left(b',b;v\right)\coloneqq-\frac{1}{h_{f}^{2}}K_{f}'\left(\frac{\xi\left(b'\right)-v}{h_{f}}\right)\xi'\left(b'\right)\int_{\underline{b}}^{\overline{b}}\frac{1}{h_{r}}K_{r}\left(\frac{\xi\left(b'\right)-\xi\left(u\right)}{h_{r}}\right)\frac{G\left(u\right)}{g\left(u\right)^{2}}\left(\frac{1}{h_{g}}K_{g}\left(\frac{b-u}{h_{g}}\right)-g\left(u\right)\right)\mathrm{d}u. \] Note that this “kernel” is different from that of the unconstrained GPV estimator. See Equation (2.6) of MMS.

Define

Note that $\mu_{\mathcal{M}}\left(v\right)=\mathrm{E}\left[\mathcal{M}_{1}\left(B_{11};v\right)\right]=\mathrm{E}\left[\mathcal{M}_{2}\left(B_{11};v\right)\right]$. Applying Hoeffding decomposition to the leading term in ((ref)) and techniques from the theories of empirical processes and U processes, we can show that \[ \widehat{f}_{RGPV}\left(v\right)-f\left(v\right)=\frac{1}{N-1}\cdot\frac{1}{N\cdot L}\sum_{i,l}\left(\mathcal{M}_{2}\left(B_{il};v\right)-\mu_{\mathcal{M}}\left(v\right)\right)+o_{p}\left(\left(Lh^{3}\right)^{-1/2}\right), \] where the remainder term is uniform in $v\in I$. $\mathcal{M}_{2}\left(B_{il};v\right)-\mu_{\mathcal{M}}\left(v\right)$, $i=1,...,N$, $l=1,...,L$ are independent, zero-mean but dependent on the bandwidths. This term can be shown asymptotically normal.

Inference

If the asymptotic variance $\mathrm{V}_{RGPV}(v)$ in ((ref)) can be consistently estimated by some estimator $\widehat{\mathrm{V}}_{RGPV}\left(v\right)$, Theorem (ref) shows that we can construct an asymptotically valid confidence interval for $f\left(v\right)$:

As shown in ((ref)), our rearrangement-based estimator has a smaller asymptotic variance than the GPV estimator. Therefore, the confidence intervals based on our estimator in ((ref)) should be shorter than those based on the GPV estimator in ((ref)) in large samples.

An estimator of $\mathrm{V}_{RGPV}\left(v\right)$ is derived based on the sample analogue of $\mathrm{V}_{\mathcal{M}}\left(v\right)$, (see ((ref))):

where we use the fact $\xi'\left(b\right)=1/s'\left(\xi\left(b\right)\right)$ and \[ \widehat{s}'\left(v\right)\coloneqq\int_{\widehat{\underline{b}}}^{\widehat{\overline{b}}}\frac{1}{h_{r}}K_{r}\left(\frac{\widehat{\xi}\left(b\right)-v}{h_{r}}\right)\mathrm{d}b. \] The integral can be approximated by an upper Riemann sum in practice. The following result provides uniform consistency of the variance estimator and also an estimate of its uniform rate of convergence. The proof uses the same arguments as in the proof of Theorem 3.1 of MMS.

An alternative approach to constructing pointwise confidence intervals is based on bootstrapping. Let

denote a set of independent random variables drawn from the original sample ((ref)) with replacement. $\widehat{G}^{*}$ and $\widehat{g}^{*}$ denote the bootstrap analogues of $\widehat{G}$ and $\widehat{g}$ respectively. Let $\widehat{\xi}^{*}$ be the bootstrap analogue of $\widehat{\xi}$. $\widehat{\xi}^{*}$ is defined by replacing $\widehat{G}$ and $\widehat{g}$ with $\widehat{G}^{*}$ and $\widehat{g}^{*}$. We further define \[ \widehat{s}^{*}\left(t\right)\coloneqq\int_{\widehat{\underline{b}}}^{\widehat{\overline{b}}}\int_{-\infty}^{t}\frac{1}{h_{r}}K_{r}\left(\frac{\widehat{\xi}^{*}\left(b\right)-u}{h_{r}}\right)\mathrm{d}u\mathrm{d}b+\widehat{\underline{b}},\;t\in\mathbb{R} \] and bootstrap analogues of $\widehat{V}_{il}^{\dagger}$, denoted by $\widehat{V}_{il}^{\dagger*}$, $i=1,\ldots,N$, $l=1,\ldots,L$ by using the pseudo inverse of $\widehat{s}^{*}$. Lastly, we construct a bootstrap analogue of $\widehat{f}_{RGPV}$: \[ \widehat{f}_{RGPV}^{*}\left(v\right)\coloneqq\frac{1}{N\cdot L}\sum_{i,l}\frac{1}{h_{f}}K_{f}\left(\frac{\widehat{V}_{il}^{\dagger*}-v}{h_{f}}\right). \] The percentile bootstrap pointwise confidence interval for $f\left(v\right)$ is \[ \left[q_{\alpha/2}^{*}\left(v\right),q_{1-\alpha/2}^{*}\left(v\right)\right], \] where $q_{\tau}^{*}\left(v\right)$ is the $\tau-$th quantile of the conditional distribution of $\widehat{f}_{RGPV}^{*}\left(v\right)$ given the original sample.

The pointwise inference results for $f(v)$ described above can be extended for the inference on the optimal reserve price, as the latter is a function of the density at the reserve price. Such an extension is discussed in Section 8 in MMS. Similarly, a function of the density $(1-F(v))/f(v)$ is of interest in applications as it represents the markup of the bidder with value $v$. Again, since relatively to GPV's our rearrangement estimator has a smaller asymptotic variance, basing inference on optimal reserve price or the markup on our estimator would result in more powerful tests and shorter confidence intervals.

Next, we show that a bootstrap-based uniform confidence band for $\left\{ f\left(v\right):v\in I\right\} $ centered at the rearrangement-based estimator can be constructed by using intermediate Gaussian approximation pioneered by chernozhukov2014gaussian,chernozhukov2014anti,chernozhukov2016empirical. Consider the following bootstrap process

Let $\mathrm{P}^{*}\left[\cdot\right]$ denote the probability conditional on the original sample and \[ \zeta_{RGPV,\alpha}\coloneqq\mathrm{inf}\left\{ z\in\mathbb{R}:\mathrm{P}^{*}\left[\left\Vert Z^{*}\right\Vert _{I}\leq z\right]\geq1-\alpha\right\} \] be the $\left(1-\alpha\right)$-quantile of the conditional distribution of $\left\Vert Z^{*}\right\Vert _{I}$ given the original sample. A uniform confidence band around the rearrangement-based estimator is

\[ CB_{RGPV}\left(v\right)\coloneqq\left[\widehat{f}_{RGPV}\left(v\right)-\zeta_{RGPV,\alpha}\sqrt{\frac{\widehat{\mathrm{V}}_{RGPV}\left(v\right)}{Lh_{f}^{2}h_{g}}},\,\widehat{f}_{RGPV}\left(v\right)+\zeta_{RGPV,\alpha}\sqrt{\frac{\widehat{\mathrm{V}}_{RGPV}\left(v\right)}{Lh_{f}^{2}h_{g}}}\right],\,\textrm{for \ensuremath{v\in I}}. \] The following theorem establishes the asymptotic validity of $CB_{RGPV}$. Its proof uses the same arguments as in the proof of Corollary 4.3 of MMS.

\[ \mathrm{P}\left[f\left(v\right)\in CB_{RGPV}\left(v\right),\textrm{ for all \ensuremath{v\in I}}\right]\rightarrow1-\alpha,\textrm{ as \ensuremath{L\uparrow\infty}}. \]

Auction-Specific Heterogeneity

In previous sections, we focused on the case of identical auctions with a fixed number of bidders. In this section, we consider auction models with auction-specific heterogeneity. 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 are denoted by \[ \left\{ V_{il}:i=1,...,N_{l},\,l=1,...,L\right\} . \]

Assume that $\left\{ \left(\boldsymbol{X}_{l},N_{l}\right):l=1,...,L\right\} $ are i.i.d. and for each $l=1,...,L$, given $\boldsymbol{X}_{l}=\boldsymbol{x}$ and $N_{l}=n$, the valuations $\left\{ V_{il}:i=1,...,n\right\} $ are i.i.d. with conditional PDF $f\left(\cdot|\boldsymbol{x}\right)$. We follow the literature and assume that the valuations and the number of bidders $N_{l}$ are conditionally independent given the observed characteristics $\boldsymbol{X}_{l}$.\footnote{For a test of this assumption, see liu2017nonparametric.} Also assume that the conditional probability mass function of $N_{l}$ given $\boldsymbol{X}_{l}$ has a known support $\left\{ \underline{n},...,\overline{n}\right\} $.

The observed bid $B_{il}$ is assumed from the Bayesian Nash equilibrium bidding for risk-neutral bidder $i$ submitted in the $l$-th auction. 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 conditional PDF. The inverse bidding strategy in this context becomes

For estimation and inference, we can generate pseudo valuations from ((ref)) by replacing the true conditional CDF and PDF by their kernel estimators. The GPV estimator can be defined analogously in this general context. Asymptotically valid pointwise confidence intervals and uniform confidence bands for $f\left(\cdot|\boldsymbol{x}\right)$ can be constructed. See Section 5 of MMS for more details.

Following haile2003nonparametric, we use a semi-parametric approach to homogenize the bids.\footnote{The homogenization approach is used extensively in the literature, e.g. athey2004csb, liu2017nonparametric, luo2017integrated, and many others.} Let

denote positive i.i.d. idiosyncratic values that are independent from the auction-specific characteristics $\boldsymbol{X}_{l}$, $l=1,...,L$. Let $F_{\epsilon}$ be its CDF and $\left[\underline{\epsilon},\overline{\epsilon}\right]$ be its support. Define \[ \widetilde{B}_{il}=s\left(\epsilon_{il},N_{l}\right)\coloneqq\epsilon_{il}-\frac{1}{F_{\epsilon}\left(\epsilon_{il}\right)^{N_{l}-1}}\int_{\underline{\epsilon}}^{\epsilon_{il}}F_{\epsilon}\left(u\right)^{N_{l}-1}\mathrm{d}u. \] Let \[ \widetilde{G}\left(b|n\right)\coloneqq\mathrm{P}\left[\widetilde{B}_{il}\leq b|N_{l}=n\right] \] be the conditional CDF and $\widetilde{g}\left(b|n\right)$ be the corresponding conditional PDF. Note that we have

The approach of haile2003nonparametric assumes that, for some parametric function $\varUpsilon$ to be specified below, $V_{il}=\varUpsilon\left(\boldsymbol{X}_{l}\right)\epsilon_{il}$ for $i=1,...,N_{l}$ and $l=1,...,L$. It then can be shown that the conditional CDF and PDF of $\varUpsilon\left(\boldsymbol{X}_{l}\right)\widetilde{B}_{il}$ (denoted by $\widetilde{G}_{\varUpsilon}$ and $\widetilde{g}_{\varUpsilon}$, respectively) satisfy \[ \widetilde{G}_{\varUpsilon}\left(b|\boldsymbol{X}_{l},N_{l}\right)\coloneqq\mathrm{P}\left[\varUpsilon\left(\boldsymbol{X}_{l}\right)\widetilde{B}_{il}\leq b|\boldsymbol{X}_{l},N_{l}\right]=\widetilde{G}\left(\frac{b}{\varUpsilon\left(\boldsymbol{X}_{l}\right)}|N_{l}\right) \] and \[ \widetilde{g}_{\varUpsilon}\left(b|\boldsymbol{X}_{l},N_{l}\right)=\widetilde{g}\left(\frac{b}{\varUpsilon\left(\boldsymbol{X}_{l}\right)}|N_{l}\right)\frac{1}{\varUpsilon\left(\boldsymbol{X}_{l}\right)}. \] It is clear from these results and ((ref)) that \[ \varUpsilon\left(\boldsymbol{X}_{l}\right)\epsilon_{il}=\varUpsilon\left(\boldsymbol{X}_{l}\right)s\left(\epsilon_{il},N_{l}\right)+\frac{1}{N_{l}-1}\frac{\widetilde{G}_{\varUpsilon}\left(\varUpsilon\left(\boldsymbol{X}_{l}\right)s\left(\epsilon_{il},N_{l}\right)|\boldsymbol{X}_{l},N_{l}\right)}{\widetilde{g}_{\varUpsilon}\left(\varUpsilon\left(\boldsymbol{X}_{l}\right)s\left(\epsilon_{il},N_{l}\right)|\boldsymbol{X}_{l},N_{l}\right)}, \] which implies that $B_{il}=\varUpsilon\left(\boldsymbol{X}_{l}\right)\widetilde{B}_{il}$, for $i=1,...,N_{l}$ and $l=1,...,L$.

Now, we write

where \[ \alpha\left(N_{l}\right)\coloneqq\mathrm{E}\left[\mathrm{log}\left(s\left(\epsilon_{il},N_{l}\right)\right)|N_{l}\right]\textrm{ and \ensuremath{U_{il}\coloneqq}\ensuremath{\mathrm{log}\left(s\left(\epsilon_{il},N_{l}\right)\right)}}-\alpha\left(N_{l}\right). \] It is easy to check that $\mathrm{E}\left[U_{il}|\boldsymbol{X}_{l},N_{l}\right]=0$. Since $N_{l}$ is discrete, we can write \[ \alpha\left(N_{l}\right)=\sum_{n=\underline{n}}^{\overline{n}}\alpha_{n}\mathbbm{1}\left(N_{l}=n\right). \] We assume that the function $\varUpsilon$ is log-linear in parameters: $\mathrm{log}\left(\varUpsilon\left(\boldsymbol{X}_{l}\right)\right)=\boldsymbol{X}_{l}^{\mathrm{T}}\boldsymbol{\beta}$ for some unknown $\boldsymbol{\beta}$. Now ((ref)) can be written as \[ \mathrm{log}\left(B_{il}\right)=\sum_{n=\underline{n}}^{\overline{n}}\alpha_{n}\mathbbm{1}\left(N_{l}=n\right)+\boldsymbol{X}_{l}^{\mathrm{T}}\boldsymbol{\beta}+U_{il}. \]

Regressing the log-bids on the covariates and the indicators for the number of bidders yields an estimator $\widehat{\boldsymbol{\beta}}$ of $\boldsymbol{\beta}$. Then, the homogenized bids are given by

These bids can be interpreted as the bid that would have been submitted by the $i$-th bidder if the covariates were equal to $\boldsymbol{x}_{0}$, in the $l$-th auction. Suppose we are interested in inference on $f\left(\cdot|\boldsymbol{x}_{0}\right)$ for some fixed $\boldsymbol{x}_{0}$.\footnote{For example, $\boldsymbol{x}_{0}$ can be taken to be the sample mean $L^{-1}\sum_{l=1}^{L}\boldsymbol{X}_{l}$. See haile2003nonparametric.}

Let

$\widehat{s}\left(\cdot,n\right)$ can also be defined analogously and $\widehat{s}^{-1}\left(\cdot,n\right)$ is its pseudo inverse. Let $\widehat{V}_{il}^{0\dagger}\coloneqq\widehat{s}^{-1}\left(B_{il}^{0},N_{l}\right)$ be the monotonicity-constrained pseudo valuations.

A semi-parametric estimator of $f\left(\cdot|\boldsymbol{x}_{0}\right)$ is \[ \widehat{f}\left(v|\boldsymbol{x}_{0}\right)\coloneqq\frac{1}{L}\sum_{l=1}^{L}\frac{1}{N_{l}}\sum_{i=1}^{N_{l}}\frac{1}{h_{f}}K_{f}\left(\frac{\widehat{V}_{il}^{0\dagger}-v}{h_{f}}\right), \] which is asymptotically normal. Its asymptotic variance can be consistently estimated by

where

and \[ \widehat{\overline{b}}_{n}^{0}\coloneqq\underset{\left(i,l\right):N_{l}=n}{\mathrm{max}}B_{il}^{0}\textrm{ and \ensuremath{\widehat{\underline{b}}_{n}^{0}\coloneqq\underset{\left(i,l\right):N_{l}=n}{\mathrm{min}}B_{il}^{0}}}. \]

A pointwise confidence interval for $f\left(v|\boldsymbol{x}_{0}\right)$ that is of the same form as ((ref)) can be proved to be asymptotically valid. For bootstrap resampling, we treat the homogenized bids ((ref)) as our observed bids and apply the two-step resampling procedure provided by Marmer_Shneyerov_Quantile_Auctions. In each bootstrap replication, we first randomly draw $L$ observations from $\left\{ N_{l}:l=1,...,L\right\} $ with replacement. Next, we randomly draw bids with replacement from bids corresponding to the selected number of bidders. If for the $l$-th observation in the bootstrap sample, we have $N_{l}^{*}=N_{l'}$, then let $\left\{ B_{il}^{0*}:i=1,...,N_{l}^{*}\right\} $ be i.i.d. draws from all the bids in auctions with number of bidders being $N_{l'}$ with replacement. Now the bootstrap sample is $\left(B_{il}^{0*},N_{l}^{*}\right)$, $i=1,...,N_{l}^{*}$ and $l=1,...,L$. Then it is straightforward to construct the bootstrap analogue of $\widehat{f}\left(\cdot|\boldsymbol{x}_{0}\right)$ and a bootstrap-based uniform confidence band for $f\left(\cdot|\boldsymbol{x}_{0}\right)$ can be constructed analogously.

Monte Carlo Simulations

In this section, we assess the finite-sample performances of the uniform confidence bands based on both the unconstrained GPV estimator and the rearrangement-based monotonicity-constrained estimator proposed in this paper. Our simulation design follows Marmer_Shneyerov_Quantile_Auctions and the DGP is described in Remark (ref). We consider $\theta=1$ and draw 2100 independent valuations from $F_{\theta}$. In all these cases, the number of bidders $N$ is constant. The number of auctions is determined by $N\cdot L=2100$. We choose $K_{r}=K_{f}$ to be the second-order triweight kernel. For estimation of the inverse bidding strategy, we use the second-order triweight kernel in the MCE. In this case, the kernel $K_{g}$ in the expressions of the asymptotic variances is the fourth-order triweight kernel.

We use the same bandwidths as in GPV. We take $h_{g}=3.72\cdot\widehat{\sigma}_{b}\cdot\left(N\cdot L\right)^{-1/5}$ when estimating the inverse bidding strategy. $\widehat{\sigma}_{b}$ is the estimated standard deviation of the observed bids. We use $h_{f}=3.15\cdot\widehat{\sigma}_{v}\cdot\left(N\cdot L\right)^{-1/5}$ as the second-step bandwidth, where $\widehat{\sigma}_{v}$ is the estimated standard deviation of the (unconstrained) pseudo valuations. The constants $3.72$ and $3.15$ are Silverman's rule-of-thumb constants corresponding to fourth-order and second-order triweight kernels. When imposing monotonicity, we take $h_{r}=h_{f}$. We consider different numbers of bidders $N\in\left\{ 3,5,7\right\} $, and also the density function over the ranges $v\in\left[0.2,0.8\right]$ and $v\in\left[0.3,0.7\right]$. When computing the bootstrap-based critical values, we set the number of bootstrap replications to 499 and use grid maximization over the grid $\left[v_{l}:0.001:v_{u}\right]$.\footnote{We also tried a finer grid $\left[v_{l}:0.0001:v_{u}\right]$, which produced similar results.}

The main result of this paper is that imposing monotonicity using smooth rearrangement results in more efficient inference. Therefore, in addition to assessing coverage accuracy, we also report the ratio of the supremum widths of the confidence bands: \[ W_{GPV}\coloneqq\underset{v\in I}{\mathrm{sup}}\,2\cdot\zeta_{GPV,\alpha}\sqrt{\frac{\widehat{\mathrm{V}}_{GPV}\left(v\right)}{Lh_{f}^{2}h_{g}}}\textrm{ and }W_{RGPV}\coloneqq\underset{v\in I}{\mathrm{sup}}\,2\cdot\zeta_{RGPV,\alpha}\sqrt{\frac{\widehat{\mathrm{V}}_{RGPV}\left(v\right)}{Lh_{f}^{2}h_{g}}}. \]

Table (ref) reports the coverage probabilities as well as the relative supremum width of the confidence bands based on the GPV and our rearrangement estimators. The coverage probabilities of both methods are similar and accurate. However, our approach can produce considerably smaller confidence with the reduction in supremum width ranging from 6.8% to 33.4%. More substantial reductions in supremum width are obtained for smaller numbers of bidders. This is due to the inverse relationship between the number of bidders and the asymptotic variance of the estimators.

Conclusion

The GPV nonparametric identification and estimation approach is an indispensable working tool in structural econometrics of auctions. This paper contributes to the literature by showing how one can reduce the asymptotic variance of GPV-type estimators by incorporating the monotonicity constraint in estimation. While monotonicity-constrained estimators have been previously considered in the auction literature, to the best of our knowledge ours is the first to obtain a reduction in asymptotic variance in this context. Our method is simple to implement, and as a by-product, it also produces a simple estimator for the bidding function. We also discuss construction of uniform confidence bands for the density of valuations. In a simulation study, we show that by applying our approach one can increase the precision of the confidence bands without sacrificing their coverage in finite samples.