Asymmetric Information, Adverse Selection and

Seller Revelation on eBay Motors∗

Gregory Lewis†

Department of Economics

University of Michigan

December 28, 2006

Abstract

Since the pioneering work of Akerlof (1970), economists have been aware of the adverseselection problem that asymmetric information can create in durable goods markets. Thesuccess of eBay Motors, an online auction market for used cars, thus poses something of apuzzle. I argue that the key to resolving this puzzle is the auction webpage, which allowssellers to reduce information asymmetries by revealing credible signals about car qualityin the form of text, photos and graphics. To validate my claim, I develop a new model ofa common value eBay auction with costly seller revelation and test its implications usinga unique panel dataset of car auctions from eBay Motors. Precise testing requires a struc-tural estimation approach, so I develop a new pseudo-maximum likelihood estimator forthis model, accounting for the endogeneity of information with a semiparametric controlfunction. I find that bidders adjust their valuations in response to seller revelations, as thetheory predicts. To quantify how important these revelations are in reducing informationasymmetries, I simulate a counterfactual in which sellers cannot post information to theauction webpage and show that in this case a significant gap between vehicle value andexpected price can arise. Since this gap is dramatically narrowed by seller revelations, I con-clude that information asymmetry and adverse selection on eBay Motors are endogenouslyreduced by information transmitted through the auction webpage.

∗I would like to thank the members of my thesis committee, Jim Levinsohn, Scott Page, Lones Smith andespecially Pat Bajari for their helpful comments. I have also benefited from discussions with Jim Adams,Tilman Borgers, Han Hong, Kai-Uwe Kuhn, Serena Ng, Nicola Persico, Dan Silverman, Jagadeesh Sivadasan,Doug Smith, Charles Taragin and seminar participants at the Universities of Michigan and Minnesota. Allremaining errors are my own.

†Email: gmlewis@umich.edu

1 Introduction

eBay Motors is, at first glance, an improbable success story. Ever since Akerlof’s classic paper,

the adverse selection problems created by asymmetric information have been linked with their

canonical example, the used car market. One might think that an online used car market,

where buyers usually don’t see the car in person until after purchase, couldn’t really exist.

And yet it does - and it prospers. eBay Motors is the largest used car market in America,

selling over 15000 cars in a typical month. This poses a question: How does this market limit

asymmetric information problems?

The answer, this paper argues, is that the seller may credibly reveal his private information

through the auction webpage. Consider Figure 1, which shows screenshots from an eBay

Motors auction for a Corvette Lingenfelter. The seller has provided a detailed text description

of the features of the car and taken many photos of both the interior and the exterior. He has

in addition taken photographs of original documentation relating to the car, including a series

of invoices for vehicle modifications and an independent analysis of the car’s performance. This

level of detail appears exceptional, but it is in fact typical in most eBay car auctions for sellers

to post many photos, a full text description of the car’s history and features, and sometimes

graphics showing the car’s condition.

I find strong empirical evidence that buyers respond to these seller revelations. The data

is consistent with a model of costly revelation, in which sellers weigh costs and benefits in

deciding how much detail about the car to provide. An important prediction of this model is

that bidders’ prior valuations should be increasing in the quantity of information posted on the

webpage. I find that this is the case. Using the estimated structural parameters, I simulate a

counterfactual in which sellers cannot reveal their private information on the auction webpage.

In that case a significant gap arises between the value of the vehicle and the expected price.

I quantify this price-value gap, showing that seller revelations dramatically narrow it. Since

adverse selection occurs when sellers with high quality cars are unable to receive a fair price

for their vehicle, I conclude that these revelations substantially limit asymmetric information

and thus the potential for adverse selection.

My empirical analysis of this market proceeds through the following stages. Initially, I run

a series of hedonic regressions to demonstrate that there is a large, significant and positive

relationship between auction prices and the number of photos and bytes of text on the auction

webpage. To explain this finding, I propose a stylized model of a common values auction

on eBay with costly revelation by sellers, and subsequently prove that in equilibrium, there

is a positive relationship between the number of signals revealed by the seller and the prior

1

Figure 1: Information on an auction webpage On this auction webpage, the seller has providedmany different forms of information about the Corvette he is selling. These include the standardizedeBay description (top panel), his own full description of the car’s options (middle panel), and manyphotos (two examples are given in the bottom panel). The right photo is of the results of a carperformance analysis done on this vehicle.

2

valuation held by bidders. In the third stage, I test this prediction. A precise test requires

that I structurally estimate my eBay auction model, and I develop a new pseudo-maximum

likelihood estimation approach to do so. I consider the possibility that my information measure

is endogenous, providing an instrument and a semiparametric control function solution for

endogeneity. The final step is to use the estimated parameters of the structural model to

simulate a counterfactual and quantify the impact of public seller revelations in reducing the

potential for adverse selection.

The first of these stages requires compiling a large data set of eBay Motors listings for a 6 month

period, containing variables relating to item, seller and auction characteristics. I consider two

measures of the quantity of information on an auction webpage: the number of photographs,

and the number of bytes of text. Through running a series of hedonic regressions, I show that

these measures are significantly and positively correlated with price. The estimated coefficients

are extremely large, and this result proves robust to controls for marketing effects, seller size

and seller feedback. This suggests that it is the webpage content itself rather than an outside

factor that affects prices, and that the text and photos are therefore genuinely informative. It

leaves open the question of why the quantity of information is positively correlated with price.

I hypothesize that this is an artifact of selection, where sellers with good cars put up more

photos and text than those selling lemons.

To formally model this, in the second stage I propose a stylized model of an eBay auction.

Bidders are assumed to have a common value for the car, since the common values framework

allows for bidder uncertainty, and such uncertainty is natural in the presence of asymmetric

information.1 Sellers have access to their own private information, captured in a vector of

private signals, and may choose which of these to reveal. The seller’s optimal strategy is

characterized by a vector of cutoff values, so that a signal is revealed if and only if it is

sufficiently favorable. In equilibrium, the model predicts that bidders’ prior valuation for the

car being sold is increasing in the number of signals revealed by the seller.

I test this prediction in the third stage by structurally estimating the common value auction

model. It is necessary to proceed structurally because the prediction is in terms of a latent

variable - the bidder’s prior value - and is thus only directly testable in a structural model.

Strategic considerations arising from the Winner’s Curse preclude an indirect test based on the

relationship between price and information measures. In estimating the model, I propose a new

pseudo-maximum likelihood estimation approach that addresses two of the common practical

concerns associated with such estimation. One of the concerns is that in an eBay context it is1I also show that this assumption finds support in the data. In an extension, to be found in the appendix, I

test the symmetric common values framework used here against a symmetric private values alternative. I rejectthe private values framework at the 5% level.

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difficult to determine which bids actually reflect the bidder’s underlying valuations, and which

bids are spurious. Using arguments similar to those made by Song (2004) for the independent

private values (IPV) case, I show that in equilibrium the final auction price is in fact equal

to the valuation of the bidder with second most favorable private information. I obtain a

robust estimator by maximizing a function based on the observed distribution of prices, rather

than the full set of observed bids. The second concern relates to the computational cost of

computing the moments implied by the structural model, and then maximizing the resulting

objective function. I show that by choosing a pseudo-maximum likelihood approach, rather

than the Bayesian techniques of Bajari and Hortacsu (2003) or the quantile regression approach

of Haile, Hong, and Shum (2003), one is able to use a nested loop procedure to maximize

the objective function that limits the curse of dimensionality. This approach also provides

transparent conditions for the identification of the model.

The theory predicts that bidders respond to the information content of the auction webpage,

but this is unobserved by the econometrician. Moreover, under the theory, this omitted content

is certainly correlated with quantitative measures of information such as the number of photos

and number of bytes of text. To address this selection problem I adopt a semiparametric

control function approach, forming a proxy for the unobserved webpage content using residuals

from a nonparametric regression of the quantity of information on exogenous variables and an

instrument. I exploit the panel data to obtain an instrument, using the quantity of information

provided for the previous car sold by that seller. As the theory predicts, I find that the quantity

of information is endogenous and positively correlated with the prior mean valuation of bidders.

My results suggest that the information revealed on the auction webpage reduces the infor-

mation asymmetry between the sellers and the bidders. One would thus expect that seller

revelations limit the potential for adverse selection. My final step is to investigate this conjec-

ture with the aid of a counterfactual simulation. I simulate the bidding functions and compute

expected prices for the observed regime and a counterfactual in which sellers cannot reveal

their private information through the auction webpage. In the counterfactual regime, sellers

with a “peach” - a high quality car - cannot demonstrate this to bidders except through private

communications. The impact of removing this channel for information revelation is significant,

driving a wedge between the value of the car and the expected price paid by buyers. I find

that a car worth 10% more than the average car with its characteristics will sell for only 6%

more in the absence of seller revelation. This gap disappears when sellers can demonstrate the

quality of their vehicle by posting text, photos and graphics to the auction webpage.

I make a number of contributions to the existing literature. Akerlof (1970) showed that in a

market with information asymmetries and no means of credible revelation, adverse selection

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can occur. Grossman (1981) and Milgrom (1981) argued that adverse selection can be avoided

when the seller can make ex-post verifiable and costless statements about the product he is

selling. I consider the intermediate case where the seller must pay a cost for credibly revealing

his private information and provide structural analysis of a market where this is the case.

Estimation and simulation allow me to quantify by how much credible seller revelation reduces

information asymmetries and thus the potential for adverse selection. The contributes to the

sparse empirical literature on adverse selection which has until now focused mainly on insurance

markets (Cohen and Einav (2006), Chiappori and Salanie (2000)).

In the auctions literature Milgrom and Weber (1982) show that in a common value auction

sellers should commit a priori to revealing their private information about product quality if

they can do so. But since private sellers typically sell a single car on eBay Motors, there is no

reasonable commitment device and it is better to model sellers as first observing their signals

and then deciding whether or not to reveal. This sort of game is tackled in the literature on

strategic information revelation. The predictions of such models depend on whether revelation

is costless or not: in games with costless revelation, Milgrom and Roberts (1986) have shown

that an unravelling result obtains whereby sellers reveal all their private information. By

contrast, Jovanovic (1982) shows that with revelation costs it is only the sellers with sufficiently

favorable private information that will reveal. Shin (1994) obtains a similar partial revelation

result in a setting without revelation costs by introducing buyer uncertainty about the quality

of the seller’s information. My model applies these insights in an auction setting, showing

that with revelation costs, equilibrium is characterized by partial revelation by sellers. I also

extend to the case of multidimensional private information, showing that the number of signals

revealed by sellers is on average positively related to the value of the object. This relates the

quantity of information to its content.

The literature on eBay has tended to focus on the seller reputation mechanism. Various authors

have found that prices respond to feedback ratings, and have suggested that this is due to the

effect of these ratings in reducing uncertainty about seller type (Resnick and Zeckhauser (2002),

Melnik and Alm (2002), Houser and Wooders (2006)). I find that on eBay Motors the size

of the relationship between price and information measures that track webpage content is far

larger than that between price and feedback ratings. This suggests that in analyzing eBay

markets with large information asymmetries, it may also be important to consider the role

of webpage content in reducing product uncertainty. I contribute to the literature here by

providing simple information measures, and showing that they explain a surprisingly large

amount of the underlying price variation in this market.

Finally, I make a number of methodological contributions with regard to auction estimation.

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Bajari and Hortacsu (2003) provide a clear treatment of eBay auctions, providing a model which

can explain late bidding and a Bayesian estimation strategy for the common values case. Yet

their model implies that an eBay auction is strategically equivalent to a second price sealed bid

auction, and that all bids may be interpreted as values. I incorporate the insights of Song (2004)

in paying more careful attention to bid timing and proxy bidding, and show that only the final

price in an auction can be cleanly linked to the distribution of bidder valuations. I provide

a pseudo maximum likelihood estimation strategy that uses only the final price from each

auction, and is therefore considerably more robust. There are also computational advantages

to my common values estimation approach relative to that of Bajari and Hortacsu (2003) and

the quantile regression approach of Hong and Shum (2002). One important potential problem

with all these estimators lies in not allowing for unobserved heterogeneity and endogeneity. For

the independent private values case, Krasnokutskaya (2002) and Athey, Levin, and Seira (2004)

have provided approaches for dealing with unobserved heterogeneity. Here I develop conditions

under which a semiparametric control function may be used to deal with endogeneity provided

an appropriate instrument can be found. In sum, I provide a robust and computable estimator

for common value eBay auctions. I also provide an implementation of the test for common

values suggested by Athey and Haile (2002) in the appendix, using the subsampling approach

of Haile, Hong, and Shum (2003).

This research leads me to three primary conclusions. First, webpage content in online auctions

informs bidders and is an important determinant of price. Bidder behavior is consistent with a

model of endogenous seller revelation, where sellers only reveal information if it is sufficiently

favorable, and thus bidders bid less on objects with uninformative webpages. Second, quantita-

tive measures of information serve as good proxies for content observed by buyers, but not the

econometrician. This follows from the theory which yields a formal link between the number

of signals revealed by sellers and the content of those signals. Finally, and most important,

the auction webpage allows sellers to credibly reveal the quality of their vehicles. Although

revelation costs prevent full revelation, information asymmetries are substantively limited by

this institutional feature and the gap between values and prices is considerably smaller than it

would be if such revelations were not possible. I conclude that seller revelations through the

auction webpage reduce the potential for adverse selection in this market.

The paper proceeds as follows. Section 2 outlines the market, and section 3 presents the re-

duced form empirical analysis. Section 4 introduces the auction model with costly information

revelation, while section 5 describes the estimation approach. The estimation results follow in

section 6. Section 7 details the counterfactual simulation and section 8 concludes.2

2Proofs of all propositions, as well as most estimation results, are to be found in the appendix.

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2 eBay Motors

eBay Motors is the automobile arm of online auctions giant eBay. It is the largest automotive

site on the Internet, with over $1 billion in revenues in 2005 alone. Every month, approximately

95000 passenger vehicles are listed on eBay, and about 15% of these will be sold on their first

listing.3 This trading volume dwarfs those of its online competitors, the classified services

cars.com, autobytel.com and Autotrader.com. In contrast to these sites, most of the sellers

on eBay Motors are private individuals, although dealers still account for around 30% of the

listings.

Listing a car on eBay Motors is straightforward. For a fixed fee of $40, a seller may post a

webpage with photos, a standardized description of the car, and a more detailed description

that can include text and graphics. The direct costs of posting photos, graphics and text

are negligible: text and graphics are free, while each additional photo costs $0.15. But the

opportunity costs are considerably higher, as it is time-consuming to take, select and upload

photos, write the description, generate graphics, and fill in the forms required to post all of

these to the auction webpage.4

Figure 2 shows part of the listing by a private seller who used eBay’s standard listing template.

In addition to the standardized information provided in the table at the top of the listing, which

includes model, year, mileage, color and other features, the seller may also provide his own

detailed description. In this case the seller has done poorly, providing just over three lines of

text. He also provided just three photos. This car, a 1998 Ford Mustang with 81000 miles on

the odometer, received only four bids, with the highest bid being $1225.

Figure 3 shows another 1998 Ford Mustang, in this case with only 51000 miles. The seller is a

professional car dealership, and has used proprietary software to create a customized auction

webpage. This seller has provided a number of pieces of useful information for buyers: a text

based description of the history of the car, a full itemization of the car features in a table, a

free vehicle history report through CARFAX, and a description of the seller’s dealership. This

listing also included 28 photos of the vehicle. The car was sold for $6875 after 37 bids were

placed on it.

The difference between these outcomes is striking. The car listed with a detailed and informa-

tive webpage sold for $1775 more than the Kelley Blue Book private party value, while that

with a poor webpage sold for $3475 less than the corresponding Kelley Blue Book value. This3The remainder are either re-listed (35%), sold elsewhere or not traded. For the car models in my data, sales

are more prevalent, with about 30% sold on their first listing.4Professional car dealers will typically use advanced listing management software to limit these costs. Such

software is offered by CARad (a subsidiary of eBay), eBizAutos and Auction123.

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Figure 2: Example of a poor auction webpage This webpage consists mainly of the standard-ized information required for a listing on eBay. The seller has provided little additional informationabout either the car or himself. Only three photos of the car were provided. In the auction, the sellerreceived only four bids, with the highest bid being $1225. The Kelley Blue Book private party value,for comparison, is $4700.

Figure 3: Example of a good auction webpage The dealer selling this vehicle has usedproprietary software to create a professional and detailed listing. The item description seen on theright, contains information on the car, a list of all the options included, warranty information, and afree CARFAX report on the vehicle history. The dealer also posted 28 photos of the car. This carreceived 37 bids, and was sold for $6875. The Kelley Blue Book retail price, for comparison, is $7100,while the private party value is $5100.

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Figure 4: Additional information The left panel shows a graphic that details the exterior con-dition of the vehicle. The right panel shows the Kelley Blue Book information for the model-year ofvehicle being auctioned.

suggests that differences in the quality of the webpage can be important for auction prices.

How can webpages differ? Important sources of differences include the content of the text-

based description, the number of photos provided, the information displayed in graphics, and

information from outside companies such as CARFAX. Examples of the the latter two follow

in Figure 4. In the left panel I show a graphic showing the condition of the exterior of the

car. In the right panel, I show the Kelley Blue Book information posted by a seller in order to

inform buyers about the retail value of the vehicle.

It is clear that many of these pieces of information, such as photos and information from outside

institutions, are hard to fake. Moreover, potential buyers have a number of opportunities to

verify this information. Potential buyers may also choose to acquire vehicle history reports

through CARFAX or Autocheck.com, or purchase a third party vehicle inspection from an

online service for about $100. Bidders may query the seller on particular details through

direct communication via e-mail or telephone. Finally, blatant misrepresentation by sellers

is subject to prosecution as fraud. In view of these institutional details, I will treat seller

revelations as credible throughout this paper. All of these sources of information - the text,

photos, graphics and information from outside institutions, as well as any private information

acquired - act to limit the asymmetric information problem faced by bidders. The question

I wish to examine is how important these sources of information are for auction prices. This

question can only be answered empirically, and so it is to the data that I now turn.

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3 Empirical Regularities

3.1 Data and Variables

I collected data from eBay Motors auctions over a six month period.5 I drop observations with

nonstandard or missing data, re-listings of cars, and those auctions which received less than

two bids.6 I also drop auctions in which the webpage was created using proprietary software.7

The resulting dataset consists of over 18000 observations of 18 models of vehicle. The models of

vehicle may be grouped into three main types: those which are high volume Japanese cars (e.g.

Honda Accord, Toyota Corolla), a group of vintage and newer “muscle” cars (e.g. Corvette,

Mustang), and most major models of pickup truck (e.g. Ford F-series, Dodge Ram).

The data contain a number of item characteristics including model, year, mileage, transmission

and title information. I also observe whether the vehicle had an existing warranty, and whether

the seller has certified the car as inspected.8 In addition, I have data on the seller’s eBay

feedback and whether they provided a phone number and address in their listing. I can also

observe whether the seller has paid an additional amount to have the car listing “featured”,

which means that it will be given priority in searches and highlighted.

Two of the most important types of information that can be provided about the vehicle by the

seller are the vehicle description and the photos. I consider two simple quantitative measures

of these forms of information content. The first measure is the number of bytes of text in the

vehicle description provided by the seller. The second measure is the number of photos posted

on the webpage. I use both of these variables in the hedonic regressions in the section below.5I downloaded the auction webpages for the car models of interest every twenty days, recovered the urls for

the bid history, and downloaded the history. I implemented a pattern matching algorithm in Python to parsethe html code and obtain the auction variables.

6I do this because in order to estimate my structural auction model, I need at least two bids per auction.Examining the data, I find that those auctions with less than two bids are characterized by extremely highstarting bids (effectively reserves), but are otherwise similar to the rest of the data. One respect in whichthere is a significant difference is that in these auctions the seller has provided less text and photos. This is inaccordance with the logic detailed in the rest of the paper.

7Webpages created using proprietary software are often based on standardized templates, and some of thetext of the item description is not specific to the item being listed. Since one of my information measures is thenumber of bytes of text, comparisons of standard eBay listings with those generated by advanced software arenot meaningful.

8This certification indicates that the seller has either hired an outside company to perform a car inspection.The results of the inspection can be requested by potential buyers.

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3.2 Relating Prices and Information

In order to determine the relationship between the amount of text and photos and the final

auction price, I run a number of hedonic regressions. Each specification has the standard form:

log(pj) = zjβ + ε (1)

where zj is a vector of covariates.9 I estimate the relationship via ordinary least squares (OLS),

including model and title fixed effects.10 I report the results in Table 1, suppressing the fixed

effects.

In the first specification, the vector of covariates includes standard variables, such as age,

mileage and transmission, as well as the log number of bytes of text and the number of photos.

The coefficients generally have the expected sign and all are highly significant.11 Of particular

interest is the sheer magnitude of the coefficients on the amount of text and photos. I find

that postings with twice the amount of text sell for 8.7% more, which for the average car in

the dataset is over $800 more. Likewise, each additional photo on the webpage is associated

with a selling price that is 2.6% higher - a huge effect.

I am not asserting a causal relationship between the amount of text and photos and the auction

price. My claim is that buyers are using the content of the text and photos to make an informed

decision as to the car’s value. The content of this information includes the car’s options, the

condition of the exterior of the vehicle, and vehicle history and usage, all of which are strong

determinants of car value. Moreover, as I will show in the structural model below, buyers will

in equilibrium interpret the absence of information as a bad signal about vehicle quality, and

adjust their bids downwards accordingly. I will also show that the amount of information is an

effective proxy for the content of that information. Thus my interpretation of this regression

is that the large coefficients on text and photos arise because these measures are proxying for

attributes of the vehicle that are unobserved by the econometrician.

In the next four regressions, I explore alternate explanations. The first alternative is that this

is simply a marketing effect, whereby slick webpages with a lot of text and photos attract

more bidders and thus the cars sell for higher prices. I therefore control for the number of

bidders, and also add a dummy for whether the car was a featured listing. I find that adding

these controls does reduce the information coefficients, but they still remain very large. A9I use the log of price instead of price as the dependent variable so that it has range over the entire real line.

10The seller may report the title of the car as clean, salvage, other or unspecified. The buyer may check thisinformation through CARFAX.

11One might have expected manual transmission to enter with a negative coefficient, but I have a large numberof convertible cars and pickups in my dataset, and for these, a manual transmission may be preferable.

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Table 1: Hedonic Regressions

(1) (2) (3) (4) (5)Age -0.1961 -0.1935 -0.1931 -0.1924 -0.1875

(0.0019) (0.0018) (0.0018) (0.0018) (0.0020)Age Squared 0.0039 0.0038 0.0038 0.0038 0.0037

(0.0000) (0.0000) (0.0000) (0.0000) (0.0000)Log of Miles -0.1209 -0.1225 -0.1221 -0.1215 -0.1170

(0.0041) (0.0040) (0.0040) (0.0040) (0.0041)Log of Text Size (in bytes) 0.0852 0.0686 0.0816 0.0794 0.0978

(0.0073) (0.0072) (0.0075) (0.0076) (0.0085)Number of Photos 0.0269 0.0240 0.0241 0.0234 0.0260

(0.0009) (0.0009) (0.0009) (0.0009) (0.0010)Manual Transmission 0.0349 0.0348 0.0328 0.0338 0.0319

(0.0124) (0.0122) (0.0122) (0.0122) (0.0121)Featured Listing 0.2147 0.2112 0.2113 0.2125

(0.0211) (0.0210) (0.0211) (0.0210)Number of Bidders 0.0355 0.0356 0.0355 0.0357

(0.0015) (0.0015) (0.0015) (0.0015)Log Feedback -0.0211 -0.0200 -0.0179

(0.0033) (0.0033) (0.0033)Percentage Negative Feedback -0.0031 -0.0031 -0.0032

(0.0015) (0.0015) (0.0015)Total Number of Listings -0.0020 -0.0019

(0.0015) (0.0015)Phone Number Provided 0.1151 0.1193

(0.0123) (0.0122)Address Provided -0.0498 -0.0510

(0.0183) (0.0182)Warranty 0.8258

(0.1638)Warranty*Logtext -0.0738

(0.0248)Warranty*Photos -0.0189

(0.0031)Inspection 0.6343

(0.1166)Inspection*Logtext -0.0654

(0.0176)Inspection*Photos -0.0054

(0.0022)Constant 9.7097 9.5918 9.5746 9.5704 9.2723

(0.0995) (0.0980) (0.0980) (0.0980) (0.1025)

Estimated standard errors are given in parentheses. The model fits well, with the R2 ranging from 0.58 in (1)to 0.61 in (5). The nested specifications (2)-(4) include controls for marketing effects, seller feedback and dealerstatus. Specification (5) includes warranty and inspection dummies and their interactions with the informationmeasures.

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second explanation is that the amount of text and photos are somehow correlated with seller

feedback ratings, which are often significant determinants of prices on eBay. I find not only

that including these controls has little effect on the information coefficients, but that for this

specific market the effects of seller reputation are extremely weak. The percentage negative

feedback has a very small negative effect on price, while the coefficient on total log feedback

is negative, which is the opposite of what one would expect. A possible reason for these

results is that for the used car market, the volume of transactions for any particular seller is

small and this makes seller feedback a weak measure of seller reputation. Another problem is

that feedback is acquired through all forms of transaction, and potential buyers may only be

interested on feedback resulting from car auctions.

Returning to the question of alternative explanations, a third possibility is that frequent sellers

such as car dealerships tend to produce better webpages, since they have stronger incentives to

develop a good template than less active sellers. Then if buyers prefer to buy from professional

car dealers, I may be picking up this preference rather than the effects of information content.

To control for this, I include the total number of listings by the seller in my dataset as a

covariate, as well as whether they posted a phone number and an address. All of these measures

track whether the seller is a professional car dealer or not. I report the results in column

(4). I find that although posting a phone number is valuable (perhaps because it facilitates

communication), both the number of listings and the address have negative coefficients. That

is, there is little evidence that buyers are willing to pay more to buy from a frequent lister

or a firm that posts its address. Again, the coefficients on text and photos remain large and

significant.

Finally, I try to test my hypothesis that text and photos matter because they inform potential

buyers about car attributes that are unobserved by the econometrician. To do this, I include

dummies for whether the car is under warranty, and has been certified by the seller as inspected.

Both of these variables should reduce the value of information, because the details of car

condition and history become less important if one knows that the car is under warranty or

has already been certified as being in excellent condition. In line with this idea, I include

interaction terms between the warranty and inspection dummies and the amount of text and

photos. The results in column (5) indicate that the interaction terms are significantly negative

as expected, while both the inspection and warranty significantly raise the expected price of

the vehicle. For cars that have neither been inspected nor are under warranty, the coefficients

on text and photos increase.

There are problems with a simple hedonic regression, and in order to deal with the problem

of bids being strategic and different from the underlying valuations, I will need a structural

13

model. I will also need to demonstrate that my measures of the quantity of information are

effective proxies for the content of that information. Yet the reduced form makes the point

that the amount of text and photos are positively related to auction price, and that the most

compelling explanation for this is that their content gives valuable information to potential

buyers.

4 Theory

Modeling demand on eBay is not a trivial task. The auction framework has a bewildering

array of institutional features, such as proxy bidding, secret reserves and sniping software.

With this in mind, it is critical to focus on the motivating question: how does seller revelation

of information through the auction webpage affect bidder beliefs about car value, and thereby

behavior and prices?

I make three important modeling choices. The first is to model an auction on eBay Motors as a

symmetric common value auction. In this model, agents have identical preferences, but are all

uncertain as to the value of the object upon which they are bidding. The bidders begin with

a common prior distribution over the object’s value, but then draw different private signals,

which generates differences in bidding behavior. This lies in contrast to a private values model,

in which agents are certain of their private valuation, and differences in bidding behavior result

from differences in these private valuations.

I believe that the common value framework is better suited to the case of eBay Motors than

the private values framework for a number of reasons. First, it allows bidders to be uncertain

about the value of the item they are bidding on, and this uncertainty is necessary for any model

that incorporates asymmetric information and the potential for adverse selection. Second, the

data support the choice of a common values model. I show this in the appendix, where I

implement the Athey and Haile (2002) test of symmetric private values against symmetric

common values, and reject the private value null hypothesis at the 5% level.12 Third, it

provides a clean framework within which one may think clearly about the role of the auction

webpage in this market. The auction webpage is the starting point for all bidders in an online

auction, and can thus be modeled as the source of the common prior that bidders have for the

object’s value. I can identify the role of information on eBay Motors by examining its effect

on the distribution of the common prior. Finally, though market participants on eBay Motors12The test is based on the idea that the Winner’s Curse is present in common values auctions, but not

in private values auctions. This motivates a test based on stochastic dominance relationships between biddistributions as the number of bidders n, and thus the Winner’s Curse, increases.

14

will in general have different preferences, it may be reasonable to assume that those bidders

who have selected into any particular auction have similar preferences. This is captured in the

common values assumption.13

The second modeling choice is to use a modified version of the Bajari and Hortacsu (2003)

model of late bidding on eBay. On eBay Motors, I observe many bids in the early stages of the

auction that seem speculative and unreflective of underlying valuations; while in the late stages

of the auction, I observe more realistic bids. With late bidding, bidders often do not have time

to best respond to opponent’s play. “Sniping” software can be set to bid on a player’s behalf

up to one second before the end of the auction, and bidders cannot best respond to these late

bids. Bajari and Hortacsu (2003) capture these ideas in a two stage model, in which bidders

may observe each other’s bids in the early stage, but play the second stage as though it were

a sealed bid second price auction. I vary from their model by taking account of the timing of

bids in the late stage, since a bid will only be recorded if it is above the current price.14

Accounting for timing is important, as an example will make clear. Suppose that during the

early stage of an auction bidder A makes a speculative bid of $2000. As the price rises and it

becomes obvious this bid will not suffice, she programs her sniping software to bid her valuation

of $5600 in the last five seconds of an auction. Now suppose that bidder B bids $6000 an hour

before the end of the auction, and so when bidder A’s software attempts to make a bid on her

behalf at the end of the auction, the bid is below the standing price and hence not recorded.

The highest bid by bidder A observed during the auction is thus $2000. An econometrician

who treats an eBay auction as simply being a second price sealed bid auction may assume that

bidder A’s valuation of the object is $2000, rather than $5600. This will lead to inconsistent

parameter estimates. I show that regardless of bid timing, the final price in the auction with

late bidding is the valuation of the bidder with the second highest private signal. This allows

me to construct a robust estimation procedure based on auction prices, rather than the full

distribution of bids.

Lastly, I allow for asymmetric information, and let the seller privately know the realizations

of a number of signals affiliated with the value of the car. One may think of these signals as

describing the car history, condition and features. The seller may choose which of these signals

to reveal on the auction webpage, but since writing detailed webpages and taking photos is

costly, such revelations are each associated with a revelation cost. In line with my earlier

description of the institutional details of eBay Motors, all revelations made by the seller are13One could consider a more general model of affiliated values, incorporating both common and private value

elements. Unfortunately, without strong assumptions, such a model will tend to be unidentified.14Song (2004) also makes this observation, and uses it to argue that in the independent private values context

(IPV), the only value that will be recorded with certainty is that of the second highest bidder.

15

modeled as being credible. This assumption prevents the model from being a signaling model

in the style of Spence (1973).15 Instead, this setup is similar to the models of Grossman (1981)

and Milgrom (1981), in which a seller strategically and costlessly makes ex-post verifiable

statements about product quality to influence the decision of a buyer. In these models, an

unravelling result obtains, whereby buyers are skeptical and assume the worst in the absence

of information, and consequently sellers reveal all their private information.

My model differs in three important respects from these benchmarks. First, I allow for revela-

tion costs. This limits the incentives for sellers to reveal information, and thus the unravelling

argument no longer holds. Second, the parties receiving information in my model are bidders,

who then proceed to participate in an auction, inducing a multi-stage game. Finally, I allow

the seller’s private information to be multidimensional. While each of these modifications has

received treatment elsewhere16 there is no unified treatment of revelation games with all these

features, and attempting a full analysis of the resulting multi-stage game is beyond the scope of

this paper. Instead I provide results for the case in which the relationship between the seller’s

private information and the car’s value is additively separable in the signals. This assumption

will be highlighted in the formal analysis below.

4.1 An Auction Model with Late Bidding

Consider a symmetric common values auction model, as in Wilson (1977). There are N

symmetric bidders, bidding for an object of unknown common value V . Bidders have a common

prior distribution for V , denoted F . Each bidder i is endowed with a private signal Xi, and

these signals are conditionally i.i.d., with common conditional distribution G|v. The variables

V and X = X1, X2 · · ·Xn are affiliated.17 The realization of the common value is denoted

v, and the realization of each private signal is denoted xi. The realized number of bidders

n is common knowledge.18 One may think of the bidders’ common prior as being based on

the content of the auction webpage. Their private signals may come from a variety of sources15In those games, the sender may manipulate the signals she sends in order to convince the receiver that she

is of high type. Here the sender is constrained to either truthfully reveal the signal realization, or to not revealit at all.

16See for example Jovanovic (1982) for costly revelation, Okuno-Fujiwara, Postlewaite, and Suzumura (1990)for revelations in a multi-stage game and Milgrom and Roberts (1986) for multidimensional signal revelation.

17This property is equivalent to the log supermodularity of the joint density of V and X. A definition of logsupermodularity is provided in the appendix.

18This assumption is somewhat unrealistic, as bidders in an eBay auction can never be sure of how manyother bidders they are competing with. The alternative is to let auction participants form beliefs about n, wherethose beliefs are based on a first stage entry game, as in Levin and Smith (1994). But dealing with endogenousentry in this way has certain problems in an eBay context (see Song (2004)), and thus I choose to go with thesimpler model and make n common knowledge.

16

such as private communications with the seller through e-mail or phone and the results of

inspections they have contracted for or performed in person.

Bidding takes place over a fixed auction time period [0, τ ]. At any time during the auction, t,

the standing price pt is posted on the auction website. The standing price at t is given by the

second highest bid submitted during the period [0, t). Bidding takes place in two stages. In

the early stage [0, τ −ε), the auction takes the form of an open exit ascending auction. Bidders

may observe the bidding history, and in particular observe the prices at which individuals drop

out of the early stage. In the late stage [τ − ε, τ ], all bidders are able to bid again. This

time, however, they don’t have time to observe each other’s bidding behavior. They may each

submit a single bid, and the timing of their submission is assumed to be randomly distributed

over the time interval [τ − ε, τ ].19 If their bid bit is lower than the standing price pt, their bid is

not recorded. The following proposition summarizes the relevant equilibrium behavior in this

model:

Proposition 1 (Equilibrium without Revelation) There exists a symmetric Bayes-Nash

equilibrium in which:

(a) All bidders bid zero during the early stage of the auction.

(b) All bidders bid their pseudo value during the late stage, where their pseudo value is defined

by

v(x, x;n) = E[V |Xi = x, Y = x,N = n] (2)

and Y = maxj 6=i Xj is the maximum of the other bidder’s signals.

(c) The final price is pτ is given by

pτ = v(x(n−1:n), x(n−1:n);n) (3)

where x(n−1:n) denotes the second highest realization of the signals Xi.

The idea is that bidders have no incentive to bid a positive amount during the early stage of

the auction, as this may reveal their private information to the other bidders. Thus they bid

only in the late stage, bidding as in a second price sealed bid auction. The second highest bid

will certainly be recorded, and I obtain an explicit formula for the final price in the auction as

the bidding function evaluated at the second highest signal. This formula will form the basis

of my estimation strategy later in the paper. In the model thus far, though, I have not allowed

the seller to reveal his private information. I extend the model to accommodate information

asymmetry and seller revelation in the next section.19Allowing bidders to choose the timing of their bid would not change equilibrium behavior, and would

unnecessarily complicate the argument.

17

4.2 Information Revelation

Let the seller privately observe a vector of real-valued signals S = S1 · · ·Sm with common

bounded support [s, s]. As with the bidder’s private information, these signals are conditionally

i.i.d with common distribution H|v and are conditionally independent of the bidder’s private

signals X1 · · ·Xn. Let V and S = S1 · · ·Sm be affiliated. The seller may report his signals to

the bidders prior to the start of the auction, and bidders treat all such revelations as credible.

The seller’s objective is to maximize his expected revenue from the auction. A (pure strategy)

reporting policy for the seller is a mapping R : Rm → 0, 1m, where reporting the signal is

denoted 1 and not reporting is denoted 0. Letting s = s1 · · · sm be a realization of the signal

vector S = S1 · · ·Sm, define I = #i : Ri(s) = 1 as the number of signals reported by the

seller. Finally, let the seller face a fixed revelation cost ci for revealing the signal Si, and let

the associated cost vector be c. Revelation costs are common knowledge.

This setup generates a multi-stage game in which the seller first makes credible and public

statements about the object to be sold, and then bidders update their priors and participate

in an auction. Since each revelation by the seller is posted to the auction webpage, there

is a natural link between the number of signals I and my empirical measures of information

content, the number of photos and bytes of text on the webpage.

Define the updated pseudo value w(x, y, s;n) = E[V |X = x, Y = y, S = s,N = n] where

Y = maxj 6=i Xj is the maximum of the other bidder’s signals. Then in the absence of revelation

costs, the equilibrium reporting policy for the seller is to reveal all his information:

Proposition 2 (Unravelling) For c = 0, there exists a sequential equilibrium in which the

seller reports all his signals (R(s) = 1 ∀s), bidders bid zero in the early stage of the auction,

and bid their updated pseudo value w(x, x, s;n) in the late stage.

This result seems counter-intuitive. It implies that sellers with “lemons” - cars with a bad

history or in poor condition - will disclose this information to buyers, which runs counter to

the intuition behind models of adverse selection. It also implies that the quantity of information

revealed on the auction webpage should be unrelated to the value of the car, since sellers will

fully disclose, regardless of the value of the car. Since I know from the hedonic regressions that

price and information measures are positively related, it suggests that the costless revelation

model is ill suited to explaining behavior in this market. Revealing credible information is

a costly activity for the seller, and thus sellers with “lemons” have few incentives to do so.

I can account for this by considering the case with strictly positive revelation costs, but this

generates a non-trivial decision problem for the seller. In order to make the equilibrium analysis

tractable, I make a simplifying assumption.

18

Assumption 1 (Constant Differences) The function w(x, y, s;n) exhibits constant differ-

ences in si for all (x, y, s−i). That is, there exist functions ∆i(s1, s0;n), i = 1 · · ·m, such

that:

∆i(s1, s0;n) = w(x, y, s−i, s1;n)− w(x, y, s−i, s0;n) ∀(x, y, s−i)

This assumption says that from the bidder’s point of view, the implied value of each signal

revealed by the seller is independent of all other signals the bidder observes.20 The extent to

which this assumption is reasonable depends on the extent to which car features, history and

condition are substitutes or complements for value. Under this assumption, one may obtain a

clean characterization of equilibrium.

Proposition 3 (Costly Information Revelation) For c > 0, there exists a sequential

equilibrium in which:

(a) The reporting policy is characterized by a vector of cutoffs t = t1 · · · tm with

Ri(s) =

1 if si ≥ ti

0 otherwise

where ti = inf t : E [∆i(t, t′;n)|t′ < t] ≥ ci.(b) Bidders bid zero during the early stage of the auction, and bid

vR(x, x, sR;n) = E[w(x, x, s;n)|si < tii∈U ]

during the late stage, where sR is the vector of reported signals and U = i : R(si) = 0is the set of unreported signals.

(c) The final auction price is pτ = vR(x(n−1:n), x(n−1:n), sR;n).

In contrast to the result with costless revelation, this equilibrium is characterized by partial

revelation, where sellers choose to reveal only sufficiently favorable private information. Bidders

(correctly) interpret the absence of information as a bad signal about the value of the car, and

adjust their bids accordingly. Then the number of signals I revealed by the seller is positively

correlated with the value of the car.

Corollary 1 (Expected Monotonicity) E[V |I] is increasing in I.

I prove the result by showing that the seller’s equilibrium strategy induces affiliation between

V and I. This result is useful for my empirical analysis, as it establishes a formal link between20A sufficient condition for the constant differences assumption to hold is that the conditional mean E[V |X, S]

is additively separable in the signals S, so that E[V |X = x, S = s] =Pm

j=1 fj(sj) + g(x) for some increasingreal-valued functions fj and g.

19

the number of signals I, quantified in the number of photos and bytes of text on the auction

webpage, and the content of that information, the public signals S. It also provides a testable

prediction: that the prior mean E[V |I] should be increasing in the amount of information

posted. I now proceed to outline the strategy for estimating this auction model.

5 Auction Estimation

Estimating a structural model is useful here. First, I am able to account for the strategic

interaction between bidders and uncover the relationship between valuations and item char-

acteristics, rather than price and characteristics. Second, I can test the model’s implications

precisely. The model predicts a positive relationship between the conditional mean E[V |I]

and I. This in turn implies that E[p|I] is increasing in I, but that positive relationship could

also hold because better informed bidders are less afraid of the Winner’s Curse and therefore

bid more.21 Estimating the structural model allows a test of the model prediction that is free

of strategic considerations on the part of bidders. Finally, and most important, structural

estimation permits simulation of counterfactuals. This will allow me to quantify the impact of

the auction webpage in reducing information asymmetries and adverse selection below.

Yet estimating a common values auction is a thorny task. There are two essential problems.

Firstly, a number of model primitives are latent: the common value V , and the private signals

X1 · · ·Xn. Secondly, the bidding function itself is highly non-linear and so even in a para-

metric model computing the moments of the structural model is difficult and computationally

intensive. Addressing endogeneity in addition to the above problems is especially tricky. My

estimation approach is able to deal with all of these problems and remain computationally

tractable, and thus may be of independent interest. Two different approaches have been tried

in recent papers. Bajari and Hortacsu (2003) employ a Bayesian approach with normal priors

and signals, and estimate the posterior distribution of the parameters by Markov Chain Monte

Carlo (MCMC) methods. Hong and Shum (2002) use a quantile estimation approach with

log normal priors and signals, and obtain parameter estimates by maximizing the resulting

quantile objective function.

My estimation approach, which is based on maximizing a pseudo-likelihood function, has a

number of advantages relative to these types of estimators. The main advantage lies in the

computational simplicity of my approach. I am able to show that my estimator may be21The Winner’s Curse is the result that bidders in a common value auction will be “cursed” if they bid their

conditional valuation of the object E[V |Xi], as they will only win when their bid exceeds everyone else’s. In thisevent, they also had the highest signal, which suggests that they were over-optimistic in their initial valuation.Rational bidders therefore shade their bids down, always bidding below their conditional valuation.

20

maximized through a nested loop, in which the inner loop runs weighted least squares (WLS)

and the outer loop is a standard quasi-Newton maximization algorithm. For most applications,

the outer loop will search over a low dimensional space, while the inner loop may be of almost

arbitrarily high dimension since the maximizer may be computed analytically. This avoids the

curse of dimensionality, and allows me to include over thirty regressors in the structural model

while still maximizing the objective function in less than thirty minutes.22 By contrast, the

existing estimation techniques would require an MCMC implementation in a thirty dimensional

parameter space, and it would typically take days of computation to get convergence of the

chain to the stationary distribution of the parameters.23

A second advantage is that my estimation approach permits a semiparametric control function

method for dealing with endogeneity. Here I consider the case where precisely one variable is

endogenous and show that if the relationship between the endogenous variable and the source

of the endogeneity is monotone, a nonparametric first stage can be used to estimate a proxy

variable that controls for endogeneity. I note that the econometric theory given in Andrews

(1994a) guarantees the consistency and asymptotic normality of the resulting estimator, even

in the complicated non-linear setting of a common value auction.

The third advantage is eBay specific, and lies in the fact that I identify the model using only

variation in auction prices, the number of bidders n, and the covariates. This contrasts with

papers that use all bids from each auction, which can lead to misinterpretation. Identification

is also extremely transparent in this setup, which is not the case with the other estimators. I

now provide the specifics of the estimation approach.

5.1 Specification

The theory provides a precise form for the relationship between the final price in each auction

and the bidders’ priors and private signals. In order to link price to the auction covariates

such as item characteristics and information, I first specify a parametric form for the interim

prior distribution F |z and the conditional signal distribution G|v.24 I may then specify the

relationship between the moments of these distributions and auction covariates z. Let F |zbe the log-normal distribution, and let G|v be the normal distribution centered at v = log v.

22Most of these are fixed effects, but this approach could accommodate thirty continuous regressors.23This remark applies to the quantile estimation approach as well, since the objective function is not smooth

and therefore MCMC is the considered to be the best approach for maximizing it. See Chernozhukov and Hong(2003) for details.

24By “interim prior”, I mean the updated prior held by bidders after viewing the webpage but before obtainingtheir private signal.

21

Then we have:

v = µ + σεv ∼ N(µ, σ2)

xi|v = v + rεi ∼ N(v, r2)

where εv, εi are i.i.d standard normal random variables for all i.25 Three variables form the

primitives of the auction model: the interim prior mean µ, the interim prior standard deviation

σ, and the signal standard deviation r.

Next I link the primitives (µ, σ, r) to the auction covariates. As argued earlier, the information

posted on the auction webpage changes the common prior distribution over values for all

bidders. Consequently, I allow the mean and standard deviation of F , µ and σ, to vary with

auction covariates z. I include in z my empirical measures of information content, the number

of photos and bytes of text. The signal variance parameter r is treated as a constant, as the

private information is by definition orthogonal to the publicly available information captured

in the covariates. The chosen specification takes the index form:

µj = αzj

σj = κ(βzj)

where κ is an increasing function with strictly positive range, so that κ(x) > 0 ∀x.26 This

specification yields admissible values for σj .

5.2 Method

The above specification governs the relationship between the parameter vector θ = (α, β, r),

bidder prior valuations, and bidder signals. Prices are a complex function of these parameters,

since they are equal to the equilibrium bid of the bidder with the second highest signal, an

order statistic. Thus even though I have a fully specified parametric model for valuations

and signals, the distribution of prices is hard to determine. This prevents a straightforward

maximum likelihood approach. I can however compute the mean and variance of log prices

implied by the structural model, E[log p|n, zj , θ] and Ω[log p|n, zj , θ].27 Notice that these

predicted moments vary with the number of bidders n. Computing these moments allows me to25The choice of the log normal distribution for v is motivated by two considerations. First, all valuations

should be positive, so it is important to choose a distribution with strictly positive support. Second, the log pricesobserved in the data are approximately normally distributed, and since the price distribution is generated bythe value distribution, this seems a sensible choice. The choice of a normal signal distribution is for convenience,and seems reasonable given that the signals are latent and unobserved.

26For computational reasons, κ(·) is based on the arctan function - specifics are given in the appendix.27Explicit formulae for these moments are to be found in the appendix.

22

estimate the model by pseudo-maximum likelihood(Gourieroux, Monfort, and Trognon 1984).

In this approach, I maximize a pseudo-likelihood function derived from a quadratic exponential

family. Since the observed price distribution is approximately log normal, and the log normal

distribution is quadratic exponential, I choose it as the basis of my pseudo likelihood function

and maximize the criterion function:

L(θ) = (−1/2)J∑

j=1

(log(Ω[log p|n, zj , θ]) +

(log pj − E[log p|n, zj , θ])2

Ω[log p|n, zj , θ]

)(4)

Provided that the model is identified, the PML estimator θ, is both consistent and asymptoti-

cally normal (Gourieroux, Monfort, and Trognon 1984).28 To see why the model is identified,

consider the case where there are no covariates apart from n, and the econometrician observes

the prices from repeatedly auctioning objects with values drawn from a common prior. The

number of bidders n in each auction is determined prior to the auction. The aim is to identify

the model primitives, the mean µ and standard deviation σ of the prior distribution, and the

signal standard deviation r. One may identify the prior mean µ from the mean prices of the

objects sold, and the variance in prices pins down σ and r though it does not separately iden-

tify them. For this, you need variation in the number of bidders n, which separately identifies

these parameters through variation in the Winner’s Curse effect.29

5.3 Computation

Due to the large number of covariates, the parameter space is of high dimension, and it is

possible that the problem could quickly become computationally intractable. Fortunately, the

following result eases the computational burden:

Proposition 4 (Properties of the Bidding Function) Let v(x, x;n, µ, σ, r) be the bidding

function under model primitives (µ, σ, r), and let v(x − a, x − a;n, z, µ − a, σ, r) be the bid-

ding function when the signals of all bidders and the prior mean are decreased by a. Then

log v(x, x;n, µ, σ, r) = a + log v(x− a, x− a;n, µ− a, σ, r) and consequently:

E[log p|n, zj , α, β, r] = αzj + E[log p|n, z, α = 0, β, r] (5)

Ω[log p|n, zj , α, β, r] = Ω[log p|n, zj , α = 0, β, r] (6)

28Those familiar with the auctions estimation literature may be concerned about parameter support depen-dence, a problem noted in Donald and Paarsch (1993). It can be shown that under this specification theequilibrium bid distribution has full support on [0,∞), so this problem does not arise here.

29The global identification condition is that if E[log p|n, zj , θ0] = E[log p|n, zj , θ1] and Ω[log p|n, zj , θ0] =Ω[log p|n, zj , θ1] ∀(n, zj) then θ0 = θ1. I have numerically verified this condition for a range of parameter values.

23

This result is extremely useful. To see this, fix the parameters β and r, and consider finding

the parameter estimate α(β, r) that maximizes the criterion function (4). Using the result of

the above proposition, we can write this parameter estimate as:

α(β, r) = Argminα

J∑j=1

1Ω[log p|n, zj , α = 0, β, r]

(log pj − E[log p|n, zj , α = 0, β, r]− αz)2

which for fixed β and r can be solved by weighted least squares (WLS) with the dependent

variable yj = log pj − E[log p|n, zj , α = 0, β, r] and weights wj = 1/Ω[log p|n, zj , α = 0, β, r].

This allows a nested estimation procedure in which I search over the parameter space of (β, r),

using WLS to calculate α(β, r) at each step. Since I choose a specification in which most of

the covariates affect the prior mean and not the variance, this space is considerably smaller

than the full space of (α, β, r), and the computation is much simpler. Further computational

details are to be found in the appendix.

5.4 Endogeneity

In the above analysis, differences between observed prices and those predicted by the structural

model result from the unobserved deviations of the latent value and bidder signals from their

mean values. There is no unobserved source of heterogeneity between cars with identical

observed characteristics. Yet I know that in fact cars sold on eBay differ in many respects

that are not captured by the variables observed by the econometrician, such as in their history

or features. When sellers have favorable private information about the car they are selling,

they will reveal such information on the auction webpage. I capture the dollar value of the

webpage content in a variable ξ. But included in my covariates z are quantitative measures

of information, and these will be correlated with the omitted webpage content value ξ. This

may cause an endogeneity problem. This section investigates when this will be the case, and

provides a control-function based solution if needed.

In this section I will assume that I have a scalar endogenous information measure I.30 For

my application, I must choose between using the number of bytes of text and the number of

photos as this measure. I choose to use the text measure because I have a large number of

missing observations of the number of photos. Fortunately, the amount of text and photos are

highly correlated in the data and thus the amount of text will serve as a good measure of the

information content of the webpage by itself.31

30I suspect that extending the approach to deal with a vector of endogenous variables is possible, but showingthis is beyond the scope of the paper.

31The correlation is 0.35, which is higher than the correlation between age and mileage (0.26).

24

Partition the auction covariates so that zj = [Ij , z′j ] where z′j is the standardized characteristic

data observed by the econometrician, and Ij is the number of bytes of text on the auction

webpage. Now, following the theory, consider a new model specification in which the mean

and standard deviation of the interim prior distribution F |sR depend on the vector of reported

signals, sR:

µj = φµsRj = αzj + ξj (7)

σj = φσsRj = κ (βzj) (8)

where ξj =∑

sRi/∈z′j

φiµsi. In other words, ξ gives the contribution of the signals observed by

the bidders but not the econometrician to the mean of the interim prior. I will sometimes

refer to ξ as the content value of the webpage. I specify that the unobserved signals make no

contribution to the standard deviation, since it is unusual in demand estimation to account

for this possibility.32 At the cost of more notation these results can be extended to cover this

case.

Under the theoretical model, the information measure I is the number of signals si above a

threshold ti. It is therefore reasonable to believe that I is positively correlated with ξ. The

information measure I will thus be a valid proxy for the contribution of the webpage content

ξ to the interim prior mean valuation µ, under standard assumptions:

Assumption 2 (Redundancy) The information measure I is redundant in the true specifi-

cation of the interim prior mean. That is, µj = E[log V |Ij , z′j , ξj ] = E[log V |z′j , ξj ].

Assumption 3 (Conditional Independence) The conditional mean E[ξ|Ij ] is independent

of the observed characteristics z′j.

The redundancy assumption is quite plausible here, as I believe that it is the webpage content

rather than the quantity of information that matters for the interim prior valuation of bidders.

The conditional independence assumption is standard for a non-linear proxy, and essentially

says that when I is included as a regressor, the remaining covariates do not generate an

endogeneity problem. This sort of assumption is typical of the demand estimation literature,

in which only the potential endogeneity of price is typically addressed. With these assumptions

in place, the amount of information I - or more generally an unknown function of it - is a valid

proxy for the contribution of the omitted content to bidder priors. This justifies the estimation

approach in the above sections where I included information measures as regressors without

accounting for endogeneity.32This does not imply that the signals observed by bidders are not informative - it just implies that the prior

variance does not vary with the realization of the signals.

25

Arguing that the potentially endogenous variable I is simply redundant seems to make sense

here, but is clearly not an acceptable solution to the more general problem of endogeneity

in common value auction estimation. I next show how a control function approach can be

used to control for endogeneity when assumption 2 fails but assumption 3 is maintained so

that there is only one endogenous variable. Suppose that I have a valid instrument W for the

endogenous variable I, so that W is independent of ξ but correlated with I. I will need two

final assumptions:

Assumption 4 (Additive Separability) The endogenous variable I is an additively sepa-

rable function of the observed characteristics z, instrument W , and the omitted variable ξ:

Ij = g(z′j ,Wj) + h(ξj)

Assumption 5 (Monotonicity) The function h is strictly monotone.

These are relatively weak assumptions. For my particular application, additive separability is

plausible since the signals are conditionally independent, and so it is not unreasonable to believe

that I depends in a separable fashion on the observed characteristics z and the unobserved

content value ξ. Monotonicity is also reasonable, as when the seller has favorable information,

the theoretical model predicts that he will reveal it.

Now, define mj as the residual Ij − g(z′j ,Wj). Then since mj is monotone in ξj , it is clear that

some function of mj could be used to control for ξj . I may estimate mj as the residuals from

a first stage nonparametric estimation procedure in which I regress Ij on the instrument Wj

and other car characteristics z′j .33 I may then use the estimated residuals mj as a control for

ξ in the second stage regression. I summarize this result in a proposition:

Proposition 5 (Endogeneity) Under assumptions 3, 4 and 5, there exists a control function

f such that:

(a) E[log p|n, z,m, θ] = αz + E[log p|n, z, α = 0, β, r] + f(m)

(b) Ω[log p|n, z,m, θ] does not depend on ξ

Under technical conditions needed to ensure uniform convergence of the estimator, the PML

estimate θ obtained by maximizing the analogue of (4) under the above moment specification

remains both consistent and asymptotically normal. The relevant conditions are provided by

Andrews (1994a). Andrews (1994b) shows that if the first stage nonparametric estimation33This approach uses ideas seen in Blundell and Powell (2003), Newey, Powell, and Vella (1999) and Pinkse

(2000). I also consider a simpler first stage OLS procedure and present the results with this specification in theresults section below.

26

procedure is kernel based, then most of these conditions can be satisfied by careful choice of

the kernel, bandwidth and trimming. A more complete discussion of these technical issues is

deferred to the appendix.

To obtain an instrument, I exploit the fact that I have panel data. My instrument W is the

number of bytes of text in the description of the previous car sold by that seller. The idea is

that the quantity of information posted on the auction webpage for the previous car should

be independent of the value of the webpage content for the current car being sold. This relies

on the unobserved content of the auction webpages ξ being randomly assigned over time for a

given seller. This is reasonably plausible, as each car listed by a given dealer is likely to have

idiosyncratic features and history. On the other hand, the data shows a strong relationship

between the amount of information posted for the current car, I, and that for the previous car

W . This makes W a valid instrument. I estimate the fitted residuals m using kernel regression

with a multivariate Gaussian kernel, bandwidth choice by cross-validation and trimming as in

Andrews (1994b). I implement the control function f as a second-order polynomial in m.34

Regardless of whether or not I is endogenous in the econometric sense, the control function

f(m) should be a better proxy than I for ξ, as it is orthogonal to the contribution of observed

characteristics z′ and firm specific effects captured in W that are included in I but irrelevant

for ξ. My model thus predicts that the control function will be significant in the regression.

In the following section, I present the results of tests for significance and analyze the results of

the structural estimation.

6 Results

The results of the demand estimation follow in Tables 3, 4, 5 and 6, to be found in the

appendix.35 In the first table, I consider estimation of demand without controlling for endo-

geneity. As noted earlier, this approach will be valid under assumptions 2 and 3, in which

case the number of bytes of text will be a valid proxy for the unobserved webpage content.

I consider five specifications, corresponding to those given in the hedonic regressions earlier

in the paper. Throughout this section, one should note that model and title fixed effects are

included but not reported.34As is usual with control function approaches, one cannot be sure what the appropriate form of the control

function is, and so a flexible function is used. I have verified that the pseudo log likelihood does not changesignificantly with the use of higher order or orthogonal (Legendre) polynomials, and so this simpler quadraticmodel was chosen.

35In all cases, standard errors are obtained from estimates of the asymptotic variance-covariance matrix. Ido not account for first stage error in constructing the standard errors for the parameters estimated using thecontrol function.

27

The coefficients on µ give the estimated relationship between the prior valuation and the

covariates. The coefficients on logged regressors may be seen as elasticities, while a one unit

change in an unlogged regressor i has a 100αi% effect on the prior valuation. It is immediately

apparent that age and mileage have significant effects on valuation in all specifications. Ignoring

the curvature introduced by the age squared term, an additional year of age is expected to

decrease bidder’s valuation by 21%, while every percentage increase in the mileage implies an

approximately 0.125% decrease in the price.

As in the hedonic regressions, the effect of the information measures on µ are significant in all

specifications and of large magnitude. I find that each additional percentage of text implies a

percentage increase of between 0.047% and 0.061% in the price (ignoring the final specification,

which includes interactions). For a typical car in the dataset, this implies that a percentage

increase in the amount of text supplied is associated with an increase in the price of about $5,

which is a big effect. But when one thinks of the range of seller revelations possible, including

important details such as car features and history, this is not really that surprising. There is

also a positive and highly significant relationship between the number of photos and the prior

valuation. I find that each photo is associated with an approximately 1.8% increase in the

prior valuation, which amounts to $180 for a typical car in my dataset. These results are as

predicted in Corollary 1, and therefore consistent with the theoretical model.

The additional covariates included in specifications (2)-(5) have coefficients of similar size and

magnitude to those in the hedonic regressions. Of interest is the sizable coefficient on featured

listings, which imply that featured cars are valued by buyers 19% more. Since these returns

vastly outstrip the costs of such a listing, one should be concerned about endogeneity. I do not

include this regressor in later estimations. Finally, I find that as before, the coefficient on the

interactions between the warranty and inspection dummies and my information measures are

significant and negative. This is consistent with the idea that the value of webpage content

such as vehicle history and photos is less valuable while the vehicle is still under warranty.

The second half of the table on the following page contains the coefficients on σ and the

constant bidder signal standard deviation r. These are harder to interpret. Note firstly that

the coefficient on text size is not significant, suggesting that the text does little to reduce prior

uncertainty about the quality of the car. The remaining coefficients for σ are significant at

the 5% level.36 To interpret the coefficients, I turn to the marginal effects. A extra year of

age is associated with a 0.013 increase in the standard deviation of the interim prior, while an

additional percentage of text is associated with a 0.00007 decrease in that deviation. Photos do36The link function used to convert rINDEX into r implies that the T-test here is against the null that

r = 1.25. So the estimate for r is relatively precise, even though rINDEX has a large standard error.

28

far more to reduce uncertainty, with each additional photo associated with a 0.0067 decrease in

the interim prior deviation. The photo content of auction webpages thus improves the precision

of bidder’s prior estimates and help them to avoid the winner’s curse.

One may wish to quantify the magnitude of this winner’s curse effect. To get at this I need

the estimated marginal effect of σ on price. I find that an increase of 1 standard deviation in

the prior uncertainty is associated with a 10% decrease in price. Thus each additional photo,

through decreasing the prior uncertainty, is expected to lead to a 0.067% increase in price, or

about $7 for a typical car. This effect, it is clear, is dwarfed by the direct effect of the photo

content. I can also use the estimates of r and σ to back out how heavily bidders will weight

their private signal as opposed to the public information in forming their posterior estimate of

car value. The relevant statistic is the ratio r/(r+σ), which indicates the weight on the public

information. This is estimated to be around 0.6 in all specifications, indicating that private

communications between buyers and sellers, while important, are less heavily weighted than

webpage content in forming posterior valuations.

In Table 4, I examine how these effects vary for different models of car. I group cars into

three groups, one consisting of “classic” cars such as the Mustang and Corvette, one consisting

of “reliable cars” such as the Honda Accord and Civic, and one consisting solely of pickup

trucks (such as the Ford F Series and GM Silverado). The vehicles in the “classic” group are

considerably older than those in the other two groups, at an average of 24 years old versus 10

years for the reliable cars and 15 for the pickup trucks. A number of differences arise. The

predicted relationship between mileage, age and price varies for the three groups. The difference

is especially marked in the case of classic cars, where mileage has a considerably strongly

negative effect. This may be because vintage cars in near mint condition command much

higher premiums. I also find differences in the coefficient on transmission, which presumably

reflects a preference for manual transmission when driving a Corvette or Mustang, but not

when driving a pickup.

Most important, I find that the coefficients on the information measures vary considerably.

For classic cars, where vehicle history, repair and condition are of great importance, the text

size and photo coefficients are both large. On the other hand, for the reliable and largely

standardized cars such as the Accord and Toyota, these effects are much less. The coefficients

for pickups fall somewhere in between, perhaps because pickups have more options (long bed,

engine size) for a given model, and more variation in wear-and–tear than do the reliable cars.

I also find that the prior standard deviation varies considerably among groups, even after

accounting for age, text and photos. There is considerable prior uncertainty about the value

of classic cars, and far less for reliable cars, which makes sense. I do find though that the

29

uncertainty is quickly increasing in age for reliable cars, which is somewhat surprising. This

may be a local effect, which cannot be extrapolated beyond a neighborhood of the “mean car”.

Finally, I find that the weighting ratio between public and private information is similar for

classic cars and pickups, at around 0.6, but is higher for reliable cars at around 0.65. This

makes sense since these cars have less idiosyncratic features and bidders will therefore weight

private signals less heavily.

In Table 5, I present a comparison of the outcomes of the structural regression with and with-

out a non-linear control function (modeled as a quadratic). The second specification gives

the control constructed with a non-parametric first stage and bandwidth h chosen by cross

validation. The third and fourth specifications are a robustness check, showing that the coef-

ficients remain similar when I vary the bandwidth. The final specification considers a control

function constructed with an OLS first stage, for comparison. The number of observations is

smaller here (around 7000, as opposed to 35000 for the preceding regressions). This is because

construction of the instrument requires a panel of at least two observations for any given seller,

and the first observation of any panel is dropped due to lack of an instrument. In all specifica-

tions I use the log text size as my information measure, using only one such measure because

I only have an instrument for this variable.

One of the predictions of the theory is that the amount of information is an endogenous

variable, in that it is chosen strategically.37 This prediction is validated here, where I find that

the controls are jointly significant in all specifications. For example, I conduct a likelihood

ratio test on the exclusion of all controls, using the log likelihoods from the first and second

specifications. I compute the likelihood ratio test statistic LR = 19.88, which exceeds the

chi-squared critical value χ20.999 = 13.81 at the 0.1% level and thus conclude that the controls

are jointly significant. After including the control function, I find a corresponding decrease in

the coefficient on text size. It is somewhat surprising though that it remains significant and

positive. I explore this in more detailed regressions below. Finally, note that the remaining

coefficients remain largely unchanged after introducing the control function. This makes sense

since the control is orthogonal to the other characteristics z′ by construction.

I consider a specification with controls and different car groups in Table 6. This specification

shows that the surprisingly large coefficient on log text, even after including controls, in Table

5 was due to the classic car group. For the other two car groups, the effect of the amount

of text is statistically indistinguishable from zero, after including controls. For both of these

groups, the controls show up as jointly significant. This validates the model prediction that it37Though it need not cause an endogeneity problem in the econometric sense - see the discussion in the section

on endogeneity.

30

is webpage content, rather than the amount of information that matters.

Yet for classic cars, both the controls and the amount of text are significant and large in

magnitude. A possible explanation for this lies in the nature of the instrument used. In order

for my instrument to be valid, I need random assignment of the content value ξ across cars

sold by the same seller. For car dealerships, who typically deal in high volume cars including

pickups and reliable cars, this assumption is reasonable. On the other hand, many of the sellers

of classic cars on eBay buy them elsewhere and restore them. In such a case, much of the value

of the car is wrapped up in the restoration process, and the unobserved content ξ will have a

large seller-specific term. The coefficient on log text may be picking up this effect.

The main results are as follows. First, the predictions of the costly revelation model hold

up in the data, across all specifications. Second, the fact that logtext is not significant after

controlling for endogeneity in two of the three specifications suggests that the amount of text

proxies for webpage content, rather than having any independent effect of its own. Third, I

find that webpage content has an important role in explaining prior valuations, which suggests

that seller revelation is important for informing bidders in this market.

7 Counterfactual Simulation

One of the strengths of structural estimation is the ability to perform counterfactual simulations

of market outcomes. In this section I return to the idea that motivated the paper, and quantify

the magnitude by which seller revelations through the auction webpage help to reduce the

potential for adverse selection. My tool in this endeavor is a stark counterfactual: a world in

which eBay stops sellers from posting their own information on the auction webpage, so that

only basic and standardized car characteristic data is publicly available.

This is hardly an idle comparison. Local classified advertisements typically provide only basic

information, while even some professional websites do not allow sellers to post their own

photos.38 In the case of local purchases, of course, potential buyers will view the cars in

person and so their private signals will be much more precise. This is typically not the case on

eBay Motors. I will show that even when private signals reflect the true value of the vehicle,

the less precise prior valuations held by bidders in the counterfactual world will lead them to

systematically underbid on “peaches” and overbid on “lemons”. Relative to the counterfactual

then, seller revelations through the auction webpage help to limit differences between prices

and valuations, and reduce adverse selection.38The South African version of Autotrader, autotrader.co.za, is such an example. Even on large online websites

such as cars.com and autobytel.com, it is unusual to find multiple photos for a vehicle.

31

Formally, let the data generating process (DGP) be as specified in the structural model pre-

sented earlier. For simplicity, I assume throughout this section that there is no unobserved

heterogeneity or endogeneity, so that I have access to all the information needed to simulate the

DGP. Since bidders in fact learn more from the auction webpage than I capture in my coarse

information measures, this assumption will lead me to underestimate the role of seller revela-

tion in my simulation results. Recall that values are log normally distributed, so that v = log v

has normal distribution with mean µ = E[v|I, z′] and standard deviation σ =√

V ar[v|I, z′].

Bidders independently observe unbiased and normally distributed private signals of the log

value v, with signal standard deviation r.

Now consider two regimes. In the true regime, bidders observe the auction webpage and

form a prior valuation based on the standardized characteristic data z′ and the amount of

information I. Their prior is correct in equilibrium, and thus parameterized by µ and σ.

In the counterfactual regime, the auction webpage contains only the standardized data and

so bidders have a less informative prior. This prior is log normal, and parameterized by

µc = E[v|z′] and σc =√

V ar[v|z′]. Bidders still obtain independent and unbiased private

signals in both regimes.

To see how prices respond to seller revelation in the equilibrium and in the counterfactual

case, I simulate the bidding function and compute expected prices under both regimes for a

random sample of 1000 car auctions in my dataset. For the equilibrium regime, this is relatively

straightforward. The estimated structural parameters imply estimates of the model primitives

µ, σ and r for each data point, and I can compute the expected price E[p|I, z′] predicted by the

model. To get the expected price under the counterfactual, E[pc|I, z′], I estimate µc and σc.

I do this by integrating out I from E[v|I, z′] and V ar[v|I, z′] over the conditional distribution

of I given z′. This gives me the counterfactual prior, and allows me to simulate the bidding

function for any signal realization xi. I obtain the expected price E[pc|I, z′] by integrating over

the distribution of ordered signal realizations x(n−1:n) specified by the DGP. Since it will be

useful to compare relative prices and values across auctions, I specify a baseline price. I define

the baseline price as the expected price E[pb|z′] which would obtain if the counterfactual prior

specified by µc and σc was in fact correct, so that µc = µ and σc = σ. One may think of this

as the case in which the seller is selling a car whose condition is “average” in the sense that

his equilibrium revelations would not cause bidders to revise their priors at all. Notice that

under the DGP the distribution of values depends on I, and thus so do the private signals and

the expected prices. This is the sense in which private signals reveal the true value of the car

even in the counterfactual world.39

39Additional details about the simulation are given in the appendix.

32

!15 !10 !5 5 10 15Rel. Value !%"

!15

!10

!5

5

10

15Rel. Price !%"

Figure 5: Simulation Results Relative value is defined as a function of the ratio E[V |I, z′]/E[V |z′],so that cars with a high relative value are revealed through seller revelations to be “peaches”, while theopposite implies the car is a lemon. The blue dots show the percentage by which the simulated pricesfor these cars exceed a baseline price E[pb|z′] under the true regime. The green dots show the relativeprice under the counterfactual in which sellers cannot reveal their private information on the auctionwebpage, but buyers still get accurate private signals.

The results are graphically depicted in Figure 5. On the x-axis, I have the relative value of

the car, which I define as the percentage by which the informed prior valuation E[V |I, z′]

exceeds the uninformed prior E[V |z′]. A simple characterization of this is that those cars with

relative value greater than zero are “peaches”, while those with relative value less than zero are

“lemons”. On the y-axis, I have the relative price of the car, which I define as the percentage

by which the expected price E[p|I, z′] or E[pc|I, z′] exceeds the baseline price E[pb|z′]. Since

the baseline price is that for an average car with characteristics z′, relative prices should be

positive when the car is better than average (i.e. a peach), and negative when the car is worse

than average (i.e. a lemon). The points in blue are expected prices under the equilibrium

model, while those in green are from the counterfactual simulation.

It is clear after examining Figure 5 that a linear model would describe the relationship between

relative value and relative price quite well in both regimes. Fitting such a model yields an

estimated slope coefficient for the equilibrium model of 1.014., while that for the counterfactual

is 0.609.40 This implies a seller with a car revealed to be worth 10% more than the baseline

can expect to fetch 10.1% more for his car than the baseline price when sold on eBay Motors.

Thus a seller with a peach can post information and inform buyers of the car’s quality and40Both are estimated extremely precisely, with t-statistics of 413 and 121 respectively.

33

hence obtain a price premium commensurate with the value of the car.41 By contrast, under

the counterfactual, that same car would fetch only 6% more than the baseline, because the

prior value that potential buyers place on the car is the baseline value. Although bidders

will on average receive favorable private signals indicating that the quality of the car is high,

such signals receive limited weight because they are noisy. The overall effect is that without

revelation on the auction webpage, the seller is on average only able to extract 61% of the

additional value of his vehicle in revenue.

These simulation results make clear the role that public seller revelations can play in reducing

adverse selection in this market. Whenever sellers with high quality cars are unable to extract

a commensurate price premium from buyers because of information asymmetries, there is

the potential for adverse selection. In the counterfactual regime without revelation, the gap

between relative value and relative price is nearly 40%, and this potential is high. It is clear

from my simulations then that seller revelations reduce the potential for adverse selection

significantly.

8 Conclusion

In a world in which ever more esoteric items are sold online, it is important to understand

what makes these markets work. This paper shows that information asymmetries in these

markets can be endogenously resolved, so that adverse selection need not occur. The required

institutional feature is a means for credible information revelation. With this in place, sellers

have both the opportunity and the incentives to remedy information asymmetries between

themselves and potential buyers.

Where such mechanisms for credible information revelation exist, it is clear from my results

that these revelations can be important determinants of buyer behavior and thus prices. The

theory developed in this paper suggests that in the face of revelation costs, sellers will reveal

only the most favorable of their private information. It follows that quantitative measures of

information, such as those provided in this paper, will have predictive power for prices. Future

research could see whether this holds true in other settings, for if it does, this provides a useful

set of proxies for unobserved differences in heterogeneous goods markets with information

asymmetries.

41In fact, the seller will do slightly better because the additional information reduces uncertainty and theWinner’s Curse, and thereby raises prices.

34

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9 Appendix

A. Proof of Proposition 1

The first two results follow from Proposition 1 in Bajari and Hortacsu (2003), after noting

that the two auction models are strategically equivalent. To obtain the final result, order the

bidders by the realization of their signals beginning with the bidder with the highest signal,

bidder 1. Now, let bidder 2 submit his bid in the late stage at randomly determined time

t2 ∈ [τ − ε, τ ] . Since bids are monotone in the signals, and the standing price is the second

highest bid submitted on [0, t2), under the equilibrium strategies the standing price pt2 cannot

exceed v(x(n−2:n), x(n−2:n);n). Thus the bid of bidder 2, v(x(n−1:n), x(n−1:n);n), will exceed the

standing price and be recorded. At some point t1 in [τ − ε, τ ], the bid of bidder 1 will likewise

be recorded. Thus at the end of the auction, the final price will be the second highest of the

recorded bids, which is v(x(n−1:n), x(n−1:n);n).

B. Proof of Proposition 2

Sequential equilibrium requires that bidders have consistent beliefs and that their behavior is

sequentially rational. So consider any equilibrium reporting policy R. On path, consistency

requires that bidders believe that reported signals SR are equal to their reports sR; and that

unreported signals SU are consistent with the reporting policy, so R(S) = r. Now, a report

vector r is off-path if there does not exist a signal realization s such that R(s) = r. In that

case, consistency still requires that bidders believe that reported signals SR are equal to their

reports sR, but places limited restrictions on the unreported signals. I specify that these

beliefs are Si = s ∀i ∈ U , so that bidders believe the worst of off-path unreported signals.

These beliefs satisfy consistency. Now, the symmetric strategies in which bidders bid zero

in the early stage of the auction and bid E[w(x, x, s;n)|R(s) = r, SR = sR] (on path) or

E[w(x, x, s;n)|SR = sR, SU = s] (off path) in the late stage are mutual best responses. No

bidder has strict incentive to deviate in the early stage. In the late stage, bidders condition

on the revealed information sR and make (correct) inferences about the unrevealed signals

based on the seller’s equilibrium reporting policy. Their play satisfies sequential rationality.

To complete the equilibrium construction, I must show that the reporting policy Ri(s) = 1 ∀iis optimal for the seller. Notice that for any deviation R′, there must exist signal realizations

s with R′i(s) = 0. But for such realizations, the seller will expect to receive (weakly) less

revenue than under the equilibrium reporting policy, since the bids under R′ take the form

E[w(x, x, s;n)|SR = sR, sU = s] and this is weakly less than the equilibrium bid w(x, x, s;n).

38

C. Proof of Proposition 3

I begin with a useful lemma.

Lemma 1 Let U be a subset of 1 · · ·m that includes element i, and let U−i be the same set

without element i. Then:

E[w(x, x, s;n)

∣∣sj < tjj∈U−i

]− E [w(x, x, s;n)|sj < tjj∈U ] = E

[∆i(si, s

′i;n)

∣∣s′i < ti]

Proof:

E[w(x, x, s;n)

∣∣sj < tjj∈U−i

]−E [w(x, x, s;n)|sj < tjj∈U ]

= E[w(x, x, s;n)− E [w(x, x, s;n)|si < ti]

∣∣sj < tjj∈U−i

]= E

[E[w(x, x, s−i, si;n)− w(x, x, s−i, s

′i;n)

∣∣s′i < ti]∣∣sj < tjj∈U−i

]= E

[E[∆i(si, s

′i;n)|s′i < ti

]∣∣sj < tjj∈U−i

]= E

[∆i(si, s

′i;n)

∣∣s′i < ti]

where the final line follows by Assumption 1.

Now, to prove the proposition, let bidders have beliefs as specified in the proof of Proposition

2 above. Let the seller’s reporting policy be as in the proposition. As before, the symmetric

strategies where bidders bid zero during the early stage of the auction and E[w(x, x, s;n)|R(s) =

r, SR = sR] in the late stage are mutual best responses (all points are on path). Under the

sellers’ reporting policy, I may write E[w(x, x, s;n)|R(s) = r, SR = sR] as vR(x, x, sR;n) =

E[w(x, x, s;n)|si < tii∈U ]. It remains to show that the seller’s strategy is optimal. To this

end, consider a deviation. Under the constant differences assumption, it will suffice to show

that for all realizations of the signal vector s, there is no signal si for which a deviation in

R(si) is profitable. So consider revealing si < ti for some i. The payoff to deviation depends

on the seller’s expected revenue, and is given by:

π =E[E[vR(x, x, sU−i ;n)− vR(x, x, sU ;n)

∣∣∣X(n−1:n) = x]∣∣∣S = s

]− ci

=E[E[E[w(x, x, s;n)|sj < tjj∈U−i

]− E [w(x, x, s;n)|sj < tjj∈U ]

∣∣∣X(n−1:n) = x]∣∣∣S = s

]− ci

= E[E[E[∆i(si, s

′i;n)

∣∣s′i < ti]∣∣∣X(n−1:n) = x

]∣∣∣S = s]− ci

= E[∆i(si, s

′i;n)|s′i < ti

]− ci

where the third line follows by Lemma 1, and the fourth by Assumption 1. Now since ti =

inf t : E [∆i(t, t′;n)|t′ < t] ≥ ci, and si < ti, this expression is negative and thus the deviation

39

is unprofitable. It can similarly be proved that the opposite deviation (not revealing si > ti)

is unprofitable. Hence the seller’s reporting policy is a best response to the bidder’s bidding

strategies. Finally, part (iii) follows immediately from the proof of proposition 1.

D. Proof of Corollary 1

By Bayes Rule:

E[V |I] =∫

vf(v|I)dv =∫

vg(I|v)∫

g(I|y)dF (y)dv =

∫v`(v, I)dv

One may show that `(I, v) is log supermodular (LSPM) - see below. It follows immediately

that v`(l, v) is also LSPM. But then the right hand side has the form EV [h(v, I)] where h(v, I)

is LSPM, and from the results in Athey (2002) it follows that EV [h(v, I)] is increasing in I.

This establishes the result.

E. Log Supermodularity of `(I, v)

A real-valued function g(x, y) : Rm+n → R is said to be LSPM if for all (x1, y1) and (x2, y2):

g(x1 ∧ x2, y1 ∧ y2)g(x1 ∨ x2, y1 ∨ y2) ≥ g(x1 ∧ x2, y1 ∨ y2)g(x1 ∨ x2, y1 ∧ y2)

where ∨ is the component-wise minimum operator, and ∧ is the component-wise maximum

operator. Since `(I, v) is two-dimensional, it will suffice to prove that for v1 < v2 and I1 < I2,

we have:

`(v1, I1)`(v2, I2) ≥ `(v1, I2)`(v2, I1)

Let pi(v) = 1−H(ti|v). By the affiliation of Si and V for all i, it follows that pi(v) is increasingin v ∀i. Let r be a vector of reports and let ||r|| = #i : ri = 1. Then we have:

g(I|v) =∑

r:||r||=I

(∏ri=1

pi(v)∏ri=0

(1− pi(v))

)

where I use the conditional independence of the signals Si to express the RHS as products.

Since the denominators on both sides cancel, it will suffice to prove that g(I1|v1)g(I2|v2) ≥

40

g(I1|v2)g(I2|v1). Take the LHS and expand it:

LHS =

∑r:||r||=I1

(∏ri=1

pi(v1)∏ri=0

(1− pi(v1))

) ∑r:||r||=I2

(∏ri=1

pi(v2)∏ri=0

(1− pi(v2))

)=

M∑j=1

N∑k=1

∏rj

i =1

pi(v1)∏

rji =0

(1− pi(v1))∏

rki =1

pi(v2)∏

rki =0

(1− pi(v2))

where j = 1 · · ·M index the

(SI1

)elements with ||r|| = I1 and k = 1 · · ·N index the

(SI2

)elements with ||r|| = I2. Now expand and index the RHS in an identical fashion, and comparethe jk-th elements. After canceling identical terms on both sides, the result is:∏

rji =0,rk

i =1

pi(v2) (1− pi(v1)) ≥∏

rji =0,rk

i =1

pi(v1) (1− pi(v2))

This inequality holds by the monotonicity of pi(v) ∀i, since that implies pi(v2) > pi(v1) and

1 − pi(v1) > 1 − pi(v2). But then since the inequality holds for all terms jk in the sum, it

holds for their summation as well. This proves the log supermodularity of `(I, v).

F. Proof of Proposition 4

By definition, v(x, x;n, µ, σ, r) = E[V |Xi = x, Y = x,N = n]. Condition on all other signals,so that we have:

v(x, x;n, µ, σ, r) = EX−i|Y <x [V |Xi = x,X−i = x−i, N = n]

= EX−i|Y <x

[exp

rµ + σ

∑ni=1 xi

r + nσ+

12

(rσ

r + nσ

)2∣∣∣∣∣Xi = x,X−i = x−i, N = n

]

= EX−i|Y <x

[exp

a +

r(µ− a) + σ∑n

i=1(xi − a)r + nσ

+12

(rσ

r + nσ

)2∣∣∣∣∣Xi = x,X−i = x−i, N = n

]

= eaEX−i|Y <x

[exp

r(µ− a) + σ

∑ni=1(xi − a)

r + nσ+

12

(rσ

r + nσ

)2∣∣∣∣∣Xi = x, X−i = x−i, N = n

]= eav(x− a, x− a;n, µ− a, σ, r)

where the second line follows from the posterior mean and variance of log v given normal

unbiased signals xi, and the fourth line follows on noting that ea is a constant and thus can

be taken outside of the expectation. Taking logs on both sides, we get log v(x, x;n, µ, σ, r) =

a + log v(x− a, x− a;n, µ− a, σ, r).

To get equation (5), note that E[log p(n, µ, σ, r)] = E[log v(x(n−1:n), x(n−1:n), n, µ, σ, r)]. It

follows that E[log v(x(n−1:n), x(n−1:n), n, µ, σ, r)] = a + E[log v(x(n−1:n), x(n−1:n);n, µ− a, σ, r)]

41

where the signals in the RHS expectation have the same conditional distribution G|v as those

on the LHS, but the latent v now has a prior with mean µ− a on the RHS. Now take a = αzj

for any observation j. Then by specification µ− a = 0, and

E[log p(n, zj , α, β, r)] = αzj + E[log p(n, zj , 0, σ, r)]

The variance result follows in similar fashion.

G. Proof of Proposition 5

To begin, one must establish the results on the moments. (b) is immediate, since by Proposition

4, Ω[log p|n, z,m, θ] does not depend on µ, and by assumption ξ enters only µ and not σ and

r. For (a), I argue as in Pinkse (2000) and write:

E[log p|n, z,m, θ] = αz + E[log p|n, z, α = 0, β, r] + E[ξ|I, z′,m]

= αz + E[log p|n, z, α = 0, β, r] + E[ξ|m]

= αz + E[log p|n, z, α = 0, β, r] + h−1(m)

= αz + E[log p|n, z, α = 0, β, r] + f(m)

where the first line follows by Proposition 2 above. The third line follows on noting that

I − m = g(z′,W ) is independent of ξ, as is z′. The existence of an inverse in the fourth line

follows by the monotonicity of h.

H. Asymptotics for the Control Function Estimator

In this section I provide a discussion of the requirements for consistency and asymptotic nor-

mality of the PML estimator with a control function. All the analysis here is based on Andrews

(1994a) and Andrews (1994b). Define the sample and limiting objective functions:

LJ(θ, m) = (−1/2J)J∑

j=1

(log(Ω[log p|n, zj , θ]) +

(log pj − E[log p|n, zj ,mj , θ])2

Ω[log p|n, zj , θ]

)

L∞(θ, m) = (−1/2)E[log(Ω[log p|n, z, θ]) +

(log p− E[log p|n, z,m, θ])2

Ω[log p|n, z, θ]

]Denote the true parameter vector as θ0 and the true residual as m0. Following Theorem

A-1 in Andrews (1994a), four requirements need to be satisfied for consistency. The first is

that LJ(θ, m) uniformly weakly converges to L∞(θ, m) on Θ × F , for F a function space for

42

m = I − E[I|W, z′]. This follows by a WLLN since the sample consists of independent draws

from the stochastic process that generates the random variables (P,Z,N, W ). The second

is that the supremum distance supθ∈Θ ||L∞(θ, m) − L∞(θ, m0)|| converges in probability to

zero. This follows from the consistency of the first stage kernel estimator for m0. Thirdly,

I need uniform continuity of LJ(θ, m), which is straightforward. Finally, I need a separation

condition, so that for every neighborhood Θ0 of θ0, infθ∈Θ/Θ0L∞(θ, m0) > L∞(θ0,m0). This

follows from identification of the model and the Kulback inequality, as in Gourieroux, Monfort,

and Trognon (1984).

Obtaining asymptotic normality is somewhat less straightforward. The idea is to first show

that√

T (θm0 − θ0) is asymptotically normally distributed, where θm0 is the maximizer of the

sample criterion function with the true residuals m0 substituted for the estimated residuals m.

Then if the convergence of m to m0 is sufficiently smooth, asymptotic normality of√

T (θ −θ) will follow. The formal requirements are given in Theorem 1 of Andrews (1994a). The

asymptotic normality of√

T (θm0 − θ0) follows immediately from the asymptotic normality

results in Gourieroux, Monfort, and Trognon (1984), under standard conditions on LJ(θ, m).

Here we will also need to extend these standard conditions to all of F so that LJ(θ, m) is twice

continuously differentiable and the vector of first order conditions satisfy uniform WLLNS for

all m ∈ F . But the key requirements are an asymptotic orthogonality condition on the vector

of first order conditions√

T ∂/∂θ(LJ(θ0, m)), and the stochastic equicontinuity of LJ around

m0. For a kernel-based estimator, Andrews (1994b) provides sufficient conditions on the first

stage estimator for these to hold. For the asymptotic orthogonality condition, it is necessary

that the kernel estimator be adaptive. I implement the kernel estimator with bandwidth chosen

by cross-validation to satisfy this. For the stochastic equicontinuity condition, it is necessary

to use kernels with smooth derivatives, and trim observations in a smooth manner. To satisfy

these conditions, I use a multivariate Gaussian kernel, and underweight outlying observations

when forming the conditional expectation m. It is important that the trimming function

has the property not only of assigning zero weight to outlying observations, but that it also

underweights observations near to the outliers, and so as suggested in Andrews (1994b) I use a

second kernel to re-weight observations based on their mahalanobis distance from other points

in the data.

I. Demand System Computation

In this section I provide a number of additional details on the demand system computation. Ido this for the case in which a control function is not used, as the case with a control functionis extremely similar apart from the first stage estimation of residuals. The first step is to

43

compute the moments E[log p|n, µ = 0, σ, r] and Ω[log p|n, µ = 0, σ, r] for a grid of values of r,σ and n (my grid has 3600 points). To do this, I use the following formulae:

v(x, x, n;µ, σ, r) =∫

eqφ((x− q)/r)2Φ((x− q)/r)n−1φ((q − µ)/σ)dq∫φ((x− q)/r)2Φ((x− q)/r)n−1φ((q − µ)/σ)dq

E[log p|n, µ = 0, σ, r] =∫ ∫

log v(x, x, n, µ = 0, σ, r)φ((x− q)/r) (1− Φ((x− q)/r))Φ((x− q)/r)n−2φ(q/σ)dxdq

E[(log p)2|n, µ = 0, σ, r] =∫ ∫

log v(x, x, n, µ = 0, σ, r)2φ((x− q)/r) (1− Φ((x− q)/r))Φ((x− q)/r)n−2φ(q/σ)dxdq

where q = log v is the latent log value of the object, and φ and Φ are the standard nor-

mal density and cumulative distribution function respectively. I compute these moments

to very high accuracy using Gauss-Kronrod quadrature. In the algorithm below, when re-

quired to provide either of these moments for a point not on the grid, I use three dimen-

sional linear interpolation, which remains highly accurate. I transform the indices βz and

rINDEX with the functions κ(βz) = σ + (σ − σ) (0.5 + arctan(βz/π)), and ν(rINDEX) =

r +(r− r) (0.5 + arctan(rINDEX/π)) where the grid boundaries for fixed n are [σ, σ] and [r, r].

These transformations constrain the indices to lie on the grid, which is convenient for compu-

tation. The estimated values do not lie near the boundaries, so my grid boundary choice does

not appear to apply constraints to the estimator.

The algorithm for finding the estimate θ given this grid of pre-computed values is then as

follows:

1. Draw an initial parameter vector (β0, r0). For each observation, generate σj under

the specification and then interpolate on the grid to find E[log p|n, µ = 0, σj , r] and

Ω[log p|n, µ = 0, σj , r].

2. As in the text, generate a temporary dependent variable yj = log pj − E[log p|n, µ =

0, σj , r] and weights 1/Ω[log p|n, µ = 0, σj , r], and find α(β, r) by WLS.

3. Evaluate the objective function, and using a quasi-Newton algorithm, pick a new point

(β1, r1). Repeat until convergence has been achieved.

J. Additional Simulation Details

There are two steps to the simulation: estimating the counterfactual prior parameters µc andσc, and computation of the expected true, counterfactual and baseline prices. For the first step,note that µc

j = E[v|z′j ] = E[E[v|Ij , z′j ]|z′j ] = E[µj |z′j ] = α′z′j +αIE[I|z′j ]. I replace the parame-

ter α with the structural estimate α, and the term E[I|z′j ] with a nonparametric kernel estimate

I to get an estimate of µcj . Similar analysis shows σc

j =√

E[σ2j |z′j ] + α2

I

(E[I2|z′j ]− E[I|z′j ]2

). I

44

estimate all unknown terms nonparametrically. Then I use the following formulas in computingthe true, counterfactual and baseline expected prices:

E[pj |I, z′] =∫ ∫

v(x, x, n, µj , σj , r)φ((x− q)/r) (1− Φ((x− q)/r))Φ((x− q)/r)n−2φ((q − µj)/σj)dxdq

E[pcj |I, z′] =

∫ ∫v(x, x, n, µc

j , σcj , r)φ((x− q)/r) (1− Φ((x− q)/r))Φ((x− q)/r)n−2φ((q − µj)/σj)dxdq

E[pbj |z′] =

∫ ∫v(x, x, n, µc

j , σcj , r)φ((x− q)/r) (1− Φ((x− q)/r))Φ((x− q)/r)n−2φ((q − µc

j)/σcj)dxdq

where the dependence on I and z of µ and σ is suppressed. I compute the integrals using

Gauss-Kronrod quadrature, evaluating all three for a sample of 2000 points from my data set.

K. Testing for Common Values

I justified the use of a symmetric common values (CV) auction model by arguing that the

common uncertainty generated by the information asymmetries between the seller and the

bidders is more important than idiosyncrasies in individual preferences. Here I provide a

formal test of symmetric common values against the alternative of symmetric private values

(PV). This test, originally proposed by Athey and Haile (2002), depends on the following

equality:2n

Pr(B(n−2:n) < b) +n− 2

nPr(B(n−1:n) < b) = Pr(B(n−2:n−1) < b) (9)

where B(i:n) denotes the i-th ordered bid in the n person auction.

This relationship holds in symmetric PV models since in those frameworks the number of

bidders n has no effect on valuations. By contrast, in CV models the Winner’s Curse effect

means that bids are on average lower as n increases than in the PV model. This implies that

the distribution on the LHS of (9) is first order stochastically dominated by that on the right

hand side.

I observe the second and third highest bids in every auction B(n−1:n) and B(n−2:n), and thus for

each n > 2 can construct the distributions on either side of (9). Let Gn(b) = 2nPr(B(n−2:n) <

b) + n−2n Pr(B(n−1:n) < b) and Hn(b) = Pr(B(n−2:n−1) < b). To test for stochastic dominance,

I use a version of the Kolmogorov-Smirnov test statistic proposed by Haile, Hong, and Shum

(2003), given by:

δT =n∑

n=n

supε

GT

n (ε)− HTn (ε)

(10)

where n and n are respectively the highest and lower number of bidders observed in all auctions,

and T =∑n

n=n Tn for Tn the number of observations of auctions with number of bidders

45

n. GTn (ε) and HT

n (ε) are calculated for each n from the empirical distribution function of

the observed residuals, Fn(ε), where the residuals ε have been obtained from separate OLS

regressions of the second and third highest bids on covariates z. I use residuals instead of

the bids in forming the test statistic because the objects being auctioned are not identical,

and a sensible null hypothesis is that of conditionally independent private values, where the

distribution of values is identical across auctions conditional on the covariates z.

Under the null hypothesis, the distributions Gn and Hn are equal for each n, whilst under

the common values alternative, first order stochastic dominance implies that ST will be large.

To obtain an asymptotically valid critical value, I use a subsampling approach as in Haile,

Hong, and Shum (2003) and Linton, Maasoumi, and Whang (2005). Define the normalized

test statistic ST =√

TδT . This differs from Haile, Hong, and Shum (2003) because they are

obliged to use a nonparametric first stage in estimating the underlying valuations, and thus

obtain a slower uniform convergence rate. I draw m samples r1 · · · rm of size b from the original

dataset without replacement, calculating Sb for each one. Forming the empirical distribution

function Ψ of the m statistics Sb, I take the critical value for a test of size α to be the (1−α)-th

quantile, S1−α of Ψ. The validity of this approach follows from results in Politis, Romano, and

Wolf (1999). In practice, I choose the number of draws m to be 1000, and since results can be

sensitive to the choice of sample size, I consider two sample sizes, sizes 30% and 40% of my

original dataset. The results follow in Table 2 below:

Table 2: Test for Common Values

Critical ValuesSample Size S0.9 S0.95 S0.99

30 % 48.592 50.184 52.59540 % 49.41 50.393 53.07Test Statistic 51.664

The test statistic is significant at the 5% level, and I can reject the null of symmetric private

values. But there is reason to view this result with caution, as in general this test will tend to

over-reject the null in an eBay context. This is because the third highest observed bid, b(n−2:n),

may not actually be the bid of the bidder with the third highest valuation (in a PV context)

or signal (in a CV context). As noted earlier, such a bidder may be too late in entering his

or her bid, and not have it recorded. It follows that the estimated distribution GTn (b) may

be stochastically dominated by the true distribution Gn(b) even in a private values context,

where “true” should be interpreted here to mean the distribution of bids in the counterfactual

case that all bids were recorded. Nonetheless this result is consistent with the assumption of

symmetric common values in the model used in the body of the paper.

46

Table 3: Structural Demand Estimation

(1) (2) (3) (4) (5)µ

Age -0.2107 -0.2094 -0.2091 -0.2088 -0.2095(0.0023) (0.0023) (0.0023) (0.0023) (0.0024)

Age Squared 0.0043 0.0043 0.0043 0.0043 0.0043(0.0001) (0.0001) (0.0001) (0.0001) (0.0001)

Log of Miles -0.1258 -0.1248 -0.1246 -0.1242 -0.1256(0.0059) (0.0058) (0.0058) (0.0058) (0.0060)

Log of Text Size 0.0537 0.0471 0.0592 0.0610 0.0785(0.0065) (0.0065) (0.0068) (0.0070) (0.0079)

Number of Photos 0.0185 0.0179 0.0179 0.0176 0.0207(0.0009) (0.0009) (0.0009) (0.0009) (0.0010)

Manual Transmission 0.0404 0.0404 0.0385 0.0378 0.0341(0.0111) (0.0111) (0.0110) (0.0110) (0.0110)

Featured Listing 0.1915 0.1897 0.1925 0.1983(0.0159) (0.0159) (0.0160) (0.0159)

Log Feedback -0.0195 -0.0178 -0.0167(0.0029) (0.0029) (0.0029)

Percentage Negative Feedback -0.0027 -0.0025 -0.0027(0.0013) (0.0013) (0.0013)

Total Number of Listings -0.0040 -0.0044(0.0011) (0.0011)

Phone Number Provided 0.0605 0.0641(0.0106) (0.0105)

Address Provided -0.0361 -0.0352(0.0154) (0.0154)

Warranty 0.4761(0.1175)

Warranty*Logtext -0.0503(0.0182)

Warranty*Photos -0.0139(0.0022)

Inspection 0.4388(0.0937)

Inspection*Logtext -0.0433(0.0142)

Inspection*Photos -0.0052(0.0018)

Constant 10.6771 10.6952 10.6726 10.6485 10.4939(0.0764) (0.0759) (0.0759) (0.0767) (0.0853)

Estimated standard errors are given in parentheses. The table gives the estimated relationship between theprior mean valuation µ and auction covariates. The nested specifications (2)-(4) include controls for marketingeffects, seller feedback and dealer status. Specification (5) includes warranty and inspection dummies and theirinteractions with the information measures, showing that the value of photos and text is lower when the car hasbeen inspected or is under warranty.

47

Table 3: Structural Demand Estimation (continued)

(1) (2) (3) (4) (5)σ

Age 0.0336 0.0337 0.0337 0.0335 0.0335(0.0022) (0.0022) (0.0022) (0.0022) (0.0022)

Log of Text Size -0.0185 -0.0206 -0.0215 -0.0221 -0.0220(0.0357) (0.0360) (0.0360) (0.0360) (0.0363)

Photos -0.0178 -0.0182 -0.0182 -0.0182 -0.0183(0.0053) (0.0054) (0.0054) (0.0054) (0.0055)

Constant -1.1490 -1.1384 -1.1344 -1.1287 -1.1345(0.2234) (0.2241) (0.2239) (0.2237) (0.2254)

rINDEX -0.0630 -0.0648 -0.0637 -0.0619 -0.0642(0.0373) (0.0364) (0.0366) (0.0374) (0.0363)

Marginal Effects for σ

Age 0.0127 0.0126 0.0126 0.0125 0.0124Log of Text Size -0.0070 -0.0077 -0.0081 -0.0083 -0.0082Photos -0.0067 -0.0068 -0.0068 -0.0068 -0.0068Mean σ 0.8472 0.8448 0.8442 0.8433 0.8402r 1.2099 1.2088 1.2095 1.2106 1.2092

Estimated standard errors are given in parentheses. The parameters in this table show the estimated relationshipbetween the prior standard deviation σ, which is a measure of prior uncertainty, and auction covariates z. Sincethe relationship between σ and z is modeled as being non-linear, the marginal effects are easiest to interpret.An estimate of r, the standard deviation of bidders private signals, is also provided.

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Table 4: Structural Demand Estimation by Car Group

Classic Cars Reliable Cars Pickupsµ

Age -0.1794 -0.1732 -0.1936(0.0029) (0.0159) (0.0038)

Age Squared 0.0038 0.0007 0.0030(0.0001) (0.0006) (0.0001)

Log of Miles -0.1370 -0.1059 -0.1097(0.0070) (0.0213) (0.0124)

Log of Text Size 0.0657 0.0329 0.0339(0.0095) (0.0206) (0.0115)

Photos 0.0202 0.0062 0.0172(0.0011) (0.0024) (0.0022)

Manual Transmission 0.2204 0.0881 -0.1757(0.0140) (0.0221) (0.0231)

Constant 10.8249 10.7823 11.1512(0.0953) (0.2557) (0.1646)

σ

Age 0.0278 0.0950 0.0321(0.0026) (0.0306) (0.0035)

Log of Text Size -0.0273 -0.0236 -0.0062(0.0400) (0.1542) (0.0853)

Photos -0.0132 -0.0169 -0.0213(0.0050) (0.0192) (0.0156)

Constant -1.1110 -2.0891 -1.1467(0.2635) (1.0739) (0.5371)

rINDEX -0.0793 -0.2537 0.0248(0.0292) (0.0290) (0.5906)

Marginal Effects for σ

Age 0.0109 0.0190 0.0109Log of Text Size -0.0107 -0.0047 -0.0021Photos -0.0052 -0.0034 -0.0072Mean σ 0.8512 0.6567 0.8065r 1.1996 1.0918 1.2658

Estimated standard errors are given in parentheses. This table gives the estimated relationship between thestructural parameters and the covariates for each of the main car groups in my dataset. In all cases the priormean is positively related to the amount of text and photos, but the magnitudes vary significantly by group.

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Table 5: Structural Demand Estimation with Controls

No Control Control Control Control OLS Control(h = 0.2) (h = 0.1) (h = 0.3)

µ

Age -0.2037 -0.2041 -0.2040 -0.2041 -0.2040(0.0037) (0.0037) (0.0037) (0.0037) (0.0037)

Age Squared 0.0043 0.0044 0.0043 0.0044 0.0043(0.0001) (0.0001) (0.0001) (0.0001) (0.0001)

Log of Miles -0.1218 -0.1221 -0.1222 -0.1221 -0.1184(0.0085) (0.0084) (0.0084) (0.0084) (0.0085)

Log of Text Size 0.0623 0.0444 0.0486 0.0406 0.0466(0.0082) (0.0104) (0.0097) (0.0112) (0.0089)

Manual Transmission 0.0565 0.0575 0.0575 0.0574 0.0579(0.0170) (0.0170) (0.0170) (0.0170) (0.0169)

Control -0.2975 0.0508 0.0440 0.0556 0.0779(0.0496) (0.0177) (0.0161) (0.0190) (0.0163)

Control Squared 0.6089 -0.0179 -0.0175 -0.0180 -0.0182(0.0418) (0.0116) (0.0116) (0.0116) (0.0114)

Constant 10.6941 10.8380 10.8074 10.8657 10.7643(0.1152) (0.1234) (0.1209) (0.1266) (0.1156)

σ

Age 0.0317 0.0316 0.0316 0.0316 0.0316(0.0035) (0.0035) (0.0035) (0.0035) (0.0035)

Log of Text Size -0.1256 -0.1248 -0.1245 -0.1250 -0.1247(0.0485) (0.0483) (0.0483) (0.0483) (0.0484)

Constant -0.6006 -0.6072 -0.6093 -0.6055 -0.6103(0.4186) (0.4176) (0.4177) (0.4174) (0.4182)

rINDEX -0.0532 -0.0534 -0.0535 -0.0534 -0.0541(0.0495) (0.0493) (0.0492) (0.0493) (0.0487)

Marginal Effects for σ

Age 0.0113 0.0113 0.0113 0.0113 0.0113Log of Text Size -0.0450 -0.0447 -0.0446 -0.0448 -0.0445Mean σ 0.8324 0.8319 0.8319 0.8319 0.8311r 1.2162 1.2160 1.2160 1.2160 1.2156

Estimated standard errors are given in parentheses. I do not account for first stage estimation error in computingthe errors. The different specifications provide estimates with and without the use of a control function, andfor varying bandwidth h in the nonparametric first stage. The final specification is estimated with an OLS firststage.

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Table 6: Demand Estimation by Car Group with Control

Classic Cars Reliable Cars Pickupsµ

Age -0.1629 -0.1815 -0.1925(0.0044) (0.0253) (0.0058)

Age Squared 0.0036 0.0013 0.0031(0.0001) (0.0010) (0.0002)

Log of Miles -0.1341 -0.0804 -0.1028(0.0097) (0.0331) (0.0163)

Log of Text Size 0.0660 -0.0216 0.0108(0.0192) (0.0428) (0.0153)

Manual Transmission 0.2499 0.0570 -0.1607(0.0223) (0.0303) (0.0334)

Control 0.0749 0.1157 0.0641(0.0305) (0.0370) (0.0319)

Control Squared -0.0293 -0.0180 -0.0003(0.0167) (0.0201) (0.0224)

Constant 10.9973 10.8878 11.3938(0.1732) (0.5546) (0.2632)

σ

Age 0.0240 0.0760 0.0331(0.0035) (0.0599) (0.0048)

Log of Text Size -0.0766 -0.2602 -0.1750(0.0529) (0.2543) (0.0915)

Constant -0.8223 -0.4439 -0.1309(0.4113) (2.4866) (0.7419)

rINDEX -0.0407 -0.0190 0.1491(0.0550) (0.5599) (0.8446)

Marginal Effects for σ

Age 0.0101 0.0124 0.0104Log of Text Size -0.0322 -0.0423 -0.0548Mean σ 0.8753 0.6064 0.7784r 1.2241 1.2379 1.3442

Estimated standard errors are in parentheses. I do not account for first stage estimation error in computingthe errors. The three specifications show the estimated relationship between the structural parameters and thecovariates for different car groups. After including the control function, I find no significant relationship betweenmy information measures and the prior mean in the case of reliable cars and pickups.

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