Estimation in English auctions with unobserved heterogeneity

• Date: 01 September 2020
• DOI: http://doi.org/10.1111/1756-2171.12343
• Author: Christopher Turansick,Daniel Quint,Cristián Hernández
• Published date: 01 September 2020

RAND Journal of Economics

Vol.51, No. 3, Fall 2020

pp. 868–904

Estimation in English auctions

with unobserved heterogeneity

Cristián Hernández∗

Daniel Quint∗∗

and

Christopher Turansick∗∗∗

We propose a framework for identification and estimation of a private values model with unob-

served heterogeneity frombid data in English auctions, using variation i n the number of bidders

across auctions, and extend the framework to settings where the number of bidders is not cleanly

observed in each auction. We illustrate our method on data from eBay Motors auctions. We find

that unobserved heterogeneity is important, accounting for two thirds of price variation after

controlling for observables, and that welfare measures would be dramatically misestimated if

unobserved heterogeneity were ignored.

1. Introduction

In many settings where auctions are used, unobserved auction-level heterogeneity has a

significant impact on valuations. For example, unobserved heterogeneity has been found to be

economically significant in highway procurement and US Forest Service timber auctions, and

structural estimation that ignored such heterogeneity would yield misleading estimates and pol-

icy conclusions.1The same holds for consumer products as well. Bodoh-Creed, Boehnke, and

∗NERA Economic Consulting; cristian.hernandez@nera.com.

∗∗University of Wisconsin-Madison; dquint@ssc.wisc.edu.

∗∗∗Georgetown University; cmt152@georgetown.edu.

The authors thank Danielle Labruzzo and Hannah O’Leary for capable research assistantship, and Aaron Bodoh-Creed,

Joachim Freyberger, Brad Larsen, Mikkel Sølvsten, seminar participants at Johns Hopkins and Penn State, the Editor,

and two anonymous referees for valuable comments. This article supplants an earlier working article, Quint (2015),

“Identification in Symmetric English Auctions with Additively Separable Unobserved Heterogeneity.”

1Workingwith bid data from Michigan highway procurement auctions, Krasnokutskaya (2011) finds that variation

in private information accounts for only one third of bid variation, and that ignoring unobserved heterogeneity would

lead to estimates of bidder markups that were more than double their actual level. Working with data from US Forest

Service timber auctions, Athey, Levin, and Seira (2011) note that allowing for unobserved heterogeneityin estimation

“appears crucial,” as they find “implausibly high bid margins when we fail to account for [it].” Working with data from

timber auctions in a different region, Aradillas-López, Gandhi, and Quint (2013) find positive correlation among bidder

valuations—possibly due to unobserved heterogeneity—“evenconditional on the rich vector of available covariates (the

presence of which is often used to defend the IPV assumption).” They find optimal reserve prices and expected seller

profit to be significantly misestimated when it is ignored: for example, they find the ForestService’s actual reserve price

868 © 2020, The RAND Corporation.

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Hickman (2018a) recently assembled an extremely detailed dataset on eBay sales of unopened

first-generation Amazon Kindle Fire tablets, and found that by combining this rich data (the en-

tire .html page of the listing) with sophisticated machine learning techniques, they could explain

42% of the variation in prices—more than three times what is explained by simpler analysis of a

more-typical subset of the variables in the dataset. Thus, even when heterogeneity across listings

is not truly unobservable, standard analysis fails to fully account for it, leaving residual variation

that is inconsistent with an independent private values model.

Although there are well-established techniques for dealing with unobserved heterogeneity

in the estimation of first-price auction models, this is much less true for English auctions. As we

discuss below, the techniques used in first-price auctions do not translate to the English auction

setting. Until recently, the empirical literature on English auctions ignored both correlation and

unobserved heterogeneity. Three recent advances offer ways to account for it, each with signifi-

cant limitations. One approach (see Aradillas-López, Gandhi, and Quint (2013)) identifies only

bounds on measures of interest, as they are not point identified, and requires wide exogenousvari-

ation in participation across auctions for the bounds to be narrow; further, the approach works

only for certain counterfactuals and not others. A second approach (Roberts (2013)) relies on an

assumption that the unobserved heterogeneity is observed by the seller, and that the reserve price

is set as a strictly increasing function of this heterogeneity. A third approach (Mbakop (2017),

Freyberger and Larsen (2017)) depends on the assumption that at least two (and, depending on

the other assumptions, as many as five) losing bidders bid up to their valuations. In the absence

of one of the latter two assumptions, we are not aware of any positive results on point identifica-

tion of an English auction model with unobserved heterogeneity. Indeed, Athey, Levin, and Seira

(2011) had data from both first-price and English auctions, but chose to estimate the structural

model using only the first-price data for exactly this reason.2

This article aims to fill this void. Focusing on a model of independent private values with

one-dimensional, separable unobserved heterogeneity,we show that the model is point identified

if there is any exogenous variation in the number of bidders across auctions—in the absence of

information-revealing reserve prices, and with only a single bidder’s bid (the highest losing bid)

being assumed to reveal her valuation. We extend this result in two ways to account for settings

where the number of bidders in each auction is not perfectly observed. Toillustrate the approach,

we apply it to data from eBay Motors car auctions. We find that after controlling for observable

covariates, auction-level unobserved heterogeneity still accounts for 67% of price variation, and

ignoring this heterogeneity would lead to a drastic (230%) overestimate of bidder surplus.

2. Related literature

The literature distinguishes the case where bidders perceive their valuations as being cor-

related (typically modeled as affiliated) from the case where bidder valuations are independent

conditional on variables they can see but the analyst cannot (unobserved heterogeneity). In first-

price auctions, equilibrium bidding depends on both a bidder’s valuation and her belief about oth-

ers’ valuations, so these are distinctly different models. In English auctions with private values,

bidding is effectively in dominant strategies, so the two models are observationally equivalent.

As noted in the Introduction, the auction literature contains well-established techniques

that allow for either unobserved heterogeneity or correlated valuations in first-price auctions. Li,

Perrigne, and Vuong (2000), Krasnokutskaya (2011), and Hu, McAdams, and Shum (2013)

build on the “measurement error” approach of Li and Vuong (1998) to estimate a model of

levels to be about as high as they could be to meet its stated policy goal of selling at least 85% of offered tracts, whereas

these reserves would seem overlycautious by a substantial margin in the absence of unobserved heterogeneity.

2Aradillas-López, Gandhi, and Quint (2013, footnote 7) note: “Referring to an earlier version of Athey,Levin, and

Seira (2011), Athey and Haile (2006, p. 33) write: ‘To account for this correlation [of bids within a first-price auction],

ALS select a model of independent private values with unobserved heterogeneity. . . . This model is not identified in data

from ascending auctions; thus, ALS focus their structural estimation on first-price auctions.’ ”

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conditionally independent values or values with unobserved auction-level heterogeneity. In a

separate approach, Li, Perrigne, and Vuong (2002) extend the estimation technique of Guerre,

Perrigne, and Vuong (2000) to affiliated private values. Compiani, Haile, and Sant’Anna (2019)

allow for both unobserved heterogeneity and affiliation of signals/interdependent values, and

discuss other approaches. However, none of these approaches work for English auctions, as they

rely on observation of multiple informative bids from each auction—either as independent noisy

estimates of the unobserved variable, or to account for the competition a bidder faces conditional

on her own valuation—which is not available in an English auction.

As discussed in Aradillas-López, Gandhi, and Quint (2013) and Roberts (2013), most of

the empirical literature on English auctions has assumed that bidder valuations are indepen-

dent (conditional on observables), ignoring both correlation and unobserved heterogeneity. Early

work modeled bidding as a button auction, where bidding revealed the exact price at which each

losing bidder stopped wanting to win. Haile and Tamer (2003) introduced a more realistic but

“incomplete” model of bidding in open-outcry ascending auctions, based on two relatively weak

assumptions about the relationship between valuations and bids: a bidder never bids more than

her valuation, and never loses an auction when she still could have bid less than her valuation.

Still assuming independent private values, they show how these assumptions lead to set identifi-

cation of the underlying primitives from bid data and estimate useful bounds.

Three recent strands of literature have moved away from the assumption of independent

private values, allowing for either correlation of values or unobserved heterogeneity. Aradillas-

López, Gandhi, and Quint (2013) use variation in the number of bidders across auctions to con-

struct bounds on relevant counterfactual measures—expected profit and bidder surplus at dif-

ferent reserve price levels, and the seller-optimal reserve. Although the model of valuations is

extremely general (except for assuming ex ante symmetry), this method requires accurate ob-

servation of the number of bidders in each auction, and the resulting bounds can be fairly wide

unless the number of bidders varies a lot. Coey, Larsen, Sweeney, and Waisman (2017) demon-

strate similar bounds for a model with asymmetric bidders. Another limitation of this method is

that it makes no attempt to fully recoverunderlying model primitives, making it useful for certain

counterfactuals but not for others.

A second approach, introduced by Roberts (2013), assumes that the seller in each auction

has access to the same information the bidders do—a one-dimensional variable that is unobserved

to the analyst—and sets a reserve price that is strictly increasing in this variable. He notes that

such behavior will often be optimal, but does not require that sellers set the optimal reserve, just

one that is monotonic in the unobserved variable. The reserve price and the transaction price

then give two separate noisy observations of the unobserved variable,identifying the model. The

assumption that reserve prices essentially reveal the unobserved characteristic of the object (and

that all sellers set them in the same way) is plausible in the environmentRoberts studies, used car

auctions, but may be less applicable in online auctions, where reserve prices are often set very

low and there is great heterogeneity in seller sophistication.

A third approach does not rely on variation in either the number of bidders or the reserve

price, but depends on the assumption that several bidders’ valuations in each auction are revealed

by their bids. Mbakop (2017) shows that a fairly general symmetric model with finite unobserved

heterogeneity is identified if five order statistics of bidders’ valuations are observed in each auc-

tion, corresponding to five losing bidders bidding up to their exact valuations; this requirement

can be reduced to three order statistics in the presence of an instrument like a varying reserve

price. Luo and Xiao (2019) show identification using three consecutive order statistics without

an instrument.3Freyberger and Larsen (2017) combine ideas from Decarolis (2018) and Song

(2004)4to show identification of a model with unobserved heterogeneity (in a separable model

3Konstantopoulos and Yuan(2019) show that an additively separable model like ours is identified from the distri-

butions of the two highest valuations, but bids in English auctions do not typically revealthe highest valuation.

4The former extends the measurement error approach discussed earlier to the case of a first-price auction where

only the winning bid is observed, using a non-binding reserve price as the second instrument. The latter uses two order

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