Order Statistics Approaches to Unobserved Heterogeneity in Auctions Yao Luo

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Order Statistics Approaches to Unobserved

Heterogeneity in Auctions

Yao Luo∗

Department of Economics, University of Toronto

and

Peijun Sang

Department of Statistics and Actuarial Science, University of Waterloo

and

Ruli Xiao

Department of Economics, Indiana University

October 10, 2022

Abstract

We establish nonparametric identiﬁcation of auction models with continuous and

nonseparable unobserved heterogeneity using three consecutive order statistics of

bids. We then propose sieve maximum likelihood estimators for the joint distribution

of unobserved heterogeneity and the private value, as well as their conditional and

marginal distributions. Lastly, we apply our methodology to a novel dataset from ju-

dicial auctions in China. Our estimates suggest substantial gains from accounting for

unobserved heterogeneity when setting reserve prices. We propose a simple scheme

that achieves nearly optimal revenue by using the appraisal value as the reserve price.

Keywords: Sieve Estimation, Nonseparable, Measurement Error, Consecutive Order Statis-

tics, Judicial Auctions

∗Contact Information: Luo: Department of Economics, University of Toronto, Max Gluskin House,

150 St. George St, Toronto, ON M5S 3G7, Canada (email: yao.luo@utoronto.ca); Sang: Department of

Statistics and Actuarial Science, University of Waterloo, 200 University Avenue West, Waterloo, ON,

Canada, N2L 3G1, Canada (email: psang@uwaterloo.ca); Xiao: Department of Economics, Indiana Uni-

versity, 100 S. Woodlawn Ave. Bloomington, IN 47405 (email: rulixiao@iu.edu). We thank the editor, an

associate editor, anonymous referees, Yingyao Hu, and Ji-Liang Shiu for their time and helpful comments,

and Qingyang Zhang for valuable research assistance. Luo acknowledges funding from the SSHRC Insight

Grant.

arXiv:2210.03547v1 [econ.EM] 7 Oct 2022

1 Introduction

In empirical auction analysis, to estimate bidder value distributions, the analyst usually

needs to pool data from auctions for similar but not identical items. However, the data

available for these auctions often lack a precise description of the auctioned item. This

situation results in auction-level unobserved heterogeneity (UH), which leads to inaccurate

estimates of bidder value distributions and, therefore, misleading policy implications. For

instance, Hern´andez et al. (2020) ﬁnds that UH accounts for two-thirds of price variation

after controlling for information provided in the eBay Motors auctions, and that ignoring

this feature would dramatically mis-estimate the welfare measures. The existing literature

adapts measurement error approaches to tackle such an issue. Suppose the analyst observes

all bids. The analyst could then identify the value distribution using observed bids as mea-

surements for the unobserved characteristics, since these bids are independent conditional

on such unobserved characteristics.

However, the conditional independence condition fails when the analyst only observes

incomplete bid data. This could occur for various reasons. First, in English or ascending

outcry auctions, the bidder with the highest value only needs to outbid the bidder with

the second-highest value to win, which means the recorded bids do not contain the highest

value. Moreover, even in ﬁrst-price sealed-bid auctions, where all bids are supposed to

be submitted to the auctioneer, the auctioneer may still not record all the bids in prac-

tice: sometimes the auctioneer only records the most competitive bids, such as the top

three bids in regular auctions or apparent low bids in procurement auctions. Thus, the

econometrician can only observe a few order statistics of the bids, i.e., incomplete bid in-

formation. For instance, the U.S. Forest Service timber auctions only record at most the

top 12 bids regardless of the number of bidders. The Washington State Department of

Transportation provides an online archive of bid opening results that are six months or

older, but only for the top three apparent low bids. Even if the auctioneer records all bids,

the most competitive bids are often more accessible to the public. For instance, The Fed-

eral Deposit Insurance Corporation resolves insolvent banks using ﬁrst-price auctions but

only publishes the top two bids and bidders’ identities (Allen et al.,2019). The three ap-

parent low bids are one-click downloadable on the website of the California Department of

Transportation. These order statistics are naturally dependent, invalidating conventional

identiﬁcation strategies.

We make three contributions in this paper. First, our paper is the ﬁrst to study iden-

tiﬁcation of auction models with continuous and nonseparable UH using incomplete bid

data. Our speciﬁcation allows for ﬂexibility in how UH aﬀects both bidder value and the

equilibrium bidding strategy, i.e., the mapping from a bidder’s private value to his/her

bid.1

Our identiﬁcation strategy adapts Hu and Schennach (2008) for nonclassical measure-

ment error models to the auction setting. This extension is nontrivial in that we only

observe order statistics of UH-contaminated bids. As a result, we cannot achieve a par-

simonious conditional independence structure as in their work.2Instead, we follow Luo

and Xiao (2022) and consider the most common case of incomplete bid data: consecutive

order statistics of bids. Their main insight is that consecutive order statistics have a semi-

multiplicatively separable joint distribution with a simple indicator function capturing the

correlation. Unlike both papers using two measurements with an instrument, we use three

consecutive order statistics of bids. Given a partition on the range of the measurements, we

again obtain a separable structure traditionally achieved under conditional independence.

This turns the identiﬁcation problem into an operator diagonalization problem, allowing

constructive identiﬁcation arguments using linear operator tools. Moreover, we use these

1Even if one assumes separable UH in the value, separability passing to the bid often requires additional

institutional features or assumptions. See, e.g., Andreyanov and Caoui (2022).

2They assume that the outcome variable is independent of the observed independent variable and an

instrument conditional on the unobserved true regressor.

tools diﬀerently by considering bounded linear operators deﬁned on a Hilbert space and

taking values in another Hilbert space. This space is smaller than the L1space adopted

in Hu and Schennach (2008), which focuses on a Banach space. While we could also work

with Banach space, using Hilbert space simpliﬁes the analysis of relevant operators and

thus our proofs thanks to many existing theoretical results.3

Second, we propose sieve maximum likelihood estimators (MLE) of the model primi-

tives and provide conditions that guarantee their consistency. The estimation of auction

models allows for counterfactual policy analysis, such as computing the optimal reserve

price. If UH is common knowledge among agents in the auction, it is a critical control in

policy analysis. Therefore, optimal policy recommendation requires estimating the joint

distribution of UH and bidder private value.4In particular, we approximate the joint

density of bids and UH using the tensor product of two univariate sieve bases. We then

represent the marginal density of the UH and the conditional distribution of the value

using the sieve-approximated joint distribution. Therefore, these distributions are all esti-

mated nonparametrically.5Hu and Schennach (2008) proposes sieve approximations to the

conditional distribution and marginal distribution. Our sieve approximation to the joint

distribution is more convenient as we just need to impose the normalization assumption on

the joint distribution approximation once.

The consistency of our estimator relies on the condition that the sieve space approxi-

mates well the joint distribution of bids and UH. To formalize this intuition, we quantify

the complexity of this space using bracket entropy and prove consistency of the sieve MLEs

for the joint, conditional, and marginal densities. We establish a concentration inequality

3For instance, it is straightforward to deﬁne the adjoint operator by using the concept of inner product

in Hilbert spaces.

4Since there is a known mapping between the bid distribution and the value distribution, we will use

the two terms interchangeably. See Guerre et al. (2000) and Athey and Haile (2002) for this mapping.

5In contrast, previous research only focuses on the estimation of the joint distribution using a semipara-

metric structure (Chen et al.,2006) and (Hu and Schennach,2008) or a nonparametric structure (Wu and

Zhang,2012).

based on the bracketing number, a similar notation to covering numbers used in Hu and

Schennach (2008). The online supplement Section S.2.2 further investigates the properties

of B-splines and Bernstein polynomials, both of which are popular in empirical applications.

Lastly, we apply our identiﬁcation and estimation method to a novel dataset from

judicial auctions conducted by a municipal court in China. By default, this court uses

70% of the appraisal value as the starting price, which also serves as a reserve price. Our

estimation results suggest substantial gains from accounting for UH when designing reserve

prices. The court can gain 5.81% more revenue using an optimal reserve price for each

item. However, this scheme is complex; the seller would need to know UH and recover the

conditional density of bidder values. Instead, we propose a simple scheme that achieves

nearly optimal revenue by using the appraisal value as the reserve price. Speciﬁcally, using

the estimated model, we ﬁnd that using the appraisal value as the reserve price achieves

98.85% of the potential gains from the optimal reserve prices.

Literature Review

The auction literature has widely applied techniques developed in the measurement error

literature for identifying auction models with UH. If the UH is continuous and has a sep-

arable structure on bidder valuations, identiﬁcation relies on the deconvolution approach

and requires two random bids for each auction. See Li and Vuong (1998), Li et al. (2000),

and Krasnokutskaya (2011), among others. If the UH is ﬁnite and discrete, which by na-

ture is nonseparable, identiﬁcation relies on the condition that the bids are independent

conditional on the UH and requires three random bids for each auction. See Hu (2008),

Hu et al. (2013), and Luo (2020).

Moreover, the literature has seen rapid growth in identifying and estimating auctions

models using order statistics of bids. Athey and Haile (2002) shows that symmetric inde-

pendent private value (IPV) auctions are identiﬁable by the transaction price and the num-

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OrderStatisticsApproachestoUnobservedHeterogeneityinAuctionsYaoLuo*DepartmentofEconomics,UniversityofTorontoandPeijunSangDepartmentofStatisticsandActuarialScience,UniversityofWaterlooandRuliXiaoDepartmentofEconomics,IndianaUniversityOctober10,2022AbstractWeestablishnonparametricidenticationofauctio...

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