A NONPARAMETRIC TEST FOR COMPARING VALUATION DISTRIBUTIONS IN FIRST‐PRICE AUCTIONS

• Author: Yao Luo,Nianqing Liu
• DOI: http://doi.org/10.1111/iere.12238
• Date: 01 August 2017
• Published date: 01 August 2017

INTERNATIONAL ECONOMIC REVIEW

Vol. 58, No. 3, August 2017

A NONPARAMETRIC TEST FOR COMPARING VALUATION DISTRIBUTIONS

IN FIRST-PRICE AUCTIONS∗

BYNIANQING LIU AND YAO LUO1

Shanghai University of Finance and Economics, China,Key Laboratory of Mathematical

Economics (SUFE), Ministry of Education of China, China; University of Toronto, Canada

This article proposes a nonparametric test for comparing valuation distributions in first-price auctions. Our

test is motivated by the fact that two valuation distributions are the same if and only if their integrated quantile

functions are the same. Our method avoids estimating unobserved valuations and does not require smooth

estimation of bid density. We show that our test is consistent against all fixed alternatives and has nontrivial

power against root-N local alternatives. Monte Carlo experiments show that our test performs well in finite

samples. We implement our method on data from U.S. Forest Service timber auctions.

1. INTRODUCTION

We propose a nonparametric test for comparing valuation distributions in first-price auctions.

One important application is to justify the exogenous participation assumption, which has

been adopted to identify various auction models, such as first-price auctions with risk aversion

(Guerre et al., 2009), ascending auctions (Aradillas-L ´

opez et al., 2013), and first-price auctions

under ambiguity (Aryal et al., 2015).2Many testing problems in auctions also reduce to the

standard form of comparing valuation distributions, such as detecting collusion (Aryal and

Gabrielli, 2012), distinguishing private value and common value auctions (Haile et al., 2003),

and testing different models of entry (Marmer et al., 2013).

Testing for the equality of distributions is a standard problem in statistics. Classic examples

of such tests are the Kolmogorov–Smirnov test, the Cramer–von Mises test, and the Anderson–

Darling test.3In first-price auctions, complications arise from the fact that valuations are esti-

mated instead of observed directly. In a seminal paper, Guerre et al. (2000) transformed the

first-order conditions (FOC) for optimal bids into an expression of bidder’s value as an ex-

∗Manuscript received September 2014; revised May 2016.

1We are deeply indebted to Quang Vuong and Isabelle Perrigne for their advice and encouragement when we

were studying at Penn State. We would like to thank the editor Hanming Fang and three anonymous referees for

comments that have greatly improved the article. We also thank Chunrong Ai, Yanqin Fan, Jiti Gao, LungFei Lee,

Arthur Lewbel, Yong Li, Jun Ma, Vadim Marmer, Joris Pinkse, Yuanyuan Wan, Adam Chi Leung Wong, Ping Yu,

and seminar/conference participants at the 2014 North American Summer Meeting of the Econometric Society, the

9th GNYEC, the 31st CESG, the 2015 World Congress of Econometric Society, the 2015 Shanghai Econometrics

Workshop, Fudan University, Shandong University, and Renmin University of China for helpful comments. Zhe

Yuan provided excellent research assistance. The authors, respectively, acknowledge financial support from the Key

Laboratory of Mathematical Economics (SUFE), Ministry of Education of China (201309KF02), and the National

Natural Science Foundation of China (NSFC-71373283 and NSFC-71463019). The usual disclaimer applies. Please

address correspondence to: Yao Luo, 322 Max Gluskin House, 150 St. George St., Toronto, Ontario M5S 3G7,

Canada. Phone: +1-416-9465288. E-mail: yao.luo@utoronto.ca;or to: Nianqing Liu, 205 School of Economics, Shanghai

University of Finance and Economics, 777 Guoding Road, Shanghai 200433, China. Phone: +86-21-35306324. E-mail:

nliu@sufe.edu.cn.

2Exogenous participation means that the valuation distribution does not depend on the number of bidders (see

Athey and Haile, 2002), namely, F(v|I1)=··· =F(v|IK)forallv∈[v,v].

3One recent related example is Barrett et al. (2014), who compared the mean-standardized Integrated-Quantile

Functions (IQFs; i.e., IQFs divided by their corresponding means) of two directly observed samples. Notice that

equality of distributions implies equality of mean-standardized IQFs, but the reverse is not true.

857

C

(2017) by the Economics Department of the University of Pennsylvania and the Osaka University Institute of Social

and Economic Research Association

858 LIU AND LUO

plicit function of the submitted bid, the bid probability density function (PDF), and the bid

cumulative distribution function (CDF). The authors then used this function to estimate each

bidder’s valuation by recovering the PDF and CDF of the bids. In principle, by treating these

estimated values as a pseudo sample, one can adapt the existing tests of distributional equality

mentioned above (e.g., Kolmogorov–Smirnov test) to obtain a two-step testing procedure in

first-price auctions. In the first step, pseudo values are estimated; in the second step, existing

tests of distributional equality are applied to the sample of pseudo values.

However, such a two-step testing procedure has at least two complications. First, it introduces

(finite sample) dependence among pseudo values by estimating the bid CDF and PDF with the

same sample of bids in the first step. Such a dependence brings technical difficulty in establish-

ing the asymptotic validity of existing tests of distributional equality. Second, besides adding

estimation error, the first-step recovery of pseudo values usually involves the complications of

choosing a bandwidth and trimming in smooth estimation of bid density.4See also discussion

in Haile et al. (2003).

The nonparametric test for distributional equality we propose here avoids the construction of

pseudo valuations in first-price auctions. Our approach is motivated by a simple yet profoundly

useful idea: Two valuation distributions are the same if and only if their IQFs are the same.

We show that the bidders’ FOC allows us to express the IQF of a valuation distribution as a

simple linear functional of the quantile function of the bids. In light of this observation, we

propose a test statistic measuring the square of the L2-distance between the sample analogues

of this linear functional for two bid samples. Consequently, our test statistic only involves the

two empirical quantile functions of the bids.

Our test has two attractive features. First, the test statistic is calculated in one step, which

allows us to characterize its asymptotic properties conveniently under regularity conditions. In

particular, we show that the test statistic converges to the square of the L2-norm of a Gaussian

process with mean zero at a parametric rate under the null hypothesis. We also show that

our test is consistent against any fixed alternative and can detect local alternatives converging

to the null hypothesis at a rate of root-N. Second, our test statistic is easy to calculate, as it

involves no density estimation. Moreover, since the empirical quantile function of bids is a step

function, the empirical counterpart of the integrated valuation quantile function is piecewise

linear. Therefore, the test statistic, that is, the square of the L2-distance between two empirical

counterparts of integrated valuation quantile functions, is simply the total area below a piecewise

quadratic curve, which has an explicit expression in the ordered bids. This feature makes our

test easy to implement in practice.

In the auction literature, there has been increasing interest in the development of statistical

tests. Examples include tests for affiliation, such as the ones proposed by Li and Zhang (2010)

and Jun et al. (2010); tests for monotonicity of inverse bidding strategy such as that proposed

by Liu and Vuong (2013); tests for discriminating entry models such as the one developed by

Marmer et al. (2013); and tests for risk aversion such as that developed by Fang and Tang

(2014).

Although our testing approach is new, we are not the first to use quantile-based approaches

in auctions. Marmer and Shneyerov (2012) and Marmer et al. (2013) proposed a quantile-based

estimator in first-price auctions and a quantile-based test for distinguishing different entry mod-

els, respectively. Liu and Vuong (2013) first introduced the idea of using the integrated valuation

quantile function and greatest convex minorant (or least concave majorant) in auctions. They

developed an integrated-quantile-based test of monotonicity of inverse bidding strategy in first-

price auctions. Based on the equivalence between monotonicity of inverse bidding strategy and

convexity of the integrated value quantile function of a bidder’s strongest competitor, they pro-

posed a test statistic comparing the difference between an estimator of such an integrated value

quantile function and its greatest convex minorant. Their test is shown to have nontrivial power

4Trimming might be avoided in the smooth estimation of bid density if boundary (bias) correction is implemented

(see, e.g., Hickman and Hubbard, 2014; Li and Liu, 2015).

COMPARING VALUATION DISTRIBUTIONS 859

against root-N local alternatives. For nonlinear pricing models, Luo et al. (2015) proposed a

quantile-based estimator, which achieves root-N consistency.

The reminder of the article is organized as follows. In the next section, we describe our

testing problem and introduce the test statistic. We then derive the asymptotic properties (i.e.,

asymptotic distribution under the null hypothesis, size, and power) of the test in Section 3. In

Section 4, we report the results of a Monte Carlo study for moderate sample sizes. In Section

5, we discuss applications of our test to auctions with an arbitrary number of samples, auctions

with endogenous entry, and auctions where only winning bids are recorded; we accommodate

the test to cases where assumptions regarding reserve price, risk aversion, unobserved hetero-

geneity, and asymmetric bidders are relaxed. Section 6 applies our method to data from U.S.

Forest Service timber auctions. Section 7 concludes the whole article. Supplementary results

are presented in Appendix A.1, whereas proofs are gathered in Appendix A.2.

2. COMPARING VALUE DISTRIBUTIONS IN FIRST-PRICE AUCTIONS

We briefly present the first-price sealed-bid auction model with independent private values.

A single and indivisible object is auctioned. Ipotential bidders are symmetric and risk neutral.

Their private values are i.i.d. draws from a common distribution F(·), which is absolutely

continuous with density f(·) and support [v, v]. With a nonbinding reserve price, the equilibrium

bid function takes the form of

b=s(v|F,I)≡v−1

F(v)I−1v

0

F(x)I−1dx.

2.1. Motivation. The seminal paper by Guerre et al. (2000) shows that a bidder’s value (v)

can be expressed as an explicit function of the submitted bid (b), the bid PDF (g(·)), and the

bid CDF (G(·))

v=ξ(b)≡b+1

I−1

G(b)

g(b).(1)

This equation can be rewritten in terms of the quantile functions of values and bids as

v(α)=b(α)+1

I−1

α

g(b(α)),(2)

where v(α) and b(α)aretheαquantiles of the valuations and bids, respectively. Marmer

and Shneyerov (2012) used the well-known formula f(v)=1/v(F(v)) to estimate f(·). Haile

et al. (2003) and Marmer et al. (2013) exploited this quantile relationship to construct tests for

detecting common value and to distinguish entry models, respectively. Note that the bid density

g(·) is involved in both Equations (1) and (2). Thus, estimating the valuation distribution

(quantile) function requires estimating both the bid distribution (quantile) function and the

density function. Recovering the latter could be troublesome, as we need to choose a tuning

parameter, namely, some bandwidth h, in density estimation.

To motivate our test, we focus on the integral of the quantile function of valuations. In

particular, we rewrite the quantile relationship as follows:

v(α)=I−2

I−1b(α)+1

I−1b(α)+α

g(b(α))

=I−2

I−1b(α)+1

I−1

d(b(α)·α)

dα.