Econometrics of first-price auctions with entry and binding reservation prices

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Journal of Econometrics 126 (2005) 173–200

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Econometrics of first-price auctions with entryand binding reservation prices

Tong Li�

Department of Economics, Indiana University, Wylie Hall, Indiana IN 47405, USA

Accepted 30 June 2004

Available online 25 August 2004

Abstract

This paper considers the structural analysis of first-price auctions with entry and binding

reservation prices. The presence of entry decisions and binding reservation prices complicates

the structural analysis. Building on the recent theoretical work on entry in auctions, this paper

assumes that each potential bidder first decides whether or not to incur an entry cost and

become an active bidder using a symmetric mixed strategy. Then each active bidder bids

optimally following the increasing Nash–Bayesian equilibrium strategy. Using the observed

bids and the number of actual bidders, we propose an MSM estimator to estimate the

parameters in the distribution of private values and the distribution of the number of active

bidders. Our approach can be used to validate the theoretical auction model, to test whether

the reservation prices are binding, and to test the mixed-strategy of entry.

r 2004 Elsevier B.V. All rights reserved.

JEL classification: C15; D44

Keywords: Endogenous participation; Independent private values; Method of simulated moments; Mixed

strategies of entry; Nash–Bayesian equilibrium

1. Introduction

This paper considers the structural analysis of first-price auctions with entry andbinding reservation prices. The presence of entry decisions and binding reservation

see front matter r 2004 Elsevier B.V. All rights reserved.

.jeconom.2004.06.002

2-855-2363; fax: +812-855-3736.

dress: [email protected] ( Tong Li).

www.elsevier.com/locate/econbase

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Tong Li / Journal of Econometrics 126 (2005) 173–200174

prices complicates the structural analysis. Building on the recent theoretical work onentry in auctions (see e.g., Levin and Smith (1994), McAfee et al. (2002), amongothers), this paper assumes that at the first stage, each potential bidder decideswhether or not to incur an entry cost and become an active bidder using a symmetricmixed strategy.1 Then at the second stage each active bidder bids optimally followingthe increasing Nash–Bayesian equilibrium strategy. One of the consequences ofthe entry decision at the first stage is that the number of active bidders is stochasticwith a distribution determined by those factors such as the heterogeneity of theauctioned items. In addition, at the second stage, only those active bidders whoseprivate values are above the reservation prices bid and hence become actual bidders.We consider the first-price sealed-bid auctions within an independent private value(IPV) paradigm, and develop a structural model that explicitly models thistwo-stage game. We propose a method of simulated moments (MSM)estimator to estimate the parameters in both the distributions of private valuesand the number of active bidders using the observed bids and the number of actualbidders.The structural analysis of auction data has been developed over the last decade.

This approach uses econometric models closely derived from the game theory withincomplete information to interpret real auction data. Various estimation methodshave been proposed since Paarsch (1992). For example, see Donald and Paarsch(1993, 1996) for the piecewise maximum likelihood estimation and constrainedmaximum likelihood estimation, respectively, Laffont et al. (1995) and Li and Vuong(1997) for the simulated non-linear least squares method, and Guerre et al. (2000) forthe non-parametric estimation method. All these methods are proposed within theIPV paradigm except that Paarsch (1992) also considers a common value paradigm.Recently Li et al. (2000, 2002) extend the structural approach using the non-parametric estimation method to more general models, namely, the affiliated privatevalue (APV) paradigm and the conditionally independent private information (CIPI)paradigm.The motivation of this paper is two-fold. First, while most of the previous work in

the structural approach assumes that the number of bidders is given or fixed, anendogenous entry process could lead the number of active bidders to be stochastic.This can be justified by the commonly observed variability in number of bidderswithin auctions for identical or similar objects (see Levin and Smith (1994) forrelated discussions, and Bajari and Hortacsu (2000), Lucking-Reiley (1999) andSmith and Levin (2002) on the experimental and empirical evidence on theendogenous entry in auctions). Therefore, it is a necessary step to take into accountthe induced entry process in the structural analysis. Second, despite the significantprogress in the structural approach, the structural analysis has focused on auctionswithout reservation prices or with nonbinding reservation prices. Exceptions areLaffont et al. (1995) and Guerre et al. (2000) with the former studying the eggplantauctions that are held in France as Dutch auctions with reservation prices and the

1Note that in the absence of an entry process, potential bidders and active bidders can be considered as

the same concept.

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Tong Li / Journal of Econometrics 126 (2005) 173–200 175

latter discussing the nonparametric estimation when the reservation price is binding.Both papers, while considering auctions without entry, however, assume that thepotential number of bidders is constant across the auctions. Also, Paarsch (1997)proposes a conditional maximum likelihood method conditional on the number ofactual bidders in studying timber auctions in British Columbia, Canada in anEnglish auction framework. Note that all these papers consider publicly announcedreservation prices, a situation considered in this paper. For secret reservation prices,see Li and Perrigne (2003) for an IPV model in timber auctions and Bajari andHortacsu (2000) for a common value model in ebay auctions. The publiclyannounced reservation prices can increase the magnitudes of bids and hence increasethe sales and profits for the sellers. In addition, if binding, they can prevent thebidders with low valuations from participating in the auctions. From an econometricpoint of view, in the presence of binding reservation prices, the problem oftruncation arises as only the bids from those active bidders whose private valuesare at least as high as the reservation prices are observed and hence the numberof actual bidders is smaller than the number of active bidders. As a result, whena binding reservation price occurs, the number of actual bidders becomesendogenous and the number of active bidders is unobservable. This could causea serious problem in the structural approach as game theory postulates the optimalbid as a function of the corresponding private value, its distribution and thenumber of active bidders. Therefore, ignoring such an issue will generally lead toinconsistent estimates resulting from the structural estimation due to the endogeneityof the number of actual bidders. While a possible approach to dealing with theendogeneity of the number of actual bidders is to use some instrumental variables(see e.g. Haile (2001) who studies the US Forest Service timber sales that are heldas English auctions), it is limited as in many cases, valid instruments are hardto construct.In this paper, we extend the structural estimation of auction models to the general

case in which the entry process is endogenous and the reservation prices are binding.This paper contributes to the structural analysis of auction data by providing aunified framework to estimate the structural parameters in first-price sealed-bidauctions with entry and binding reservation prices. It is worth noting that this paperclarifies three types of bidders and exploits the information from the observednumber of actual bidders, which has been ignored by most of the previous work.Despite the complication arising from the endogeneity of the number of actualbidders due to the entry and binding reservation prices, this paper shows that theinformation from the number of actual bidders is indeed useful in estimating andtesting the structural models of auctions with entry and binding reservation prices.We also establish several sets of moment conditions, some of which depend on theequilibrium bidding strategy while the others depend only on the random entryprocess or the rationality of the reservation prices. The difference among thesemoment conditions can thus be exploited to test some important hypotheses as wellas to validate the structural model. Lastly, our estimator is in spirit similar to thesimulated nonlinear least squares (SNLLS) estimator proposed by Laffont et al.(1995) as it does not require the computation of the bidding function in the

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estimation. This is a considerable advantage given the computational difficultyarising from the Nash–Bayesian bidding strategies.The paper is organized as follows. In Section 2 we present the first-price sealed-bid

auction model with entry within the IPV paradigm. In Section 3 we derive themoment conditions associated with the game-theoretic model presented in Section 2.In Section 4 we propose an MSM estimator to estimate the parameters in both thedistributions of private values and the number of active bidders. Section 5 gives somefurther discussions on how our approach can be used for validation of the theoreticalmodels on both the entry and bidding processes and testing for the bindingreservation prices. Section 6 presents some Monte Carlo results. Section 7 concludes.The technical proofs are included in the Appendix.

2. The first-price sealed-bid IPV model with entry

The entry process modeling in this section essentially follows Levin and Smith(1994).2 A single and indivisible object is auctioned to N potential bidders who arerisk-neutral. The auction involves two stages. At the second stage, all bids arecollected simultaneously from those n active bidders who have decided to incur anentry cost c and participate in the auction. The object is sold to the highest bidderwho pays his/her bid to the seller, provided that the bid is at least as high asa reservation price p0: At the first stage, each potential bidder decides whether ornot to incur c and bid at the second stage. In the IPV paradigm, each bidderi ði ¼ 1; . . . ;NÞ is assumed to have a private value vi for the auctioned object. Whenforming his/her bid, each bidder knows his/her private value but does not knowothers’ private values. On the other hand, each bidder knows that all private valuesincluding his/her own are drawn independently from a probability distribution F ð�Þ;which is absolutely continuous with respect to Lebesgue measure, with density f ð�Þ

and support ½v; v� � ½0;þ1Þ: Therefore all bidders are identical a priori and the gameis symmetric. Also, N is common knowledge to all potential bidders and n is revealedprior to stage 2.3

We will focus on the strictly increasing differentiable symmetric Nash–Bayesianequilibrium strategies. Denote by E½p j n� each potential bidder’s ex ante expectedgain from entering, paying c; learning n and bidding following the Nash–Bayesianequilibrium strategy implied by n: Then as shown by Levin and Smith (1994), forq� 2 ð0; 1Þ; at a mixed-strategy equilibrium, a potential bidder’s ex ante expected gain

2Levin and Smith (1994) study symmetric mixed strategies in entry, which is different from the previous

work (e.g. McAfee and McMillan (1987)), which assumes that the N potential bidders use pure strategies

leading to a deterministic and asymmetric equilibrium in which exactly n bidders enter and N n do not

enter.3As discussed in Levin and Smith (1994), while active bidders in some auctions do not know n at the

time of their bidding in stage 2, the assumption that n is revealed prior to stage 2 is not as strong as it seems

to be. In fact, Levin and Smith (1994) show that within the IPV paradigm, the equilibrium analysis and

results hold for both cases. We will maintain this assumption throughout the paper, however, for

convenience from an econometric point of view. See also the related discussion in the conclusion.

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must be zero, which leads to

XN

n¼1

N 1

n 1

� �ðq�Þ

n 1ð1 q�Þ

N nE½p j n�

� �¼ 0: ð1Þ

q� that satisfies (1) characterizes equilibrium in mixed strategies. Also, the number ofactive bidders n follows a binomial distribution denoted by hð�Þ with mean equal toq�N and variance equal to q�ð1 q�ÞN:4 Note that one assumption of the two-stageentry model is that bidders do not know their private values at the first stage untilthey have decided to participate in the auction. The same assumption is also made inthe ‘‘random participation’’ model in Hendricks et al. (2003).In stage 2, each active bidder bids according to the increasing Nash–Bayesian

equilibrium strategy denoted by sð�Þ: As shown by Riley and Samuelson (1981)among others, the (unique) symmetric differentiable Nash–Bayesian equilibriumstrategy satisfies the first-order differential equation

s0ðviÞ ¼ ðvi biÞðn 1Þf ðviÞ=F ðviÞ ð2Þ

for all vi 2 ½p0; v� subject to the boundary condition sðp0Þ ¼ p0: The solution of (2) is

bi ¼ sðviÞ ¼ vi 1

ðF ðviÞÞn 1

Z vi

p0ðF ðxÞÞn 1 dx if viXp0; ð3Þ

otherwise bi ¼ sðviÞ can be any value b0 (say) strictly smaller than p0:The above theoretical model leads to a closely related structural econometric

model. Specifically, because vi is distributed as F ð�Þ; then the equilibrium strategybi ¼ sðviÞ given from (3) is also a random variable with a distribution, say, Gð�Þ: Ingeneral, bids are observed while private values are unobserved. On the other hand,bids are determined, as given in (3), by the reservation price, the number of activebidders, and the distribution of private values. The structural analysis of auctiondata has been developed to estimate the distribution of the latent private values usingthe observed bids based upon the fact that the distribution of bids Gð�Þ is related tothe distribution of private values F ð�Þ through (3). Parametric estimation proceeds byspecifying a parametric family for F ð�Þ: This is, however, complicated by the highnonlinearity of (3) in the latent distribution F ð�Þ; whenever the estimation procedureinvolves computing (3) within each iteration of an optimization procedure. This isindeed the case for MLE for which (3) needs to be inverted for each b and y at eachiteration. In addition, computation of the jacobian of the bid transformation that isrequired in the implementation of the MLE makes it more difficult computationally.Using the Revenue Equivalence Theorem (see Vickrey (1961)), Laffont et al. (1995)propose a simulated nonlinear least squares (SNLLS) method to estimate the latentdistribution of private values in Dutch auctions where only the winning bids areobserved. Their method has the advantage of simulating the moments of bidswithout numerically determining (3). Moreover, Guerre et al. (2000) have proposed

4In case that N is large, which occurs in some internet auctions, hð�Þ can be approximated by a Poisson

distribution or variants of Poisson distribution, as a Poisson distribution is the limit of a binomial

distribution when N goes to 1:

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an indirect nonparametric estimation procedure by noting that (2) can be rewritten as

v ¼ b þ1

n 1

G�ðbÞ

g�ðbÞþ

1

n 1

F ðp0Þ

1 F ðp0Þ

1

g�ðbÞif bXp0; ð4Þ

where G�ðbÞ ¼ ½GðbÞ Gðp0Þ�=½1 Gðp0Þ�; which is the conditional distribution ofthe observed bids given that the corresponding private values exceed the reservationprices with Gðp0Þ being the discrete probability of binding reservation price, and g�ð�Þ

is its density. Using (4), Li and Vuong (1997) derive a relationship between the meansof observed bids and private values to propose an SNLLS estimator using all bids infirst-price auctions without entry assuming that the number of potential bidders isconstant across the auctions and known to the econometrician.5 In this paper, wewill adopt the parametric framework and extend the structural analysis to the case inwhich there is an entry process as described above and the reservation price isbinding. Note that a nonparametric approach without specifying the functional formof the private value distribution prevents one from identifying the entire distributionof private values due to the truncation of the observed bids. See, e.g., Guerre et al.(2000). Such a nonparametric identification problem in truncated data has beenobserved by Flinn and Heckman (1982) in a seminal paper analyzing structuralmodels of labor force dynamics.

3. The structural econometric model

In an econometric investigation, one often considers more than one auction, andthe statistical inference is usually based on the assumption that the number ofauctions approaches infinity. Dealing with more than one auction raises two issues,one of which is the possible heterogeneity across the auctions.6 Consequently, thedistribution of private values depends on the heterogeneity of auctioned objects andvaries across auctions. Specifically, let F ‘ð�Þ denote the distribution of private valuesfor the ‘th auction, ‘ ¼ 1; . . . ;L; where L is the number of auctions, and assume thatF ‘ð�Þ ¼ F ð� j z‘; yÞ; where y is an unknown parameter vector in Y � Rk; z‘ is a vectorof variables affecting bidders’ valuations, and n‘ is the number of active bidders inthe ‘th auction. Let n�

‘ denote the number of bidders who actually bid in the ‘thauction, i.e. whose private values vi‘ are at least p0‘ : Assume that the number ofpotential bidders denoted by N‘ is observed across the auctions. This assumptioncan be justified in many auctions where the researcher has the knowledge on thenumber of potential bidders, or has enough information to get a proxy for it.Otherwise, in some cases, the number of potential bidders can be identified and

5In first-price sealed-bid auctions without entry, when the number of potential bidders is a constant

across all the auctions, the assumption that it is known is not restrictive as otherwise, the maximum of the

numbers of actual bidders across auctions can be used to estimate the number of potential bidders.6Another issue, as discussed in Laffont et al. (1995), is the dynamic feature related to the repetition of an

auction, which may result in underbidding during the first auctions. To preserve the equilibrium strategy

(3), following Laffont et al. (1995), we assume that bidders draw new independent private values at each

auction.

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estimated from the number of actual bidders in this two-stage game. For example, ifthe number of potential bidders is a constant across the auctions, then it can beidentified provided that there are auctions in which the reservation prices are notbinding.7 Moreover, assume that the probability given by (1) for a potential bidderto enter in the ‘th auction is q�

‘ ¼ q�ðx‘; gÞ; where g is an unknown parameter vectorin G � Rq and x0

‘ � ðp0‘ ; z0‘Þ: Then the number of active bidders in the ‘th auction n‘ is

distributed with a probability mass function hð� jx‘; gÞ; which is a binomialdistribution with mean q�

‘N‘ and variance q�‘ ð1 q�

‘ ÞN‘; as described in Section 2.Our approach to modeling q� merits some discussion. Since q�

‘ is a solutionto (1), in principle, it is a function of N‘; p0‘ ; F ð� j z‘; yÞ and the entry cost c‘: If c‘;‘ ¼ 1; . . . ;L; are observed, then they can also be included in x‘: In manyapplications, however, they are not observed. Our approach can then be used torecover these entry costs as discussed later. Moreover, due to the complex feature of(1), the functional form for q�

‘ in terms of N‘; p0‘ ;F ð� j z‘; yÞ and c‘ has no explicitsolution. Since our primary interest is to estimate the distribution of the privatevalues by taking into account endogenous entry, we propose to use a flexible discretechoice probability functional form for q�

‘ such as a logistic or normal distribution. Aspointed out by a referee, essentially q�

‘ and N‘ are not too important per se, but servemostly the role of providing a distribution for the number of active bidders. Thusalternative direct flexible parametrization for the distribution of the number of activebidders could also be considered. Nonetheless, (1) imposes some testable restrictions,which could be tested in principle using the estimators developed in this paper, asdiscussed later.For each ‘ ¼ 1; . . . ;L; and i ¼ 1; . . . ; n�

‘ ; (4) now becomes

vi‘ ¼ bi‘ þ1

n‘ 1

G�ðbi‘ j n‘;x‘; y0Þ

g�ðbi‘ j n‘;x‘; y0Þþ

1

n‘ 1

F ðp0‘ j z‘; y0Þ

1 F ðp0‘ j z‘; y0Þ

�1

g�ðbi‘ j n‘;x‘; y0Þ

if bi‘Xp0‘ ; ð5Þ

where y0 is the unknown true parameter vector, and G�ð� j n‘;x‘; yÞ is the distributionof bi‘ given bi‘Xp0‘ ; n‘ and x‘: Now integrating both sides of (5) with respect to thedensity g�ðbi‘ j n‘;x‘; y

0Þ on its support ½p0‘ ; bðn‘; x‘; y

0Þ� gives

E½vi‘ j vi‘Xp0‘ ; n‘;x‘; y0� ¼ E½bi‘ j bi‘Xp0‘ ; n‘;x‘; y

0� þ

1

n‘ 1

Z bðn‘ ;x‘ ;y0Þ

p0‘

G�ðb j n‘;x‘; y0Þdb þ

1

n‘ 1

F ðp0‘ j z‘; yÞ

1 F ðp0‘ j z‘; y0Þ

� ½bðn‘;x‘; y0Þ p0‘ �:

7In this case, the numbers of actual bidders observed in those auctions with non-binding reservation

prices are identical to the numbers of active bidders, which follow a binomial distribution with mean q�N

and variance q�ð1 q�ÞN: Thus N is identified and can be estimated by the maximum of these numbers of

actual bidders.

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Noting that bðn‘;x‘; y0Þo1 when the private value distribution has a finite first

moment,8 and when n‘42; integrating by parts and rearranging terms give9

E½bi‘ j bi‘Xp0‘ ; n‘; x‘; y0� ¼

n‘ 1

n‘ 2E½vi‘ j vi‘Xp0‘ ; n‘; x‘; y

0� þ

p0‘n‘ 2

F ðp0‘ j z‘; y0Þ

1 F ðp0‘ j z‘; y0Þ

bðn‘; x‘; y

0Þ

n‘ 2

1

1 F ðp0‘ j z‘; y0Þ

� mðn‘; x‘; y0Þ: ð6Þ

When the number of active bidders n‘ ¼ 2; we can derive the following (conditional)moment condition for the observed bids,

E½bi‘ j bi‘Xp0‘ ; n‘ ¼ 2;x‘; y0� ¼

p0‘F ðp0‘ j z‘; y0Þ

1 F ðp0‘ j z‘; y0Þlog F ðp0‘ j z‘; y

0Þ

E½vi‘ logF ðp0‘ j z‘; y0Þ j vi‘Xp0‘ ; n‘ ¼ 2;x‘; y

0�

� mð2;x‘; y0Þ: ð7Þ

For the derivation of (7), see Appendix. Furthermore, if the number of active biddersn‘ ¼ 1; the optimal strategy for the single active bidder is to bid the reservation priceprovided that his/her private value is not less than the reservation price. This leads tothe following (conditional) moment condition:

E½bi‘ j bi‘Xp0‘ ; n‘ ¼ 1;x‘; y0� ¼ p0‘

� mð1;x‘; y0Þ: ð8Þ

Note that (6)–(8) are conditional on the number of active bidders n‘;which is not observed and is a random variable with the probability massfunction h‘ð�Þ ¼ hð� jx‘; gÞ: The next proposition gives the moment condition for theobserved bids at the ‘th auction with the (unobserved) number of active bidders‘‘integrated out’’.

Proposition 1. For n�‘X1; we have

E½bi‘ j bi‘Xp0‘ ; n�‘ ;x‘; Z0� ¼

XN‘

n‘¼n�‘

mðn‘;x‘; y0ÞPr½n‘ j n�

‘ ; x‘; Z0�; ð9Þ

8This can be seen from bðn‘;x‘; y0Þ ¼ E½maxðp0‘ ; v‘Þðn‘ 1ÞFn‘ 2ðv‘ j z‘; y0Þ�pðn‘ 1ÞE½maxðp0‘ ; v‘Þ�:9Note that (6) is derived from (4). Alternatively, it can be derived directly from (3). The author is

grateful to L.-F. Lee for suggesting this alternative derivation.

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where Z0 ¼ ðy0; g0Þ and

Pr½n‘ j n�‘ ; x‘; Z0� ¼

N‘ n�‘

n‘ n�‘

!F ðp0‘ j z‘; y

0Þq�

‘

F ðp0‘ j z‘; y0Þq�

‘ þ 1 q�‘

!n‘ n�‘

�1 q�

‘

F ðp0‘ j z‘; y0Þq�

‘ þ 1 q�‘

!N‘ n‘

:

While Eq. (9) gives one moment condition for the observed bids, in first-pricesealed-bid auctions, we also observe the number of actual bidders n�

‘ for each auction‘: An interesting and natural question is whether the number of actual bidders isinformative in the estimation of the parameters y and g: To address this issue, notethat at the ‘th auction, an active bidder j will bid for the object if and only if his/herprivate value vj‘ is not lower than the reservation price p0‘ : Therefore, at the ‘thauction, the number of actual bidders n�

‘ ¼Pn‘

j¼11Iðvj‘Xp0‘ Þ is a binomial variablewith parameters ðn‘; 1 F ðp0‘ j z‘; y

0ÞÞ:10 We have the following lemma, which gives

the distribution for the number of actual bidders.

Lemma 1. For the number of actual bidders n�‘ ; we have

Pr½n�‘ jx‘; Z0� ¼

N‘

N‘ n�‘

!fq�

‘ ½1 F ðp0‘ j z‘; y0Þ�gn�‘

� f1 q�‘ ½1 F ðp0‘ j z‘; y

0Þ�gN‘ n�‘ : ð10Þ

Lemma 1 can be used to obtain the moments of n�‘ conditional on n�

‘X1: The nextproposition gives the first two moments.

Proposition 2. We have

E½n�‘ j n�

‘X1;x‘� ¼N‘q

�‘ ½1 F ðp0‘ j z‘; y

0Þ�

1 f1 q�‘ ½1 F ðp0‘ j z‘; y

0Þ�gN‘

� mf1gðx‘; Z0Þ; ð11Þ

E½n�2‘ j n�

‘X1;x‘� ¼N‘q

�‘ ½1 F ðp0‘ j z‘; y

0Þ�f1þ ðN‘ 1Þq�

‘ ½1 F ðp0‘ j z‘; y0Þ�g

1 f1 q�‘ ½1 F ðp0‘ j z‘; y

0Þ�gN‘

� mf2gðx‘; Z0Þ: ð12Þ

10Donald and Paarsch (1996), Paarsch (1997) and Donald et al. (1999) develop a binomial model for the

bidders’ participation, while the number of actual bidders has not been used in the inference. I am thankful

to P.A. Haile for bringing Donald et al. (1999) to my attention after he read a previous version of my

paper.

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It is worth noting that we only use the first two moments for a binomialdistribution as given in (11) and (12). It is possible, on the other hand, that somehigher order moments for a binomial distribution are used to improve efficiency.Alternatively, an immediate result of Lemma 1 gives the probability mass functionfor n�

‘ conditional on n�‘X1 as follows

xðn�‘ j n�

‘X1;x‘; Z0Þ

¼N‘

n�‘

!f1 q�

‘ ½1 F ðp0‘ j z‘; y0Þ�gN‘ n�‘ fq�

‘ ½1 F ðp0‘ j z‘; y0Þ�gn�‘

1 f1 q�‘ ½1 F ðp0‘ j z‘; y

0Þ�gN‘

: ð13Þ

Thus one can use the moment condition for the gradient of the log-likelihoodfunction of n�

‘ : In what follows, as an illustration, we use (9), (11) and (12) as themoment conditions to construct a structural econometric model.11 As mentionedearlier, various methods have been proposed for the structural estimation of auctiondata, some of which are based upon some moment conditions (see, e.g. Paarsch(1992), Laffont et al. (1995) and Li and Vuong (1997)). Our moment conditions,however, differ from the others as they clearly take into account the effects of entryand binding reservation prices, then use the information from the observables suchas the bids and the numbers of actual bidders. As will be shown, Eqs. (9), (11) and(12) can be jointly used to estimate the parameters y and g; to which we now turn.

4. An MSM estimator

Given the moment conditions (9), (11), (12), in principle, following Hansen (1982),a generalized method of moments (GMM) estimator can be proposed. Specifically,let yi‘ðZÞ ¼ yðbi‘;x‘; ZÞ ¼ ðbi‘

PN‘n‘¼n�

‘mðn‘;x‘; yÞPr½n‘ j n�

‘ ;x‘; Z�; n�‘ mf1gðx‘; ZÞ;

n�2‘ mf2gðx‘; ZÞÞ

0: Then the moment conditions (9), (11) and (12) can be written as

E½yi‘ðZ0Þ jx‘� ¼ 0;

where Z0 ¼ ðy0; g0Þ:12 Let X ‘ be a K � 3 matrix function of x‘ with KXk þ q; thedimension of Z ¼ ðy; gÞ: Then it follows from the preceding equation that

E½X ‘yi‘ðZ0Þ� ¼ 0: ð14Þ

11The moment conditions from (9), (11), and (12), as well as (13) are conditional upon that the number

of actual bidders is at least one, and thus are valid when the sample used consists of only auctions with at

least one actual bidder. This will be the scenario considered in the paper. Alternatively, as pointed out by a

referee, one could derive the modified moment conditions that are valid for using the entire sample with all

auctions including those with no actual bidders. This approach, however, raises a difficulty in the GMM

and MSM estimation proposed in the next section as n�; the number of actual bidders, appears in a

denominator in both (15) and (16) below. I am grateful to the same referee whose careful review and

constructive comments lead to the clarification of this issue and the correct characterization of the moment

conditions in (9), (11), (12) and (13).12Note that the conditioning in this moment condition actually involves the additional conditioning on

n�‘ in its first component in view of (9), and on n�‘X1 in its last two components in view of (11) and (12).

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As a result, a GMM estimator can be defined by

ZGMM ¼ arg minZ

XL

‘¼1

1

n�‘

Xn�‘i¼1

X ‘yi‘ðZÞ

0@

1A

0

OXL

‘¼1

1

n�‘

Xn�‘i¼1

X ‘yi‘ðZÞ

0@

1A; ð15Þ

where O is a K � K symmetric positive-definite matrix.It is computationally intensive in practice, however, to implement the GMM

estimator defined by (15). Such a computational difficulty arises mainly becausebðn‘; x‘; yÞ ¼ sðv‘;F ð� jx‘; yÞ; n‘Þ is difficult to compute as it involves the Nash–Baye-sian equilibrium strategy (3). Therefore, the implementation of the GMM estimator(15) requires the computation of bðn‘;x‘; yÞ from the Nash–Bayesian equilibriumstrategy for each trial value y at each iteration. This makes the GMM estimator (15)less attractive from a computational viewpoint. The computational burden of theGMM estimator (15), on the other hand, can be alleviated by extending theapproach adopted in Li and Vuong (1997). The idea here is to simulate bðn‘;x‘; yÞwithout computing the Nash–Bayesian equilibrium strategy provided that bðn‘;x‘; yÞcan be written in terms of some moment equations. This can be achieved, indeed, bynoting that from Laffont et al. (1995), bðn‘; x‘; yÞ ¼ E½maxðUmax

ðn‘ 1Þ‘; p0‘ Þ�; where

Umaxðn‘ 1Þ‘

is the largest order statistic in n‘ 1 independent draws from F ð� j z‘; yÞ withdensity f ‘ð�; yÞ ¼ f ð� j z‘; yÞ: Note now that the term involving bðn‘; x‘; yÞ in the firstcomponent of yi‘ðZÞ; if denoted by Aðx‘; ZÞ; can be written as

Aðx‘; ZÞ ¼XN‘

n‘¼n�‘

bðn‘;x‘; yÞn‘ 2

1

1 F ðp0‘ j z‘; yÞPr½n‘ j n�

‘ ;x‘; Z�

¼XN‘

n‘¼n�‘

Zmaxðumaxðn‘ 1Þ‘

; p0‘ ÞYn‘ 1t¼1

f ‘ðut‘; yÞYn‘ 1t¼1

dut‘Pr½n‘ j n�

‘ ; x‘; Z�ðn‘ 2Þð1 F ðp0‘ j z‘; yÞÞ

¼ En‘ Eðu1‘ ;...;uðn‘ 1Þ‘Þ j n‘ maxðumaxðn‘ 1Þ‘; p0‘ Þ

Yn‘ 1t¼1

f ‘ðut‘; yÞr‘ðut‘Þ

" #"

�Pr½n‘ j n�

‘ ; x‘; Z�=g‘ðn‘Þ

ðn‘ 2Þð1 F ðp0‘ j z‘; yÞÞ

�;

where r‘ð�Þ; ‘ ¼ 1; . . . ;L; is a known density function with a support including that off ‘ð�; yÞ; g‘ð�Þ is a known probability mass function with a support equal to that ofPr½� j n�

‘ ; x‘; Z�; En‘ ½�� denotes the expectation with respect to n‘; which is now arandom draw from g‘ð�Þ; and Eðu1‘ ;...;uðn‘ 1Þ‘Þ j n‘ ½�� denotes the conditional expectationof n‘ 1 independent draws from r‘ð�Þ conditional on n‘:

13 The last expressionsuggeststhat the method of importance sampling can be used to simulate Aðx‘; ZÞ:Specifically, for each ‘ ¼ 1; . . . ;L; we have S independent draws denoted as n

ðsÞ‘ from

the known g‘ð�Þ; for s ¼ 1; . . . ;S: Then for each nðsÞ‘ ; draw T independent samples,

13Note that Aðx‘; ZÞ only holds when n�‘42; as it comes from (6). As a result, the support of g‘ð�Þ used to

draw nðsÞ‘ should be equal to the support of Pr½� j n�‘ ;x‘; Z� to avoid drawing some n

ðsÞ‘ not greater than 2. I

owe this point to a referee.

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each of size nðsÞ‘ 1; denoted as u

ðs;tÞ1‘ ; . . . ; uðs;tÞ

ðnðsÞ‘ 1Þ‘

; where uðs;tÞi‘ is independently drawn

from a known density r‘ð�Þ for t ¼ 1; . . . ;T : It then follows that we can estimate

Aðx‘; ZÞ by A‘ðZÞ ¼ 1S

PSs¼1As‘ðZÞ; where

As‘ðZÞ ¼1

T

XT

t¼1

maxðuðs;tÞmax

ðnðsÞ‘ 1Þ‘

; p0‘ ÞYnðsÞ‘ 1

k¼1

f ‘ðuðs;tÞk‘ ; yÞ

r‘ðuðs;tÞk‘ Þ

24

35

�Pr½n

ðsÞ‘ j n�

‘ ;x‘; Z�=g‘ðnðsÞ‘ Þ

ðnðsÞ‘ 2Þð1 F ðp0‘ j z‘; yÞÞ

;

and uðs;tÞmax

ðnðsÞ‘ 1Þ‘

is the highest value among uðs;tÞ1‘ ; . . . ; uðs;tÞ

ðnðsÞ‘ 1Þ‘

: It is important to note that

A‘ðZÞ is an unbiased simulator for Aðx‘; ZÞ because Eufs;tg;nðsÞ

‘½A‘ðZÞ� ¼ Aðx‘; ZÞ:

For the sake of simplicity, we assume that all the other terms in yi‘ can be obtainedexplicitly.14 Let yi‘ðZÞ ¼ yðbi‘; x‘; ZÞ denote yi‘ after replacing Aðx‘; ZÞ by A‘ðZÞ: Thena method of simulated moments (MSM) estimator ZMSM can be proposed as

ZMSM ¼ arg minZ

XL

‘¼1

1

n�‘

Xn�‘i¼1

X ‘yi‘ðZÞ

0@

1A

0

OXL

‘¼1

1

n�‘

Xn�‘i¼1

X ‘yi‘ðZÞ

0@

1A: ð16Þ

For the consistency of ZMSM; we make the following assumptions.A.1: The true parameter Z0 ¼ ðy0; g0Þ lies in the interior of a convex compact space

Y� G:A.2: Z0 is uniquely identified in the sense that E½X ‘yi‘ðZÞ� ¼ 0 if and only if Z ¼ Z0:A.3: Denote the joint distribution of bi‘; x‘ and n�

‘ by H‘ð�Þ: There exists a distributionfunction Hð�Þ such that 1

L

PL‘¼1H‘ð�Þ ! Hð�Þ for all continuity points of Hð�Þ:

A.4: supLð1L

PL‘¼1E½supZ2Y�G j

1n�‘

Pn�‘i¼1X ‘yi‘ðZÞ j

1þd�Þo1 for some d40:

Assumptions A.1 and A.2 are the standard ones that are made in establishing theconsistency of extremum estimators (see, e.g., Newey and McFadden (1994)). Inaddition, A.3 and A.4 are made following the assumptions in Theorem 2.7.2 ofBierens (1994) which establishes a (weak) uniform law of large numbers forheterogenous data. Our MSM estimator ZMSM as defined in (16), however, issomewhat different from the conventional extremum estimators in the sense that theobjective function to be minimized involves a random sum where the number of thesummands is the number of actual bidders, which is an endogenous variable.Nonetheless, under assumptions A.1–A.4, consistency of ZMSM; which is stated in thenext theorem, can still be established.15

14If not, then the method of importance sampling can be applied to the simulation of the other terms by

noting that for instance,PN‘

n‘¼n�‘ðn‘ 1ÞE½vi‘ j vi‘Xp0‘ ; n‘; x‘; y�=ðn‘ 2ÞPr½n‘ j n�‘ ; x‘; Z� ¼

PN‘n‘¼n�

‘

Rðn‘

1Þv1IðvXp0‘ Þf ðv j z‘; yÞPr½n‘ j n�‘ ;x‘; g�=½ðn‘ 2Þð1 F ðp0‘ j z‘; yÞÞ�dv:15Intuitively, why it is not a problem to have a random endogenous n�‘ as the number of summands in

the objective function is because the moments in (16) are conditional on actual bidding. I owe this insight

to a referee.

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Theorem 1. Assume A.1–A.4. Then for any fixed S and T and as L ! 1; ZMSM

converges in probability to Z0:

On the other hand, for the asymptotic distribution of ZMSM; to simplify the

presentation, let Bðuðs;tÞ; nðsÞ‘ ; ZÞ ¼ maxðu

ðs;tÞmax

ðnðsÞ‘ 1Þ‘

; p0‘ ÞQn

ðsÞ‘ 1

k¼1 ½f ‘ðuðs;tÞk‘ ; yÞ=r‘ðu

ðs;tÞk‘ Þ�; and

CðnðsÞ‘ ; ZÞ ¼ Pr½n

ðsÞ‘ j n�

‘ ;x‘; Z�=½g‘ðnðsÞ‘ Þðn

ðsÞ‘ 2Þð1 F ðp0‘ j z‘; yÞÞ�: Use E0½�� and Var0½��

to denote the expectation and variance with respect to b;x; n�; respectively. We alsomake the following assumption.A.5: mðn‘; p0‘ ; z‘; yÞ is twice continuously differentiable in y 2 Y; mf1gðx‘; ZÞ; and

mf2gðx‘; ZÞ; and Pr½n‘ j n�‘ ;x‘; Z� are twice differentiable in Z 2 Y� G:

Then we have the following result.

Theorem 2. Assume A.1–A.5. For any fixed S and T ;ffiffiffiffiL

pðZMSM Z0Þ converges in

distribution to a normal Nð0; ðD0ODÞ 1D0OSOD0ðD0ODÞ

1Þ as L ! 1; with

D ¼ limL!1

1

L

XL

‘¼1

E0 X ‘@yi‘ðZ

0Þ

@Z0

� �;

S ¼ limL!1

1

L

XL

‘¼1

E01

n�‘

X ‘M‘X0‘

� �;

and M‘ is a 3� 3 matrix with the following entries:

M‘;11 ¼ y21‘;1 þ n�‘Q;

M‘;22 ¼ n�‘y21‘;2; M‘;33 ¼ n�

‘y21‘;3;

M‘;23 ¼ M‘;32 ¼ n�‘y1‘;2y1‘;3;

M‘;12 ¼ M‘;21 ¼ M‘;13 ¼ M‘;31 ¼ 0;

where yi‘;k is the kth component of yi‘ for k ¼ 1; 2; 3; and Q ¼ VarnðsÞ‘½Cðn

ðsÞ‘ ; Z0Þ

Euðs;tÞ ½Bðuðs;tÞ; nðsÞ

‘ ; Z0Þ��=S þ EnðsÞ‘½C2ðn

ðsÞ‘ ; Z0ÞVarufs;tg ½Bðu

ðs;tÞ; nðsÞ‘ ; Z0Þ��=ðSTÞ:

As indicated by Theorem 1, one advantage of the MSM estimator is that for anyfixed S and T ; it is consistent. Consequently, the choices of S and T do not affectconsistency. On the other hand, as shown by Theorem 2, the MSM estimator has alarger asymptotic variance–covariance matrix than the GMM estimator by theterms of order 1=S and 1=ðSTÞ due to the simulations. Also, when the numbers ofsimulations S and T approach infinity, the MSM estimator has the same asymptoticefficiency as the GMM estimator. Moreover, if N‘; ‘ ¼ 1; . . . ;L; are not large, thenthe sum with respect to the number of active bidders in Aðx‘; ZÞ can be directlycalculated without the need to simulate n

ðsÞ‘ ; s ¼ 1; . . . ;S: In this case, we only need

to have the simulations with respect to the private values. It would be interestingto compare the asymptotic variance of the resulting MSM estimator with that givenin Theorem 2 for the MSM estimator when both n

ðsÞ‘ and ui‘ are simulated. Indeed,

it can be shown that for the former, without simulating nðsÞ‘ ; all the terms in

the asymptotic variance expression in Theorem 2 remain the same, except thatQ is different. The new Q is now ð1=TÞ

PN‘n‘¼n�

‘fVaruðtÞ ½

~BðuðtÞ; Z0Þ� ~C2ðn‘; Z0Þg;

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where ~BðuðtÞ; ZÞ ¼ maxðuðtÞmaxðn‘ 1Þ‘

; p0‘ ÞQn‘ 1

k¼1 ½f ‘ðuðtÞk‘; yÞ=r‘ðu

ðtÞk‘Þ�; and ~Cðn‘; ZÞ ¼

Pr½n‘ j n�‘ ; x‘; Z�=½ðn‘ 2Þð1 F ðp0‘ j z‘; yÞÞ�: As a result, if S values nðsÞ are not

simulated, there may be a potential efficiency gain over the MSM estimator when

nðsÞ are simulated.

Theorem 2 can be used to make inferences on the parameter Z0: For this purpose,

D and S need to be consistently estimated. As shown in the Appendix, D ¼

limL!11L

PL‘¼1E0½

1n�‘

Pn�‘i¼1X ‘

@yi‘ðZ0Þ

@Z0 � and S ¼ limL!11L

PL‘¼1Var0½

1n�‘

Pn�‘i¼1X ‘yi‘ðZ

0Þ�:

Thus, the next result provides consistent estimators for D and S; respectively.

Corollary. Define

D ¼1

L

XL

l¼1

1

n�‘

Xn�‘i¼1

X ‘@yi‘ðZMSMÞ

@Z0;

S ¼1

L

XL

l¼1

1

n�‘2

Xn�‘i¼1

X ‘yi‘ðZMSMÞXn�‘i¼1

y0i‘ðZMSMÞX 0

‘:

Then For any fixed S and T ; and as L ! 1; D D and S S converge in probability

to 0, respectively.

One usefulness of our approach is to recover the entry cost c‘ for auction ‘;‘ ¼ 1; . . . ;L: To see how, note that following Levin and Smith (1994), the zero exante expected gain condition (1) leads to

c‘ ¼XN‘

n¼1

N‘ 1

n 1

� �ðq�

‘ Þn 1

ð1 q�‘ Þ

N‘ nTnðp0‘ ÞðV n;‘ W n;‘Þ=n

� �;

where Tnðp0‘ Þ is the probability of a trade given n active bidders and the reservation

price p0‘ ; V n;‘ is the expected value of the ‘th item to the highest bidder and W n;‘ isthe expected payment this bidder makes to the seller given that the trade occurs. As

shown by Levin and Smith (1994), V n;‘ W n;‘ ¼ nðV n;‘ Vn 1;‘Þ and V n;‘ ¼R v

vvnf ðv j z‘; yÞF ðv j z‘; yÞ

n 1dv: Moreover, it can be readily shown that Tnðp0‘ Þ ¼

1 ðR p0‘

vf ðv j z‘; yÞdvÞn: Therefore, c‘ can be estimated as soon as the estimates of the

parameters in both hð� jx‘; gÞ and f ð� j z‘; yÞ are obtained.Although our MSM estimator and the SNLLS estimator proposed by Laffont

et al. (1995) share a common feature in the sense that both use some momentconditions derived from economic theory to simulate some computationally difficultterms without solving the Nash–Bayesian equilibrium strategy, ours differssignificantly from theirs in several ways. First, while the Laffont et al. (1995)estimator can be applied to the auctions without entry and with the number ofpotential bidders being a constant, ours can be applied to the auctions with thenumber of active bidders depending on the heterogeneity of auctioned objects andvarying across auctions because of the entry process. In particular, different from the

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previous estimation methods, our estimator exploits the information contained in theobserved number of actual bidders. As it turns out, the moment conditions we usehave different implications on the bidders’ bidding and participating behaviors, andhence can be employed for testing purposes as discussed in the next section. Second,our estimator is a simulated GMM estimator which does not need the bias correctionas required in the simulated nonlinear least squares estimator to achieve consistencyfor a fixed number of simulations.

5. Testing important hypotheses

We have shown that the entry process and binding reservation prices in first-priceauctions lead to a structural model that involve both observed bids and the number ofactual bidders, which can be jointly used to estimate the parameters in the distributionsof private values and the number of active bidders. Another important insight thispaper attempts to offer is that our approach proposed here can be used to addressseveral issues related to the validation of economic theory and policy. One issue, whichis of great interest but has not been fully resolved yet in the literature, is how to test thevalidity of the game-theoretic auction model. Another issue related to policy analysis,is whether the reservation prices are binding. Also, in auctions involving entryprocesses, it is interesting to test whether the zero ex ante expected gain for a potentialbidder as given by (1) is a valid assumption. In addition to the estimation methodproposed for the first-price auctions with entry and binding reservation prices, thispaper makes another contribution to the structural approach as the approachproposed here can be used to address these three issues as discussed below.

5.1. Test of binding reservation prices

In the real world, many auctions are organized by sellers as first-price sealed-bidauctions with publicly announced reservation prices. It is an important issue inpractice for the seller to address whether reservation prices are binding, as itindicates the extent to which the reservation prices are effective in screening bids.From an econometric viewpoint, effectively binding reservation prices make theinference of auction models more involved; this is one of the main messages thispaper attempts to convey. In theory, a reservation price p0 is binding if it is above thelower support of the private value distribution F ð�Þ; as in this case, a bidder has apositive probability of F ðp0Þ to have a private value below p0 and hence not to bid.From an econometric point of view, however, even when the reservation price p0 isabove the lower support of the private value distribution, it is still possible that foran auction data set under study, the overall binding effect of reservation prices is notstrong enough, in the sense that the reservation prices are too low to provide enoughinformation in efficiently estimating the structural model that takes into account theeffect of binding reservation prices. If this is the case, we can say that the reservationprices are virtually non-binding. It would be useful for practitioners if a formal testcan be provided to assess the effectiveness of reservation prices. This objective can be

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indeed achieved by exploiting (13), which gives the probability mass function for thenumber of actual bidders conditional on that the number of actual bidders is at leastone when the reservation price is present. We have the following proposition, whichis a direct result of (13).

Proposition 3. When F ðp0‘ j z‘; y0Þ ¼ 0; xðn�

‘ j n�‘X1; x‘; Z0Þ given in (13) is identical to

hðn�‘ j n�

‘X1;x‘; g0Þ ¼ hðn�‘ jx‘; g0Þ=½1 hðn� ¼ 0 jx‘; g0Þ�:

As stated in Proposition 3, the probability mass function xð� j �X1;x‘; Z0Þ for thenumber of actual bidders n� given in (13) reduces to hð� j �X1;x‘; g0Þ when thereservation price is not binding, i.e., F ðp0‘ j z‘; y

0Þ ¼ 0: This important fact can be

used as a basis for testing binding reservation prices. Specifically, we follow Vuong(1989) to consider the following non-nested hypotheses:

H0 : ET logxðn�

‘ j n�‘X1; x‘; Z0Þ

hðn�‘ j n�

‘X1;x‘; g0Þ

� �¼ 0;

against

Hf : ET logxðn�

‘ j n�‘X1;x‘; Z0Þ

hðn�‘ j n�

‘X1;x‘; g0Þ

� �40;

or

Hh : ET logxðn�

‘ j n�‘X1; x‘; Z0Þ

hðn�‘ j n�

‘X1;x‘; g0Þ

� �o0;

where the expectation ET ½�� is taken with respect to the true probability massfunction of n�

‘ conditional on n�‘X1: Note that under H0; the two competing models,

namely, the probability mass functions xð� j �X1; x‘; Z0Þ and hð� j �X1;x‘; g0Þ;provide statistically equivalent results when used to model n�; the (observed)number of actual bidders. On the other hand, Hf means that xð� j �X1;x‘; Z0Þ ispreferred to hð� j �X1; x‘; g0Þ to model n�; while Hh means that hð� j �X1; x‘; g0Þ ispreferred to xð� j �X1;x‘; Z0Þ to model n�: Therefore, for an auction data set underconsideration, if H0 is rejected in favor of Hf ; then Hh must be misspecified,meaning that the binding effect of reservation prices is strong enough and has to betaken into account. If H0 is rejected in favor of Hh; it means that the binding effectof reservation prices is not strong enough, and the model with non-bindingreservation prices produces more meaningful results.

5.2. Test of validity of the theoretical auction model

As is well known, the structural analysis of auction data is crucially based on theassumption that the bidders bid optimally by following the Nash–Bayesianequilibrium strategies given by (3). As a result, validity of the structural approachrelies on whether the observed bids are equilibrium bids. If the bids are notequilibrium bids, then the structural approach is inappropriate. Therefore, it isimportant to test whether the bidders bid optimally following (3). Due to the

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complexity and difficulty of the structural approach, most of the work in thestructural approach has mainly focused on estimation methods, assuming thatbidders bid optimally following (3). Exceptions are Paarsch (1992), Li et al. (2002),Guerre et al. (2000), Hendricks et al. (2003), Athey and Haile (2002) and Haile et al.(2000). Among these papers, Paarsch (1992) uses the data on tree planting contractauctions in British Columbia to estimate structural models from both common valueand private value paradigms and finds out that the common value is preferred as theestimates obtained from the private value model are quite sensitive to differentmoment conditions. Guerre et al. (2000) consider the first-price sealed-bid auctionswithout entry and assume that the number of potential bidders is constant across theauctions. They establish the restrictions imposed by economic theory on thedistribution of observed bids, which, in principle, could be tested nonparametrically.Li et al. (2002) extend the Guerre et al. (2000) method to the affiliate private valueparadigm. Extending the estimation methods developed in Guerre et al. (2000) andLi et al. (2002), Hendricks et al. (2003) and Haile et al. (2000) develop nonparametrictests of equilibrium bidding in a common values model. While focusing onnonparametric identification of auction models, Athey and Haile (2002) also discusssome tests that can distinguish between private and common value models. Relevantto the first-price sealed-bid auctions within the private value paradigm that is studiedin this paper, the test proposed in Guerre et al. (2000) is robust to parametricmisspecification. It has, however, limited applicability by now due to the difficulty innonparametrically estimating a function under monotonicity constraint, which iscrucial in the implementation of the nonparametric test. Given the importance oftesting for the optimality of bids, a novel contribution of this paper is that the MSMestimator proposed in the paper can be used to propose an easy-to-implement testingprocedure for the optimality of observed bids. Such a procedure can be proposedintuitively by first noting that the MSM estimator assumes that the bids areequilibrium bids leading to the moment condition (9). On the other hand, themoment conditions (11) and (12) always hold whether or not the bids are optimalprovided that the bidders optimally participate in the auctions. Therefore, one canuse (11) and (12) to propose a GMM estimator, say, Z; for Z0 ¼ ðy0; g0Þ; which isconsistent whether or not the bids are optimal.16 As a result, a Hausman type test ofoptimality of bids can be derived by comparing ZMSM with Z:17

An alternative approach to validating the optimality of bids can be taken by firstusing n�

‘ to obtain MLE estimates from the distribution given by (13), which containsthe information on y0 when the reservation prices are binding. Then a GMM test fortesting the moment condition for the optimal bid (9) can be implemented following

16Note that (11) and (12) are valid only when the reservation prices are binding. Consequently, such a

GMM estimator and the test for the optimality of bids can only be proposed when the reservations prices

are binding.17See Newey (1985) for the Hausman type test of testing hypotheses on subsets of moment conditions in

a GMM framework. Also, to implement this test, the weighting matrices used in MSM and GMM

estimators need to be (asymptotically) optimal, which in practice can be estimated by an initial MSM

(GMM) estimator using any weighting matrix.

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Pagan and Vella (1989) using the MLE estimates of y0 and g0 obtained from thedistribution of the number of actual bidders.

5.3. Test of mixed-strategies of entry

As assumed in Section 2, we consider the symmetric entry equilibrium involvingmixed strategies: each potential bidder incurs a cost c and enters the auction withprobability q� and does not do so with probability 1 q�; where q� satisfies (1), thezero ex ante expected gain restriction. Such a restriction, if valid, has some importantimplications. For example, under this restriction, the induced entry makes allexpected gain to the bidders driven out and hence the seller’s expected revenue andthe total social welfare become identical. A resulting proposition concerning theequilibrium analysis, as given in Levin and Smith (1994), is that unrestricted entry isoptimal and use of reservation prices is not desired in IPV auctions.18 It is clear thattesting whether the zero ex ante expected gain restriction is valid is an interestingtask. Since the MSM estimator proposed in this paper enables us to estimate theparameters in both distributions of bids and numbers of active bidders withoutimposing (1), one could test (1) using our MSM estimator. Alternatively, a restrictedMSM estimator can be proposed using the restriction implied by (1).19 Then sucha restricted MSM estimator and the unrestricted MSM estimator can be comparedso as to validate (1). The testing procedures and their properties are studied in detailsin Li (2004).

6. Monte Carlo results

The Monte Carlo studies in this section serve two purposes. First, this paperstresses that when there is an entry process and the reservation prices are binding, thenumber of active bidders is unobserved, while the active bidders’ equilibrium biddingstrategies are derived based on the number of active bidders. In this case, given theunobservability of the number of active bidders, it would be interesting to evaluatethe bias of the estimates resulting from using the number of potential bidders toreplace the number of active bidders in the structural model. Second, we evaluate thefinite sample performance of Vuong’s (1989) test in testing whether the reservationprices are binding. We conduct such a study and report the results mainly because, asdiscussed earlier, the MSM estimator as well as the testing procedures for validatingthe optimality of bids are based on the assumption of binding reservation prices,which, in addition, is also a policy related issue of interest. Furthermore, one ofthe main messages that this paper attempts to convey is that the observed numberof actual bidders is useful in the structural inference of the auction models.

18This result is not true, however, for a common value paradigm. See Levin and Smith (1994).19This raises some challenging issues as discussed previously such as the unobservability of entry costs.

An approach that Li (2004) in progress explores to adopt is to treat the entry costs as unobserved

heterogeneity with some parametrically specified distribution so as to make the use of (1) feasible.

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In particular, when the reservation prices are non-binding, the number of actualbidders contains the information on the parameters of the distribution of number ofactive bidders, while the number of actual bidders contains information onparameters in both distributions of private values and number of active bidders inthe case of binding reservation prices. Since Vuong’s test requires the estimation ofthese two non-nested cases using the MLE, Monte Carlo studies on the performanceof Vuong’s test can actually shed some lights on how practically the number of actualbidders can be used to make inferences. This is an important issue, as most of theliterature on structural analysis of auction models has used the observed bids only.The experiments are designed as follows. The probability for entry is modeled by

q�‘ ¼

expðg0 þ g1z‘Þ1þ expðg0 þ g1z‘Þ

; ‘ ¼ 1; . . . ; 100;

where g0 ¼ ðg0; g1Þ ¼ ð0:3; 0:1Þ and z‘; ‘ ¼ 1; . . . ; 100; are generated from a Uniformdistribution on (0, 5). This means that we consider the number of auctions as 100,which is about the usual sample size for auctions. Also, the variable z is used to controlfor some observed heterogeneity of the auctioned objects. The number of potentialbidders in our design is fixed across all auctions, and is chosen as N ¼ 15: Then foreach ‘; ‘ ¼ 1; . . . ; 100; we generate the number of active bidders from a binomialdistribution with parameters ðN; q�

‘ Þ: Lastly, to generate the number of actual bidders,we need to specify the distribution of private values as well as the reservation prices. Atthe ‘th auction, the density of private values is specified as the exponential form

f ðv‘ j z‘Þ ¼1

y‘exp

1

y‘v‘

� �;

where y‘ ¼ expðy0 þ y1z‘Þ for y0 ¼ ðy0; y1Þ ¼ ð0:8; 0:3Þ: We also set the reservationprice constant across all auctions as p0 ¼ expð0:8þ 0:3� 5=2Þ ¼ 4:44; which isabout the (unconditional) mean of the private value distribution f ðv‘Þ; and hence highenough to be binding. We then generate the number of actual bidders n�

‘

at the ‘th auction, ‘ ¼ 1; . . . ; 100; from a binomial distribution with parameters ðn‘; 1 F ðp0 j z‘ÞÞ ¼ ðn‘; expð p0=y‘ÞÞ: The number of replications is 500 in this Monte Carlostudy.Table 1 reports the Monte Carlo results on the estimation of the structural model

when the entry process is ignored, that is when the number of potential biddersðN ¼ 15Þ is used in place of the (unobserved) number of active bidders. Theestimation method used here is the simulated nonlinear least squares estimatorproposed in Li and Vuong (1997). It is evident that the estimates for y0 ¼ 0:8 andy1 ¼ 0:3 are both severely biased. These results demonstrate clearly the need fortaking into account the entry process, and hence reinforce the message of the paper.Table 2 reports the Monte Carlo results from the MLE of the distribution of n�

given in (13) that involves both g0 and y0 assuming that the reservation price isbinding, in which case n; the number of active bidders is unobserved. Since thereservation price p0 is set at 4.44, high enough to be binding, the distribution given in(13) is actually the true one for n�: As can be seen from Table 2, the MLE yields niceestimates for both g0 and y0:

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Table 1

Estimates Mean Median Std. Dev.

y0 0.5054 0.5048 0.0296

y1 0.1019 0.1022 0.0133

Table 2


g0 0.2780 0.2637 0.1126

g1 0.0862 0.0898 0.0306

y0 0.6957 0.7000 0.0310

y1 0.2621 0.2606 0.0097

Table 3


g0 0.5032 0.4905 0.1296

g1 0.1716 0.1665 0.0521


Table 3 reports the Monte Carlo results from the MLE of hð� j �X1;x; gÞ usingobservations on n�: This is based on the assumption of non-binding reservationprices under which hð� j �X1;x; gÞ becomes the distribution for the number of actualbidders conditional on that this number is at least one as given in Proposition 3.Also, in this case, only g0 are estimated. As expected, the estimation results for g0;which will be referred to as the ‘‘naive’’ estimates, are seriously biased, as the truedata generating process is that the reservation price is binding.To assess how Vuong’s test can perform in testing binding reservation prices in

practice, we use the MLE estimates from Tables 2 and 3. Note that here we want todistinguish between two strictly non-nested models. In this case, the null hypothesisof Vuong’s test is that the two models are indistinguishable, and the alternative is bi-directional in that the first alternative in our setting is that the true distribution ispreferred, meaning that the binding effect of reservation prices is dominant in thesample, and the second is that the ‘‘naive’’ distribution is preferred and hence thereservation price is virtually non-binding. It turns out that in 500 replications, at 5%significance level, Vuong’ test is in favor of the first alternative 500 times, that is, thetrue model is preferred.20 This means that Vuong’s test has a remarkable empiricalpower, which is 100% in our case.

20Note that the critical values of Vuong’s test depend on whether the two models are strictly non-nested,

overlapping or nested. Since the two models considered here are strictly non-nested, the critical value for

the test is chosen from the standard normal distribution at 5% significance level.

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Table 4


g0 0.2660 0.2645 0.0155

g1 0.0900 0.0901 0.0042

y0 0.0005 0.0008 0.0283

y1 0.0009 0.0007 0.0079

Table 5


g0 0.2561 0.2578 0.0005

g1 0.0906 0.0906 0.0002


To evaluate the power of Vuong’s test in detecting the second alternative, that is,the reservation price is virtually non-binding, we change the reservation price p0 inour Monte Carlo design to 1, low enough to be essentially non-binding. Then we usethe observed number of actual bidders to estimate the two models. Again, the formermodel uses the distribution xð� j �X1; x; ZÞ given in (13) while the latter useshð� j �X1;x; gÞ: Table 4 reports the MLE estimates using the general distribution andTable 5 reports the MLE estimates for the second model, which will be referred to asthe true model as the true data generating process is that the reservation price is toolow to be effectively binding. Table 4 shows that even in the virtually non-bindingreservation price case, the distribution in (13) is general enough to give goodestimates for g0; the parameters in the distribution of the number of active bidders.Moreover and more interestingly, the estimates for the parameters y0 in the privatevalue distribution are all insignificant as evidenced by the large standard deviations.This is reasonable as the general model for the number of actual bidders does notcontain enough information to estimate y0 as soon as the reservation price is non-binding. On the other hand, in the second model, if one treats the observed numberof actual bidders n� as the number of active bidders n, which is indeed the case as thereservation price is non-binding, then the MLE estimates are very close to the true g0:Lastly, at 5% significance level, out of 500 replications, Vuong’s test prefers thesecond alternative 500 times, which is the non-binding reservation price hypothesis.Therefore, the empirical power of Vuong’s test is 100% in this case.

7. Conclusion

In this paper, we consider first-price sealed-bid auctions with entry and reservationprices. We extend the structural estimation of auction models to a more general andrealistic framework in which entry decision is involved and there is a publiclyannounced reservation price that is possibly binding. We develop a Method of

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Simulated Moments estimator to simultaneously estimate the structural parametersin both the entry and bidding processes. Our estimator also has the computationaladvantage that it does not require the computation of Nash–Bayesian equilibriumstrategy in the estimation procedure. This paper demonstrates how the informationfrom the observed number of actual bidders can be helpful in the structural inferenceof auction data. In particular, it shows through Monte Carlo studies the necessity oftaking into account the entry process in the structural estimation and that Vuong’s(1989) test for non-nested models can be implemented using the MLE estimatesobtained from the two alternative models for the number of actual bidders to testwhether the reservation prices are binding. Since our MSM estimator is proposedassuming that the reservation prices are binding, in practice, one can first useVuong’s test to check whether the reservation prices are binding following themethodology developed in this paper. If the binding reservation prices are detected,then one can use the MSM estimator to estimate more efficiently the structuralmodel. Also, whether the bids are optimal according to the Nash–Bayesianequilibrium strategies can be tested.An interesting extension would be to drop the assumption that the active bidders

learn the number of active bidders n prior to stage 2 and to allow the active biddersto bid without knowing n: See, e.g., the theoretical model developed in McAfee et al.(2002) to study optimal reserve prices in real estate auctions and the nonparametricanalysis of a similar model within a common value paradigm using the outercontinental shelf (OCS) auctions data by Hendricks et al. (2003). While thetheoretical analysis from a game-theoretic viewpoint yields the same results whetheror not the active bidders know n before they bid, the econometric implementation ismore involved. In particular, the symmetric increasing Nash–Bayesian biddingstrategy is more complicated than (3). Although one can still derive momentconditions similar to (6)–(8), and propose a GMM estimator, the computationaladvantage of the MSM estimator in this paper that is to estimate the structuralmodel without computing the Nash–Bayesian equilibrium is no longer possible.Instead, while the basic idea in this paper still applies, one has to compute theNash–Bayesian equilibrium strategy for each trial value of the underlyingparameters, which is computationally intensive.To summarize, this paper makes two-fold contributions to the structural analysis

of auction data. On one hand, it shows how the complexity in the structuralestimation arising from the entry and binding reservation prices can be circumventedby using the information on both bids and numbers of actual bidders. Such amethodological contribution, which deals with two jointly distributed continuous-discrete variables, can also be useful for other econometric models, in particular, forthose selection models with participation decisions, which have been one of thecentral issues in econometrics since Heckman (1979). On the other hand, this papershows that despite the complexity introduced by the entry and binding reservationprices, the information from the observed numbers of actual bidders not only can beused to improve the efficiency of the structural estimation, but also can be used fortesting important policy related issues. Such an approach could open a door forfuture research on estimation and testing on more general structural models.

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Acknowledgements

I am grateful to the Editor and four referees for their helpful comments thatgreatly improved the paper. I also wish to thank S. Athey, M. Baye, P. Haile, J.Heckman, H. Hong, L.-F. Lee, H. Paarsch, I. Perrigne, P. Trivedi and Q. Vuong fortheir helpful comments and suggestions. Earlier versions of this paper have beenpresented at the Clarence W. Tow Conference on Applied Research concerningAuctions and Mechanism Design, Iowa City, Iowa, May 2000, the 8th WorldCongress of the Econometric Society, Seattle, August 2000, the MidwestEconometrics Group Annual Meeting, Chicago, October 2000, and at IndianaUniversity, University of Kansas, Ohio State, Penn State, Purdue, Rice, andUniversity of Texas at Austin seminars. Financial support from the National ScienceFoundation (SES-0001663) is gratefully acknowledged.

Appendix: Derivation of (7).

For ease of exposition, we omit the subscripts i and ‘ in the derivation. When thenumber of active bidders n ¼ 2; the Nash–Bayesian equilibrium strategy given in (3)becomes

b ¼ v 1

F ðv j z; y0Þ

Z v

p0F ðu j z; y0Þdu: ðA:1Þ

Using integration by parts, (A.1) is reduced to

b ¼p0F ðp0 j z; y0Þ

F ðv j z; y0Þþ

1

F ðv j z; y0Þ

Z v

p0uf ðu j z; y0Þdu:

It then follows that

E½b j bXp0; n ¼ 2; x; y0� ¼ p0F ðp0 j z; y0ÞE1

F ðv j z; y0Þj vXp0; n ¼ 2;x; y0

� �

þE1

F ðv j z; y0Þ

Z v

p0uf ðu j z; y0Þdu j vXp0; n ¼ 2; x; y0

� �

¼ p0F ðp0 j z; y0ÞZ �v

p0

f ðv j z; y0Þ

F ðv j z; y0Þð1 F ðp0 j z; y0ÞÞdv

þ

Z �v

p0

f ðv j z; y0Þ

F ðv j z; y0Þð1 F ðp0 j z; y0ÞÞ

Z v

p0uf ðu j z; y0Þdudv:

Note thatR �v

p0f ðv j z; y0Þ=F ðv j z; y0Þdv ¼ log F ð�v j z; y0Þ log F ðp0 j z; y0Þ ¼

log F ðp0 j z; y0Þ because F ð�v j z; y0Þ ¼ 1; andZ �v

p0

f ðv j z; y0Þ

F ðv j z; y0Þð1 F ðp0 j z; y0ÞÞ

Z v

p0uf ðu j z; y0Þdu dv

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¼1

1 F ðp0 j z; y0Þ

Z �v

p0

Z v

p0uf ðu j z; y0ÞdudlogF ðv j z; y0Þ

¼ 1

1 F ðp0 j z; y0Þ

Z �v

p0v log F ðv j z; y0Þf ðv j z; y0Þdv;

where the last equality follows from the integration by parts. Then (7) follows aftersome algebra. &

Proof of Proposition 1. Throughout the proofs of Propositions 1 and 2 as well asLemma 1, the subscript ‘ is omitted to simplify the notation. Define

Di ¼1 if bidder i is active

0

�D�

i ¼1 if bidder i is actual

0

�

so that the numbers of active and actual bidders are n ¼PN

i¼1Di and n� ¼PN

i¼1D�i ;

respectively. Also, since bidder i is an actual bidder if and only if bidder i is activeand viXp0; it follows that D�

i ¼ Di1IðviXp0Þ: We then have

Pr½n j n�;x; Z0�

¼ PrXN

j¼1

Dj ¼ n�� XN

j¼1

D�j ¼ n�;x; Z0

" #

¼ PrXN

j¼1

Dj ¼ n��D�

1 ¼ 1; . . . ;D�n� ¼ 1;D�

n�þ1 ¼ 0; . . . ;D�N ¼ 0; x; Z0

" #

¼ PrXN

j¼n�

Dj ¼ n n��D�

1 ¼ 1; . . . ;D�n� ¼ 1;D�

n�þ1 ¼ 0; . . . ;D�N ¼ 0;x; Z0

" #

¼ PrXN

j¼n�

Dj ¼ n n��D�

n�þ1 ¼ 0; . . . ;D�N ¼ 0;x; Z0

" #; ðA:2Þ

where the second equality follows from symmetry, the third equality follows fromDj ¼ 1 whenever D�

j ¼ 1 and fourth equality follows from the mutual independenceof the ðD�

j ;Dj ; vjÞ: The right-hand side of (A.2) is the probability that there are n n�

active bidders among N n� bidders who have not submitted a bid. On the otherhand, using Bayes rule, we have

Pr½Dj ¼ 1 jD�j ¼ 0; x; Z0� ¼ Pr½D�

j ¼ 0 jDj ¼ 1;x; Z0�Pr½Dj ¼ 1 jx; Z0�Pr½D�

j ¼ 0 jx; Z0�

¼F ðp0 j z; y0Þq�

F ðp0 j z; y0Þq� þ 1 q�; ðA:3Þ

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since Pr½D�j ¼ 1 jx; Z0� ¼ ½1 F ðp0 j z; y0Þ�q�: Thus combination of (A.2) and (A.3)

gives the expression for Pr½n j n�; x; Z0� in Proposition 1, that is

Pr½n j n�; x; Z0� ¼N n�

n n�

!F ðp0 j z; y0Þq�

F ðp0 j z; y0Þq� þ 1 q�

� �n n�

�1 q�

F ðp0 j z; y0Þq� þ 1 q�

� �N n

: ðA:4Þ

Now to derive the moment condition (9), we have

E½bi j biXp0; n�;x; Z0� ¼ E½bi jD�i ¼ 1; n�;x; Z0�

¼XN

n¼n�

E½bi jD�i ¼ 1; n; n�;x; Z0�Pr½n jD�

i ¼ 1; n�;x; Z0�

¼XN

n¼n�

E½bi jD�i ¼ 1; n; x; Z0�Pr½n jD�

i ¼ 1; n�;x; Z0�

¼XN

n¼n�

E½bi j biXp0; n;x; Z0�Pr½n jD�i ¼ 1; n�;x; Z0�; ðA:5Þ

where the first and the fourth equalities follow from 1IðbiXp0Þ ¼ Di1IðviXp0Þ ¼ D�i ;

the second equality follows from Pr½n jD�i ¼ 1; n�;x; Z0� ¼ 0 for non�; and the third

equality follows from vi and hence bi being independent of ðDj ; vjÞ jai and hence ofn� given ðD�

i ; nÞ: Moreover, for n�X1; we have by symmetry

Pr½n jD�i ¼ 1; n�;x; Z0� ¼ Pr½n j n�;x; Z0�; ðA:6Þ

where the right-hand side is given by (A.4). Then (9) follows from (A.5), (A.6) and(A.4). &

Proof of Lemma 1. Knowing that n jx � BinomialðN; q�Þ and n� j n; x �

Binomialðn; 1 F ðp0 j z; y0ÞÞ; we have that the joint distribution of ðn; n�Þ jx hassupport on the triangle 0pn�pnpN and is given by

Pr½n; n� jx� ¼N!

ðN nÞ!

1

n�!ðn n�Þ!q�n½1 F ðp0 j z; y0Þ�n

�

� ð1 q�ÞN nF n n� ðp0 j z; y0Þ:

As a result, (10) in Lemma 1 is obtained by noting that Pr½n� jx� ¼PNn¼n� Pr½n; n

� jx�: &

Proof of Proposition 2. First, we note for any rX0; E½n�r jx� ¼

E½n�r j n�X1;x�Pr½n�

X1 jx� þ E½n�r j n� ¼ 0; x�Pr½n� ¼ 0 jx�; which leads toE½n�r j n�

X1;x� ¼ E½n�r jx�=f1 Pr½n� ¼ 0 jx�g because E½n�r j n� ¼ 0;x� ¼ 0: (11)and (12) in Proposition 2 then follow because Pr½n� ¼ 0 jx� ¼ ½1 q� þ

q�F ðp0 j z; y0Þ�N from Lemma 1. &

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Proof of Theorem 1. To prove consistency, denote the objective function in (16) byMS;T ;LðZÞ: It then follows from A.3 and A.4 and Theorem 2.7.2 of Bierens (1994)

that 1L

PL‘¼1

1n�‘

Pn�‘i¼1X ‘yi‘ðZÞ converges in probability uniformly in Z 2 Y� G to

limL!11L

PL‘¼1E½

1n�‘

Pn�‘i¼1X ‘yi‘ðZÞ�: Using the law of iterated expectations, we have

E1

n�‘

Xn�‘i¼1

X ‘yi‘ðZÞ

24

35 ¼ En�

‘½E½X ‘yi‘ðZÞ j n�

‘ ��

¼ E0½X ‘yi‘ðZÞ�:

It then follows that 1L

PL‘¼1

1n�‘

Pn�‘i¼1X ‘yi‘ðZÞ converges in probability uniformly in

Z 2 Y� G to limL!11L

PL‘¼1E0½X ‘yi‘ðZÞ�; which is continuous in Z because of the

continuity of yi‘ð�Þ: It then can be readily shown by using the triangle andCauchy–Schwartz inequalities following the proof of Theorem 2.6 of Newey and

McFadden (1994) that MS;T ;LðZÞ=L2 converges in probability to

limL!11L

PL‘¼1E0½ðX ‘yi‘ðZÞÞ

0�O limL!1

1L

PL‘¼1E0½ðX ‘yi‘ðZÞÞ� uniformly in Z; which

is uniquely minimized at the true parameter Z0 following (14) provided that Z0 isuniquely identified as assumed in A.2. Therefore, ZMSM that minimizes MS;T ;LðZÞ asdefined in (16) converges in probability to Z0: &

Proof of Theorem 2. Note first that the first order conditions for the minimizationproblem defined in (16) are

0 ¼1

LffiffiffiffiL

pXL

l¼1

1

n�‘

Xn�‘i¼1

@y0i‘ðZMSMÞ

@ZX 0

‘OXL

l¼1

1

n�‘

Xn�‘i¼1

X ‘yi‘ðZMSMÞ:

A Taylor expansion of yi‘ðZMSMÞ around Z0 yields

0 ¼1

L

XL

l¼1

1

n�‘

Xn�‘i¼1

@y0i‘ðZMSMÞ

@ZX 0

‘O1ffiffiffiffiL

pXL

l¼1

1

n�‘

Xn�‘i¼1

X ‘ðyi‘ðZ0Þ

þ@yi‘ðZ

�Þ

@Z0ðZMSM y0ÞÞ;

where Z� is between ZMSM and Z0:Note that as L ! 1; ZMSM converges in probability to Z0: It follows that Z�

converges to Z0 as well. Therefore, as L ! 1; we have

0 ¼ D0O1ffiffiffiffiL

pXL

l¼1

1

n�‘

Xn�‘i¼1

X ‘yi‘ðZ0Þ þ D0OD

ffiffiffiffiL

pðZMSM Z0Þ;

where

D ¼ limL!1

1

L

XL

‘¼1

E1

n�‘

Xn�‘i¼1

X ‘@yi‘ðZ

0Þ

@Z0

24

35

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¼ limL!1

1

L

XL

‘¼1

E X ‘@yi‘ðZ

0Þ

@Z0

� �

¼ limL!1

1

L

XL

‘¼1

E0 X ‘@yi‘ðZ

0Þ

@Z0

� �;

by using the law of iterated expectations.On the other hand, as L ! 1;

1ffiffiffiffiL

pXL

l¼1

1

n�‘

Xn�‘i¼1

X ‘yi‘ðZ0Þ

converges in distribution to Nð0;SÞ; where

S ¼ limL!1

1

L

XL

‘¼1

Var1

n�‘

Xn�‘i¼1

X ‘yi‘ðZ0Þ

24

35

¼ limL!1

1

L

XL

‘¼1

E1

n�‘2

Xn�‘i¼1

X ‘yi‘ðZ0ÞXn�‘i¼1

y0i‘ðZ

0ÞX 0‘

24

35

¼ limL!1

1

L

XL

‘¼1

E01

n�‘

X ‘M‘X0‘

� �;

where the second equality follows from that E½ 1n�‘

Pn�‘i¼1X ‘yi‘ðZ

0Þ� ¼ 0; the third onefollows from a tedious but straightforward algebra, and M‘ is a 3� 3 matrix definedin Theorem 2. &

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Econometrics of first-price auctions with entry and binding reservation prices

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