JOURNAL OF APPLIED ECONOMETRICS
J. Appl. Econ. 23: 925– 948 (2008) Published online 7 November 2008 in Wiley InterScience (www.interscience.wiley.com) DOI: 10.1002/jae.1036

ECONOMETRICS OF AUCTIONS BY LEAST SQUARES
LEONARDO REZENDE*
PUC-Rio, Rio de Janeiro, Brazil; and University of Illinois at Urbana–Champaign, Illinois, USA

SUMMARY I investigate using the method of ordinary least squares (OLS) on auction data. I find that for parameterizations of the valuation distribution that are common in empirical practice, an adaptation of OLS provides unbiased estimators of structural parameters. Under symmetric independent private values, adapted OLS is a specialization of the method of moments strategy of Laffont, Ossard and Vuong (1995). In contrast to their estimator, here simulation is not required, leading to a computationally simpler procedure. The paper also discusses using estimation results for inference on the shape of the valuation distribution, and applicability outside the symmetric independent private values framework. Copyright  2008 John Wiley & Sons, Ltd. Received 15 September 2006; Revised 1 July 2008

1. INTRODUCTION The field of econometrics of auctions has been successful in providing methods for the investigation of auction data that are well grounded in economic theory and allow for inference on the structure of an auction environment. Today, a researcher has a number of alternative structural methods, especially within the independent private-values paradigm (IPVP); an excellent reference to this literature is the book by Paarsch and Hong (2006). To name a few alternatives, it is possible to use maximum likelihood (Donald and Paarsch, 1996), nonparametric methods (Guerre et al., 2000), simulated nonlinear least squares (Laffont et al., 1995) and bounds estimation of incomplete models (Haile and Tamer, 2003). However, it is still common to find empirical studies of auction markets that do not use these techniques and instead run regressions of the following form: p D Xˇ C ε 1

where p is the transaction price (or its logarithm). For example, several studies have recently investigated the importance attributed by consumers to the reputation of sellers in eBay using specifications of this sort (Houser and Wooders, 2006; Lucking-Reiley et al., 2007; McDonald and Slawson, 2002; Melnik and Alm, 2002). These authors identify interesting empirical relationships, but their work is not structural, in the sense of identifying parameters from the valuation distribution. This is a drawback, since these estimates cannot be used for counter-factual analysis of changes in the institutional environment.
Ł Correspondence to: Leonardo Rezende, Department of Economics, Catholic University at Rio de Janeiro, Rua Marquˆ s e de S˜ o Vicente, 225 SL 210F, Rio de Janeiro 22453-900, Brazil. E-mail: lrezende@econ.puc-rio.br a

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What makes equation (1) not structural is not the right-hand side of regression—covariates may indeed impact consumer preferences in a linear fashion—but rather the left-hand side. A superior starting point for an empirical analysis would be a specification such as Vi D Xˇ C εi 2

where Vi is consumer i’s valuation, or willingness to pay, for the product being auctioned. Unlike price, valuation is a demand concept, which reflects the consumer’s preferences in isolation of supply or market institutions effects. A bidder will never elect to pay his or her own valuation for the product—doing so would guarantee that the bidder would not gain anything from participating in an auction. Because of that, one knows that prices and valuations are not the same. Therefore, whenever the objective is to measure consumers’ preferences, a regression such as equation (1) would suffer from misspecification bias. The main point of this paper is that applying the broad method of moments approach from Laffont et al. (1995) to a standard specification such as equation (2) leads to the possibility of structural estimation by an adaptation of ordinary least squares. Laffont et al. (1995) exploited the fundamental property of symmetric, independent, privatevalues auctions, expected revenue equivalence, as a moment condition that can be used to estimate parameters that appear in the valuation distribution in a general way. To obtain a mapping between parameters and moments for estimation purposes, they use computer simulation. They show that due to simulation error standard nonlinear least squares on simulated moments is biased; they propose instead a simulated nonlinear least squares estimator (SNLLS) that minimizes an objective function that corrects for the simulation bias. In this paper, I argue that once one specializes the analysis to parameterizations that are commonly made in empirical work, the method simplifies dramatically. In particular, simulation of the sort used in the SNLLS is no longer necessary, which reduces the computational cost of the algorithm and the interpretation of its statistical properties. I also show that, while revenue equivalence is helpful, it is not crucial for the method. The key property is that auction games are ‘linear’, in the sense that a change of location or scale of all valuations within an auction leads to a corresponding change in the expected price in equilibrium. Using this property, it is possible to generalize the method to some environments where revenue equivalence does not hold. Of course, the idea (or the practice) of estimation with auction data through OLS is not new. What I do is to propose an adjustment, an additional artificial regressor, that makes the standard OLS estimators unbiased for the parameters of the valuation distribution. In exercise 3.2.b of Paarsch and Hong (2006), the same adjustment is proposed in the context of a second-price auction. Here, I argue that the adjustment is applicable to a wider class of auctions and, when properly generalized, to contexts where revenue equivalence does not hold. I hope my discussion of the opportunities and limits of OLS for the structural estimation of auction models can help guide researchers interested in estimating factors that affect bidder valuations in a way that is simple to implement and easy to interpret, as well as theoretically sound; conversely, it may also help inform researchers of the potential pitfalls of simple OLS on auction data in many contexts. I do not see the adjusted OLS method proposed here as a true substitute for more complex estimation procedures found in the literature; the computational simplicity comes with
Copyright  2008 John Wiley & Sons, Ltd. J. Appl. Econ. 23: 925–948 (2008) DOI: 10.1002/jae

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a cost. First, while unbiased, the estimators are not as efficient as those obtained via the method of maximum likelihood. Second, the method depends on a parameterization where only the location and scale of valuations vary across auctions (and the variation is the same for every bidder). This is too restrictive in some circumstances; for example, in a study of an asymmetric auction environment the focus may be on whether a given covariate affects bidder valuations differently in the same auction. Such effect cannot be accounted for in the framework discussed here. Even then, I hope the methods discussed here can find a place in the toolbox of applied economists, as they may help guide preliminary work on auction data. The paper is organized as follows: Section 2 describes the setting. Section 3 introduces the method. The possibility of exploring the information obtained from the method for inference about the shape of the valuation distribution is discussed in Section 4. Section 5 discusses whether the method can be generalized for contexts where revenue equivalence does not hold. A practical application of the method to a sample of iPod mini eBay auctions is illustrated in Section 6. Section 7 concludes.

2. THE SETTING I am interested in estimating the parameters that determine the location and scale of distributions of bidder valuations in a sample of L auctions. Let Vi be the valuation of the ith bidder in the be the mean and the standard deviation of valuations in auction . As the th auction. Let subscript suggests, they may vary across auctions. I shall assume that variation across auctions only affects the location and scale of valuations, not other aspects of the valuation distribution: Assumption 1 Vi D distribution F. C εi , where the εi are independently and identically distributed with

/ , has a common distribution F that does not vary The standardized valuation, εi D Vi across auctions or bidders. For the moment, I also impose independence, both across auctions and bidders. Independence across different auctions is an assumption made here for convenience. Relaxing it in what follows would have the same effect of having non-spherical disturbances in a linear regression model: it would affect the efficiency, but not the unbiasedness of the estimators.1 Independence across bidder valuations within an auction is a requirement of the benchmark model in auction theory, the symmetric independent private values auction model. Section 5 investigates the possibility of extending the method to settings where valuations are not independent. I consider the problem of a researcher interested in evaluating the effect of exogenous covariates and . Let X be the vector of covariates that affect the expected valuation and Z the on vector of covariates that affect . It is possible to have regressors that appear in both X and Z . We assume that [X , Z ] are either deterministic or otherwise publicly known by all bidders before auction starts. Linearity is imposed:
1 In any case, spherical disturbances in equation (2) are not enough to guarantee efficiency of OLS, since the regression will be heteroskedastic even under this assumption.

Copyright  2008 John Wiley & Sons, Ltd.

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Assumption 2

D X ˇ and

D Z ˛.

This is imposed solely in the interest of simplicity. All that follows would still hold if these relationships were nonlinear, substituting nonlinear least squares for OLS. Linearity in the specification of is unusual, since the standard practice is to specify a form that restricts it to be non-negative. This can be done here as well, and again would lead to nonlinear least squares.2 I intend to obtain estimates of ˇ and ˛ from a sample of auctions where the covariates [X , Z ], the number of bidders n and the winning price p are observed. I require Assumption 3 The number of bidders, n , is exogenous and common knowledge. Bidders are risk neutral, and maximize their profits at each auction in isolation. Here exogeneity is meant both in the game-theoretic sense—n is taken as given, and is not determined by each bidder decision-making process—and in the econometric sense—εi and n are independent. The number of bidders is also assumed to be publicly known before bidding. We consider a standard auction, in the following sense: Assumption 4 The auction rules are such that the good is always assigned to the bidder with the highest value, and the lowest valuation bidder expects to pay nothing. This is a condition satisfied by the English auction, the sealed bid first-price auction, the secondprice auction, and also by the all-pay auction.3 The method uses the central result in auction theory, the expected revenue equivalence theorem (Vickrey, 1961; Myerson, 1981):4 Theorem 1 (expected revenue equivalence) Under Assumptions 1, 3 and 4, the expected payment for the good in auction is E[V 2:n ]. Here the expectation is taken with respect to all information that is publicly available at the time of the auction; therefore the expected revenue equivalence theorem establishes that E[p jX , Z , n ] D E[V 2:n jX , Z , n ]. 3. ESTIMATION In this section, I propose two alternative estimation procedures. The first one requires information about F, that is, knowledge of the valuation distribution up to location and scale. The second method is applicable when F is unknown.
2 It may be worth pointing out that the role played by the specification of in what follows is not the usual one. When is used to correct for heteroskedasticity in a GLS procedure, allowing for negative values is the functional form of computationally problematic, since the objective function of the estimation procedure may become ill behaved. This is will appear in the least squares computation in a different way. Hypothetical negative values not a problem here, since would not lead to any computational problems, although they are hardly justifiable on economic grounds. for 3 In the case of an all-pay auction, p should be interpreted as the sum of the prices paid by all bidders, rather than the amount paid by the winner. 4 In what follows the notation x k : n represents the kth-highest order statistic from an i.i.d. sample of n observations fxi gn . iD1

Copyright  2008 John Wiley & Sons, Ltd.

J. Appl. Econ. 23: 925–948 (2008) DOI: 10.1002/jae

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3.1. Estimation when F is Known Suppose a dataset of auctions is available with information about the final selling price p , the number of bidders n and covariates [X , Z ]. Then, under the assumptions made, I can write E[p jX , Z , n ] D E[V 2:n jX , Z , n ] D E[ D C C ε 2:n jX , Z , n ] E[ε 2:n ]

where the first equality is due to the expected revenue equivalence theorem, the third by linearity of the conditional expectation, and the second is a useful property of order statistics: when an increasing affine transformation is applied to all variables in a sample, the order statistics of the new sample are the same affine transformation of the original order statistics. Paarsch and Hong (2006) use this property to propose an OLS estimator for and in the context of a second-price auction, since then the winning price is the second-order statistic itself.5 Defining a n Á E[ε 2:n ] D n n I obtain E[p jX , Z , n ] D X ˇ C a n Z ˛. 3 1 tF t n 1

1

F t dF t

Note that the conditional expectation of the winning bid is linear in ˇ and ˛. This means that OLS is an unbiased, consistent estimation method for these coefficients. This observation gives rise to a straightforward procedure to estimate ˇ and ˛ when F is known: Method 1 Using the standardized value distribution F, compute a n for all values of n in the sample. Construct the set of regressors [X , a n Z ], and run OLS of the observed winning bids on these regressors. It is important to notice that Method 1 is not the same as introducing n as an additional regressor, as is sometimes done in the literature, but rather introducing a n , a nonlinear function of n . In fact, ignoring the nonlinearity is never correct: Proposition 1 There is no non-degenerate distribution F such that a(n) is affine in n. Because the proof of this proposition depends on tools that will be developed in Section 4, it will be deferred to Appendix B. The first step in Method 1 requires the computation of a n D E[ε 2:n ]. Values of a(n) are shown for some standardized distributions from selected location-scale families in Table I. By standardizing, I mean working with parameter values that lead to a distribution with mean 0 p p and variance 1; thus, in Table I, values labeled ‘uniform’ are from the U[ 3, 3] distribution;
5 This property has also been discussed and utilized by Thiel (1988) in the context of a common values auction. Thiel’s objective was to obtain for estimation purposes a linear relationship between two different objects, a bidder’s own signal and the bid. This procedure was corrected by Levin and Smith (1991) and applied by Paarsch (1992).

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‘logistic’ are from thep distribution F t D e t/ 3 / 1 C e t/ 3 (that is, the logistic distribution with scale parameter 3/ ); ‘Laplace’ is for the Laplace or double exponential distribution p p p with location 0 and scale 1/ 2, with density f t D e jtj 2 / 2;p and ‘Gumbel’ is p Gumbel the 6 / and scale 6/ , with or Extreme Value Type I distribution with location parameter p t/ 6 , where D 0.577216 . . . is the Euler–Mascheroni constant. F t De e Of these five distributions, in four a(n) can be written in closed form for all n ½ 2: ž For the uniform distribution, a n D p n 3 . 3 p nC1 n 2 1 ž For the logistic distribution, a n D 3 kD1 k n ž For the Laplace distribution, a n D nω1:n 1 k p

p

1 . 1 ω1:n , where n
2

ω1:n

n D p 2 2

n 1 tD1 k kD0

t

2 k! k C 1

1 p n2 2 n for all n ½ 1 (including ω1 : 1 D 0). ž For the Gumbel or extreme value type I distribution, p an D 6 [n log n 1 n 1 log n ]

p For the normal distribution, closed forms for a(n) exist only for n Ä 5: a 2 D 1/ , a 3 D 0 5 3 (Jones, 1948), a 4 D p [1 6 arcsin 1/3 ] and a 5 D p [1 6 arcsin 1/3 ]; see Godwin 2 2
Table I. Values of a(n) for selected distributions n 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Uniform 0.57735 0 0.34641 0.57735 0.74231 0.86603 0.96225 1.03923 1.10221 1.15470 1.19911 1.23718 1.27017 1.29904 1.32451 1.34715 1.36741 1.38564 1.40214 Normal 0.56419 0 0.29701 0.49502 0.64176 0.75737 0.85222 0.93230 1.00136 1.06192 1.11573 1.16408 1.20790 1.24794 1.28474 1.31878 1.35041 1.37994 1.40760 Logistic 0.55133 0 0.27566 0.45944 0.59727 0.70754 0.79943 0.87819 0.94710 1.00836 1.06350 1.11362 1.15956 1.20197 1.24135 1.27811 1.31256 1.34500 1.37563 Laplace 0.53033 0 0.24307 0.40511 0.53033 0.63419 0.72373 0.80278 0.87369 0.93807 0.99703 1.05144 1.10196 1.14910 1.19330 1.23489 1.27418 1.31139 1.34675 Gumbel 0.54044 0.09184 0.18367 0.38495 0.54410 0.67588 0.78842 0.88665 0.97383 1.05219 1.12336 1.18857 1.24872 1.30456 1.35665 1.40548 1.45142 1.49480 1.53590

Copyright  2008 John Wiley & Sons, Ltd.

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(1949). For higher values of n numerical integration can be used. The results reported in Table I have been computed using Mathematica’s NIntegrate algorithm. Tabulations with more accuracy and for higher values of n can be found in Harter (1961) (5 decimal places, n up to 100), Yamauti et al. (1972) (10 places, n up to 50) and Parrish (1992) (25 places, n up to 50). The method can be applied with an arbitrary distribution F with mean 0 and variance 1. In this case, numerical integration to evaluate a(n) will be required. Unbiasedness of the least squares regression requires this integration to be accurate, as computational error will lead to error-in-variables bias. Laffont et al. (1995) deal with this problem by modifying the objective function of their estimator to compensate for the computational error bias. In our case, the issue is less problematic because: (1) closed-form solutions for a(n) exist in some cases; (2) even when they do not exist, the computation of a(n) must be done only once and, therefore, it is possible to require a high level of accuracy. (In the context of Laffont et al., 1995, the computation must be redone for every evaluation of the objective function, and therefore accuracy is much more costly.) Assuming that the hypothesis concerning F is correct, and that the calculation of a(n) is made without errors, the estimators from Method 1 have the standard properties of least-squares p estimators with non-spherical (and non-normal) disturbances; they are unbiased and L-consistent, but they are not efficient. Even with independence across auctions, there is heteroskedasticity across auctions with different numbers of bidders and efficiency can be gained by implementing generalized least squares. For some distributional families, closed-form solutions for the variance of ε 2:n are available; 6 n n 1 [log n for example, for the standardized Gumbel distribution, var[ε 2:n ] D 1 2 log n 1 ]2 . If is constant across auctions, the variance of the residuals in Method 1 regression is a multiple of var[ε 2:n ], and thus weighted least squares can be done with little additional computational cost. Also, once F is imposed the model is fully specified, so the method of maximum likelihood (ML) is available. Maximum-likelihood estimators for the symmetric independent private-values framework have been studied by Donald and Paarsch (1996). For first-price auctions, the support of the bid distribution depends on parameters; as Donald and Paarsch (2002) put it, ‘in such situations the maximum-likelihood estimator is often difficult to calculate and usually has a nonstandard limiting distribution that depends on nuisance parameters’ (p. 305). Donald and Paarsch (1993, 2002) propose alternative estimation procedures that are computationally easier and have simpler asymptotic properties. Similarly, the method proposed here trades off computational simplicity for efficiency. Appendix A provides evidence from Monte Carlo experiments that compare the OLS estimators with maximum likelihood in two simple settings where the ML estimator is computationally less involved. In both cases, there is a loss of efficiency from using OLS, as expected. OLS estimates for the mean tend to have acceptably low variances, in fact smaller than the variance of the estimator a researcher would obtain observing a sample of L valuations directly. The same is not true for the estimator of , which is unbiased, but imprecise. This is confirmed with field data in section 6. 3.2. Estimation when F is not Known The method proposed in the previous section requires introducing an appropriately chosen nonlinear function of the number of bidders as an additional regressor, and the appropriate function
Copyright  2008 John Wiley & Sons, Ltd. J. Appl. Econ. 23: 925–948 (2008) DOI: 10.1002/jae

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depends on the shape of the valuation distribution. This is a drawback since there may not be a priori information about F. In that case, it is possible to substitute dummy variables to flexibly estimate the unknown function a n in the conditional expectation of p . For every number of bidders k observed in an auction in the sample, I construct a dummy variable dk for the event n D k. I consider two situations in turn: the homoskedastic case where D , and then the general case where is a function of variable regressors as well. When D (that is, Z contains only a constant), writing m D X ˇ D ˇ0 C x ˇ1 , I can write the conditional expectation of p in terms of these variables as follows: E[p jX , fdk g] D x ˇ1 C k dk [ˇ0 C a k ] dk υk k D x ˇ1 C

where the summation over k is over the empirical support of n and υk D ˇ0 C a n . Again I obtain a linear relationship that can be estimated using OLS. The OLS estimates allows us to recover all ˇ coefficients except the intercept. Both and the intercept ˇ0 will be absorbed in the dummy coefficients, and thus cannot be identified.6 When Z contains other variables, the conditional expectation will involve interactions of all these variables (including the constant) with the dk . Adding those in a linear regression is enough to obtain unbiased estimates of the coefficients for regressors that appear on X but not on Z. To see that, let Y be the set of regressors common to both X and Z, and let x and z be the sets of regressors that appear on X but not on Z and vice versa. Write ˇ D ˇ1 , ˇ2 and ˛ D ˛1 , ˛2 so that Xˇ D Yˇ1 C xˇ2 and Z˛ D Y˛1 C z˛2 . Then E[p jX , Z , fdk g] D x ˇ2 C k dk Y [ˇ1 C a k ˛l ] C k dk z [a k ˛2 ];

4

D x ˇ2 C k dk Y υk C k dk z υk2

where υk1 D ˇ1 C a k ˛1 and υk2 D a k ˛2 . As before, ˇ2 can be estimated from OLS coefficients from regression (4). If there is no information about a Ð , ˛ is not identified from these coefficients alone, since multiplying all coefficients in ˛ and dividing all a k by the same scalar would lead to the same conditional expectation for p . Normalizing one of the coefficients in ˛1 to unity, the remaining coefficients can be identified in a sample where two different observed values for n are observed. Suppose for concreteness that ˛21 (the first coefficient in ˛2 ) is set to one; then the first coefficient in υk2 is υk21 D a k , and a k is identified. The remaining coefficients in ˛2 are identified from υk2 for a single value for k, and ˇ1 and ˛1 can be identified from υk1 from two different values for k. In practice, there might be more than two different values for n in the sample, in which case the model will be overidentified (after the normalization of a coefficient in ˛). To estimate the
6 If the researcher is willing to impose the condition that F is symmetric, then ˇ can be identified in the following way: 0 for any symmetric distribution F (with finite expectation), a 3 D 0. Thus, the coefficient of d3 is υ3 D ˇ0 C a 3 D ˇ0 .

Copyright  2008 John Wiley & Sons, Ltd.

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coefficients using the whole sample would involve estimating expression (4) using the method of nonlinear least squares, treating a k as additional coefficients to be estimated. We thus propose the following method: Method 2 For every number of bidders k observed in an auction in the sample, construct dummy variables dk for the event n D k. ž To obtain estimates of ˇ parameters for the set of regressors x in X that do not appear in Z , run OLS of p on x and all interactions of Z and the dummies dk . ž To obtain estimates of the other parameters, normalize one of the parameters in ˛ to 1, and run nonlinear least squares of p on the function in the right-hand side of equation 4, treating a k as parameters to be estimated. An interesting feature of Method 2 is that it may be possible to obtain estimates of moments of order statistics a n of the underlying distribution F. This information can be used for inference O about F. Section 4 discusses this possibility and Section 6 provides a practical illustration.

4. IDENTIFYING DISTRIBUTIONS FROM LEAST SQUARES COEFFICIENTS As mentioned in the previous section, with Method 2 it may be possible to obtain estimates for the a n s up to scale, the first moments of the second-order statistics of the F distribution. This section discusses ways to explore this information. It establishes that knowledge of second-order statistics expectations (for all n) is enough to fully identify the valuation distribution: in principle, it is possible to obtain a nonparametric estimator of the valuation distribution from the estimated coefficients of a least squares regression! It also provides a straightforward test for specific distributional assumptions. The methodology is illustrated in the next section with an application to a dataset of auctions of iPod minis from eBay. 4.1. Full Identification of F from fa(n)g A probability distribution can be fully identified from knowledge of its a n s. This has been shown in several versions in the statistics literature (Hoeffding, 1953; Chan, 1967; Pollak, 1973). Here a constructive proof is provided, which directly shows how to compute F from fa n g1 . nD2 Theorem 2 Suppose that F has a finite expectation. Then there is a one-to-one mapping between F and the sequence fa n g1 . nD2 The overall strategy of the proof follows Pollak (1973): the construction of F from a n will be made in two steps, which I state as lemmas. Lemma 1 (recurrence relation) Let ω k : n D E[ε k : n ], for all n D 2, 3, . . . and all k D 1, 2, 3, . . . , n 1. Then the following recurrence relation holds: ω k:n
Copyright  2008 John Wiley & Sons, Ltd.
1

D

n n

k

ω k:n C

k ω kC1:n n
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Proof See Appendix B. Corollary 1 Knowledge of a n D E[ε 2:n ] for every n D 2, 3, . . . is sufficient to know all ω k : n D E[ε k : n ], for all n D 2, 3, . . . and all k D 2, 3, . . . , n. Proof See Appendix B. The recurrence relation is valid for any random variable (with finite expectation). If the mean D ω 1 : 1 of it is known, then one can further compute ω 1:n using the recurrence formula. However, this will not be needed in what follows.7 Another remarkable fact is that the same recurrence relation is valid for the expectation of any measurable function g of the order statistics: E[g ε kC1:n ] D n E[g ε k : n 1 ] k n k E[g ε ]. So the same lemma applies to other moments, such as the variance of the k:n k order statistics.8 Lemma 2 Let fkn g be a sequence of integers such that 1 F 1 ˛ in probability. Proof See appendix B. Proof of Theorem 2 To obtain the quantile F 1 ˛ for any ˛ε[0, 1], select a sequence fkn g such that kn ½ 2 and kn /n ! 1 ˛. Use the recurrence formula to compute ω kn :n from the a n s, and then take the limit. The converse is immediate. In principle Theorem 2 provides a way to obtain a non-parametric estimate of the distribution of F by ordinary least squares. From Method 2 I obtain estimators a n that are consistent for a n . O Since the recurrence formula is linear, the corresponding ω k : n are also consistent (for a fixed n, O as the number of auctions goes to infinity). Finally, for large n, from Lemma 2 the expectation of these estimators converges to the quantiles of the original distribution. However, the linear combinations that arise from the recurrence formula have large coefficients of opposing signs. Therefore the variance of ω k : n can be very large even if the a n are estimated O O precisely. So this estimation strategy is likely to perform very poorly even in a very large dataset. To see that, using the recurrence formula the following closed-form expression for ω k : n can be obtained: ω k:n D n! k!k k 2

kn /n ! ˛ε[0, 1]. Then ε kn ,n !

n

1!

1 jD0 k j 2

k j! k

2! 2 j! n

an j j n j

1

where ω k : n is a linear combination of the k 1 previous a n s, with coefficients that alternate signs. As k and n grow, not only the number of terms in the sum grow, but so do the coefficients. This means that the estimator for ω k : n is likely to have a variance too large to be of practical use.
7 The recurrence relation can also be used to facilitate computation of a n ; for example, see the closed form solution for the Laplace distribution in Section 3.1. 8 Another application of the recurrence relation in Economics can be found in Athey and Haile (2002). Using the relation with g x D fx < tg one obtains a recurrence relation between distributions of order statistics. Athey and Haile (2002) use this relation to investigate ways of testing for the winner’s curse.

Copyright  2008 John Wiley & Sons, Ltd.

J. Appl. Econ. 23: 925–948 (2008) DOI: 10.1002/jae

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Consider, for example, the case where D 0 and D 1. In this case the estimators for a n O are means of different auction subsamples, and are therefore independent. In this case, the varik 2 2 O O ance of ω k : n is simply var ω k : n D jD0 cjkn var a n j , with cjkn D n k n!k 1 ! ð O ! k 2! j! k 2 j ! n j n j 1 . Suppose I am interested in estimating the median of the valuation distribution, and for that I use ω n/2:n for some large choice of n. Take the first coefficient c0kn . For k D n/2 I find O c0kn D n! n k ! k 1 !n n n! !1 ' n/2 ! 2 n!1 D n 2! k!k 1!

1

n

Thus the variance of ω n/2:n would be arbitrarily large for large values of n even if I had a O sample that allowed us to estimate a n well. For this reason, one should not expect to obtain accurate estimates of quantiles of F using this method. However, Theorem 2 indicates that there is a significant amount of information in fa n g; in particular, about aspects of the shape of the upper tail that are of interest to study revenue and surplus issues. The next section discusses a practical way to explore this information. 4.2. Hypothesis Testing about Distributions The information contained in the a n estimators can be used to test for the hypothesis of a specific O distribution F. Under the null hypothesis that a given F is the standardized valuation distribution, Method 1 would be appropriate. Under the alternative hypothesis, Method 2 would be. But the regression by Method 1 is a restricted version of the regression in Method 2, where the a n terms are required to follow the shape specified by F. So one can simply test the hypothesis by running both methods and applying an F-test on the R2 difference.9 This procedure is illustrated in Section 6.

5. OTHER SETTINGS The presentation of the method was done under the assumption of the benchmark model in auction theory, the independent private values framework with symmetric bidders and exogenous participation. In many applications, some or many of the underlying hypotheses in this model do
9 If F is known up to a family of distributions indexed by a set of parameters Â, a similar idea can be used in a two step procedure to estimate Â. Assume that εi is drawn from a distribution FεfF g , where fF g is a family parameterized by Â. I am interested in finding  such that F D F . Then, for every n observed in the sample, I can use the moment condition

an D

n! n 2!

xFÂ

n 2

1

FÂ dFÂ

for each n to run a GMM method using as the left-hand side the estimates that come from Method 2. Further research is needed to investigate if this approach is feasible and yields good estimators in small samples. Copyright  2008 John Wiley & Sons, Ltd. J. Appl. Econ. 23: 925–948 (2008) DOI: 10.1002/jae

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not hold. This section briefly discusses to what extent the method works when applied in more general contexts. Given that the presentation relied on the expected revenue equivalence theorem, it may come as a surprise that the method is to some extent applicable even in contexts where the theorem does not hold. The key to the method is a general property of auction games: when an affine transformation is applied to all valuations, then the new equilibrium bids will be an affine transformation of the original equilibrium bids. More formally, consider the class of auctions where a bidder action can be represented by a number bi (i’s ‘bid’). Let b D b1 , . . . , bn . Given an auction rule, let Wi b be i’s probability of winning the item given that bidders in the auction played b, and let Pi b be the expected payment given b, conditional upon winning. Note that these are functions of the realized bids of all participants, and thus are determined solely by the auctions rules and not by the data-generating process of the valuations. For example, in a first-price auction, Wi b D 1 if bi is the sole highest bid and 0 if bi is not the highest bid10 and Pi b D bi . Let vi be i’s valuation, not necessarily independent or symmetric. Let v D v1 , . . . , vn , and let and > 0 be constants. Then I have the following proposition, which generalizes results found in Bajari and Horta¸ su (2003), Deltas and Chakraborty (2001) and Krasnokutskaya (2004): c Proposition 2 Suppose bidders are risk neutral and the auction rule is such that (i) Wi C b D Wi b , (ii) Pi C b D C Pi b , for all i and (iii) losing bidders pay nothing. Then, if bŁ is a Nash equilibrium of an auction when bidders have valuations v, then C bŁ is a Nash equilibrium of the auction when valuations are C v. As a consequence, if the expected selling price in the former auction is pŁ , then the expected price in the latter auction will be C pŁ . Proof For bŁ to be a Nash equilibrium under v, it must be that it prescribes i to bid
Ł bi 2 argmax

bi , bŁ i D argmax E[ vi bi

Pi bi , bŁ i Wi bi , bŁ i ]

bi

where this expectation is taken to be conditional on all information available to i (including the contingency of winning). In the game with valuations C v, suppose the other bidders play C bŁ i . By following Ł C bi bidder i obtains Q C bŁ D E[ D E[ C vi C vi Pi
Ł

Pi

C b Ł Wi C Pi bŁ

C bŁ ]

Wi b Ł ]

D E[ vi D bŁ

Wi B Ł ]

Ł Ł Thus, if bi is a best response to bŁ i in the former game, C bi is a best response to C bŁ i in the latter game. As for the expected selling price, this is a consequence of this result combined with the affinity of Pi .
10 If

bi is tied for highest bid Wi b is the probability of i winning, as determined by the auction tie-breaking rule. J. Appl. Econ. 23: 925–948 (2008) DOI: 10.1002/jae

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The three conditions for the proposition are satisfied by standard auction rules. The condition on Wi is satisfied by any rule that assigns a winner based on a ordering of the bids.11 The condition on Pi is also very easily met, as typically the payment for the winner is either the bidder’s own bid (as in a first price auction) or somebody else’s bid (as in the Vickrey or English auctions). The condition on losing bidders paying nothing rules out all-pay auctions, but is otherwise usually satisfied. Consider now a setting with three sets of exogenous variables: X , Z , R . As before, let X and Z be regressors that affect the location and scale of bidder valuation. Let R be variables that capture all changes in conditions across auctions: for example, changes in auction rules and variation in participation levels (both total number of bidders and the presence of specific bidders if their valuations are asymmetric). Let V D V1 , . . . , Vn be the vector of valuations for bidders in auction . I impose the following generalization of assumption 1: C ε , where ε , the vector of standardized valuations, conditional on R Assumption 5 V D has the same distribution across auctions (and are independent across auctions). In other words, after a suitable standardization, the joint distribution of valuations is the same across auctions with the same characteristics R . Using Proposition 2, I obtain E[p jX , Z , R ] D X ˇ C Z ˛Q R a 5

where a R D E[p jR ], where p is the equilibrium selling price in an auction with rules R and Q Q Q D 0 and D 1). standardized valuations (that, is, if Conceptually, this is a direct generalization of equation (3); in practice, however, what is lost is the easy statistical interpretation of the additional regressor a n that is possible when the expected revenue equivalence theorem holds. To compute a R , one needs to fully specify the Q auction game being played, then solve for the equilibrium bids. Thus, generalizing Method 1 becomes computationally burdensome. It is possible to generalize Method 2. What is needed is to add dummy variables for all interactions of variables in R rather than just the number of bidders n . Here are two examples. As a first example, consider an affiliated valuations model, and suppose the joint distribution of valuations across auctions varies only in location and scale. Because valuations are not independent, the expected price in the auction depends not only on the number of bidders but also on the specific auction rule utilized; for example, if it is an English auction the expected price will be higher than in a first-price auction. In this case, the successful application of Method 2 will require a set of dummies that interact the number of bidders with the particular auction rule used, if it changes across the sample. Another example would be a situation where valuations are independent, but asymmetric: there are two types of bidders with valuation from different distributions. Now ‘location’ and ‘scale’ D X ˇ and coefficients are interpreted as follows: when evaluated at the true parameters, D Z ˛ are such that for all bidders i of the first type, Vi / is i.i.d. F1 and for bidders / is i.i.d. F2 , F1 6D F2 (note that this definition is only meaningful of the second type, Vi
11 It is important to point out that this assumption implicitly imposes a restriction on reserve prices. For W to have the desired property, either the reserve price r must be trivial (in the sense that the probability of vi and C vi be below r is zero) or it must change along with v (so that in the second auction the reserve price is C r).

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with variable regressors: the intercepts of and are not identified). In this case one would need all interactions of numbers of bidders for each separate type, plus dummies for changes in the auction rules, since in this case the expected revenue equivalence theorem is invalid as well. Appendix A shows a Monte Carlo experiment involving asymmetric auctions with bidders coming from three different distributions. In general, two important requirements should be met for this approach to be valid. First, coefficients that can be directly estimated have to impact all bidders in the same way. In an asymmetric auction setting, for example, it is not possible to have different coefficients in the location specification for different bidder types.12 This makes the method unappealing if the focus of the research is investigating the nature of bidder asymmetry. Second, R should be exogenous, and must contain all changes in rules and settings that modify the game being played. If equilibrium behavior is affected by any variable that is observed by bidders but is absent from R , then estimates from the method will be biased from spurious correlation between X or Z and this variable. An important practical setting where this problem may occur (even in the symmetric independent private values case) is when the number of bidders is not observed or is not exogenous. In conclusion, it is possible to apply Method 2 in some circumstances outside the symmetric independent private values environment, but one should keep in mind that the approach is just a way to obtain estimates for location and scale coefficients controlling for ‘auction effects’, not a way to investigate properties of auctions in wider settings. In the example reported in Appendix A, the coefficient of a exogenous regressor on can be estimated, but the coefficients of the ‘auction controls’ are very difficult to interpret, and it is not obvious how to use this information for inference on the nature of the asymmetry between bidders. It would also be impossible to apply the method without information on the number of bidders of each type. 6. ILLUSTRATION: iPOD MINI AUCTIONS IN eBAY This section illustrates the methods proposed in this paper using a sample of online auctions for Apple iPod mini players. The iPod is a popular portable device that stores and plays music files, as well as storing other electronic data.13 This product is a convenient choice for this illustration for two reasons: First, because of its success and size the online market for iPods is very liquid; currently in eBay alone there are several thousand listings of Apple iPods. Second, since the design of all iPods is under the control of the same company, the amount of product heterogeneity is limited. Out of the several different types of iPods, the analysis here will focus on the iPod mini. Compared to the original iPods, the minis provide the same capability, but are smaller and come in several different colors.14 At the time the data were collected iPod minis were no longer actively promoted by Apple, but were still commonly found in stores and in online markets such as eBay. It was a mature product in the technology adoption life cycle, in the sense that its characteristics were already well known by the public, but it was not yet perceived as obsolete. There is both a vertical and a horizontal dimension of differentiation among versions of the iPod mini. The vertical element (among new devices) is memory size, which can be 4 GB or
12 This

also implies that reservation prices move along with valuations as regressors change. models of iPod can also display pictures and play video. 14 Apple later released smaller devices: the iPod shuffle and the iPod nano.
13 Newer

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6 GB. The horizontal element is color (silver, blue, green or pink). In eBay, a significant fraction of the market is second-hand; even non-working devices are sold, often to be harvested for parts or used as a learning tool by technicians. This introduces an obvious additional source of differentiation. There may also be differentiation among auctions due to the seller characteristics, such as reputation. 6.1. Data Data were collected from 1225 completed eBay auctions from June 27 to July 18, 2006. The data include only auctions listed under the Apple iPod mini category in eBay. Auctions for accessories or parts as well as lots of devices were not included. However, the sample does include devices described as defective. We also excluded 225 auctions with one single bidder, since the methodology does not apply in this case.15 While the majority of auctions in the sample has a symbolic posted reserve price,16 some sellers elect to place a nontrivial reserve price. Analysis of the latter auctions is problematic, since we observe neither the bids nor the number of bidders below the reservation price.17 I elected to focus attention only on auctions with a posted reserve price of US $17 or less. This avoids auctions with reserve prices around $20 where the truncation effect starts to be visible. Fortunately, the restricted sample still contains the majority of the observations, with 654 auctions. For each auction I collected the final price, the number of bidders (defined as the number of different buyers that placed bids), product characteristics such as color, memory size and condition (new, used, refurbished or other), and seller feedback statistics. In addition to stated condition, a dummy variable BROKEN was created for items described as either non-operational or with a serious defect. Usually such a defect is short battery life, but includes issues such as a broken screen or headphone jack (which makes the device useless for its intended purpose). I did not classify as broken devices with scratches and other defects of cosmetic nature. In eBay, after every transaction the parties involved are asked to provide feedback on each other. Two statistics are widely reported: feedback score, which is the number of positive feedback ratings minus the negatives; and the positive feedback percentage. Instead of using these figures directly as regressors, I used POS D log2 P C 1 and NEG D log2 N C 1 , where P and N are the numbers of positive and negative ratings not withdrawn. 6.2. Results We report four specifications involving both indicators of product characteristics and seller reputation. Specification I simply assumes that both and are constant. Specification II allows to be a function of several characteristics that might affect willingness to pay, namely BROKEN,
15 In eBay, completed auctions with a single bid are almost always due to ‘Buy It Now’ bids. A popular auction rule in eBay allows sellers to post a ‘Buy It Now’ price, at which any buyer can purchase the item before bidding starts. Reynolds and Wooders (2009) provide a theoretical analysis of this auction rule. The ‘Buy It Now’ price is not a reserve price; if taken, it immediately ends the auction. Furthermore, once regular bidding starts, the ‘Buy It Now’ option is no longer available, and thus does not affect bidding in the auctions in the sample. 16 Confusingly, in eBay open reserve prices are called ‘starting bids’. 17 Note that the same problem does not exist for hidden reserve prices, since for those participation below the reserve is public information.

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dummies for used, new and refurbished condition,18 dummies for color19 and the number of gigabytes of memory. In addition, POS and NEG have been included in order to evaluate the impact of seller reputation in the bidders’ average willingness to pay. In specification III the variables found to be insignificant in specification II are dropped. Finally, in specification IV BROKEN, NEW, POS and NEG are allowed to affect both and . For each of the specifications, regressions were ran using Method 2 and Method 1 for each of the five distributions used in Table I. Table II reports the results of tests on the shape of F. In all cases, the Gumbel distribution provides a better fit. There is a theoretical reason to favor this distributional choice as well, as it is one of three possible limiting distributions of maxima of random variables. Suppose valuations in the general population are independent and identically distributed with an arbitrary distribution that satisfies a right tail condition;20 split the general population into several independent subgroups of the same size, and let bidders that go to eBay be those with the highest value within each group. Then according to the Fisher–Tippett–Gnedenko theorem, the distribution of the bidders’ valuations will be approximately Gumbel. For both reasons, I select the Gumbel distribution and report the coefficients obtained for each specification under this assumption in Table III. The coefficients obtained under other distributional assumptions are similar and have been omitted for brevity. The figures in parentheses are heteroskedasticity-consistent standard error estimates. Coefficients on the mean valuation have the expected sign. In specification II, I find that broken iPods are worth US $39 less, while iPods reported to be new are worth US $60 more. Other condition variables and color are not significant, and were dropped in specification III. Seller feedback has a significant effect on : roughly every doubling of positive ratings increases the buyers’ willingness to pay by US $2. A doubling of both positive and negative ratings leads to a decrease of US $1.20.
Table II. Functional form specification tests
Statistic I R2 F-statistic p-value R2 F-statistic p-value R2 F-statistic p-value R2 F-statistic p-value Unrestricted 0.2126 Uniform 0.1654 2.3811 0.9982 0.4402 2.4349 0.9986 0.4200 2.1646 0.9947 0.4228 1.4052 0.9788 Normal 0.1735 1.9743 0.9872 0.4445 2.1200 0.9935 0.4240 1.8761 0.9801 0.4273 1.3305 0.9554 Logistic 0.1755 1.8736 0.9799 0.4456 2.0346 0.9903 0.4251 1.7982 0.9720 0.4286 1.3094 0.9456 Laplace 0.1792 1.6827 0.9545 0.4479 1.8683 0.9794 0.4272 1.6489 0.9478 0.4311 1.2671 0.9205 Gumbel 0.1822 1.5327 0.9174 0.4490 1.7824 0.9701 0.4281 1.5850 0.9326 0.4322 1.2500 0.9079

II

0.4731

III

0.4502

IV

0.5074

omitted dummy variable is no condition reported. silver, blue, pink and green. Again, the omitted category is no color reported. 20 Namely, let G(x) be the distribution in question; then a function b(x) should exist such that, for all t > 0, F C tb limx!1 1 1 x F x x D e t .
19 Namely,

18 The

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ECONOMETRICS OF AUCTIONS BY LEAST SQUARES

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Table III. Estimated coefficients, F Gumbel
Regressor I On Constant BROKEN POS NEG New Used Refurbished Silver Blue Pink Green # GB On Constant BROKEN POS NEG New II OLS III IV I WLS II III

65.9728 (20.2600)

67.8343 (61.6944) 39.3721 (15.1692) 1.9572 (0.7003) 3.2092 (0.7145) 59.8633 (36.4966) 1.0429 (10.3430) 8.9332 (19.7835) 5.9684 (21.7494) 4.0321 (20.4925) 6.6170 (27.6143) 15.7220 (28.2819) 0.9592 (0.1522) 34.2180 (24.0819)

72.6487 (48.1675) 40.9110 (13.9647) 1.9924 (0.7031) 3.1656 (0.7192) 61.5885 (27.7271)

68.6269 (222.9034) 45.8312 (43.4235) 2.1551 (4.6552) 2.1885 (4.3013) 74.0562 (167.2685)

65.8326 (14.5745)

0.8072 (0.1738) 33.8169 (24.1540)

0.7739 (0.1695) 39.7211 (337.1709) 12.4026 (99.6003) 0.1737 (7.2029) 1.4749 (6.7277) 14.0496 (174.1814) 55.1977 (20.9938)

67.6936 (55.0027) 39.2975 (18.9874) 1.9562 (0.6954) 3.2107 (0.7354) 59.8172 (34.1179) 1.0479 (11.7372) 8.9111 (30.8950) 6.0242 (17.2327) 4.0017 (19.5332) 6.6749 (21.9380) 15.7465 (26.5944) 0.9547 (0.2544) 34.3841 (17.0459)

72.5584 (42.9749) 40.8381 (18.5294) 1.9924 (0.6761) 3.1688 (0.7455) 61.5575 (29.0913)

0.8027 (0.2595) 33.9580 (17.4974)

55.0320 (27.4887)

Why would that be? It is obvious why buyers are willing to pay more for sellers with positive feedback and less for sellers with negative feedback, but it is less clear why the latter effect is greater. In a simple framework of Bayesian updating, increasing both POS and NEG amount to more precise information about the odds of a bad deal, and this is not expected to make buyers less willing to buy. Specification IV suggests an explanation for this puzzle. In it, POS and NEG (as well as BROKEN and NEW) are allowed to affect valuation dispersion as well as location. Point estimates for both POS and NEG are negative, albeit insignificant. This suggests an explanation for why longer feedback histories seem to be worse: as POS and NEG grow, bidders obtain more information about the prospect of a good deal, and as a result their valuations become less dispersed (presumably because their heterogeneous beliefs converge). As a result, the transaction price decreases. Omitting this effect introduces a bias in the estimated mean effect. When this effect
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is accounted for, I find virtually identical effects of POS and NEG in , which is compatible with a theory where bidders care only about the odds of a bad deal. Similar biases have been found for the BROKEN and NEW variables. There is more uncertainty about the value of broken iPods and the there is less uncertainty about iPods classified as new. This leads to substantial biases on the coefficients on the mean when this effect is omitted; in specification IV I find that broken iPods are worth US $5 less and new iPods are worth US $13 more than previously estimated. While suggestive, point estimates of coefficients in specification IV are for the most part not significant. This is probably due to near multicollinearity for regressors used both for and . This combined with the findings from the Monte Carlo experiments suggests that the method may lead to imprecise estimates for rich specifications of . As discussed in Section 3.1, theory predicts heteroskedasticity of a form that depends on the F distribution. In the case of a standardized Gumbel distribution, the variance of the ε 2:n is 6 n n 1 [log n log n 1 ]2 , and the square root of this expression can be used as a 1 2 D . Table III reports weighted least squares estimates for weight to improve efficiency if the specifications where this requirement is met. Here, conventional standard errors are reported. Results are similar to those reported in the previous table.

7. CONCLUDING REMARKS This paper attempts to combine the strengths of two distinct branches of the empirical literature on auctions with the goal of obtaining a method at once computationally accessible and theoretically sound. In a nutshell, the main finding of the paper is that one can estimate parameters that impact the location and scale of the value distribution of all bidders in a simple and unbiased way, provided one controls for variables that affect bidding behavior in a flexible way. Under the framework investigated in this paper, the method provides a way to separate the effect of regressors that affect bidding through the value distribution from those that affect bidding strategically. It can be a useful tool for researchers interested in measuring the former while properly controlling for the latter; conversely, it helps identify the misspecification bias that occurs when the latter are not accounted for. The method is not designed for investigation of deeper aspects of the strategic interaction of players in general auctions. It is, however, an acceptable way to identify covariates that affect the location or scale of all bidder valuations. It can therefore be used as a tool to control for variation across auctions, such as product characteristics. As such, I hope the method might play a complementary role to existing structural methods, as a preliminary exploratory tool for researchers interested in applying more ambitious structural econometric models to auction data.

APPENDIX A: MONTE CARLO EXPERIMENTS I report results from three Monte Carlo experiments involving the estimation method proposed in this paper. In the first two, the objective is to compare its small-sample performance with that of the maximum likelihood estimator in simple settings; the third experiment is designed to illustrate the applicability of Method 2 in the context of asymmetric auctions.
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In the first experiment, I consider second-price auctions with independent private values where valuations are normally distributed with mean D 3 and standard deviation D 1. Participation is exogenous and random: in each auction, the number of bidders n is drawn from the uniform distribution over f2, 3, 4, 5, 6g. No reserve price is imposed. I seek to estimate both and from samples with 50, 100 or 200 auctions where I observe the selling price p and number of bidders n . As discussed in Donald and Paarsch (1996), because this is a Vickrey auction the likelihood of the winning price is hB p ; n D n 2 p p p 1 n n 1 1  , where  and are the standard normal distribution and density. One can obtain (efficient) estimates maximizing  hB p , n for and ; alternatively, one can run OLS of p on a constant and the artificial regressor for the normal distribution reported in Table I. In Table IV, I report statistics from 1000 replications of this experiment. Both estimators for show no bias and the variance of the least-square estimator is around 50–60% larger. The loss of efficiency in the estimator for is much larger; this seems to be a general weakness of the method. On the other hand, O LS does not show the downward bias that is present in the maximum-likelihood estimator. In the second experiment I compare the least-squares and maximum-likelihood estimator in the context of a first-price auction. As discussed in Donald and Paarsch (1996), using maximum likelihood with first-price auction data is more complex for two reasons: first, computing the maximum likelihood is harder, since it may require solving a differential equation for each observation and each parameter evaluation. Second, the maximum-likelihood estimators are not asymptotically normally distributed, so inference is not straightforward.21
Table IV. Monte Carlo experiment: least squares v. maximum likelihood, second-price auctions, valuations normally distributed
True value O LS 3 O LS 1 O ML 3 O ML 1

Sample size D 50 Mean Variance Mean squared error Jarque–Bera p-value Sample size D 100 Mean Variance Mean squared error Jarque–Bera p-value Sample size D 200 Mean Variance Mean squared error Jarque–Bera p-value

2.9978 0.0125 0.0125 2.4072 0.6999 3.0015 0.0063 0.0063 0.7573 0.3152 3.0006 0.0031 0.0031 2.0148 0.6348

1.0087 0.0579 0.0580 0.5250 0.2309 0.9907 0.0284 0.0285 2.1903 0.6655 0.9963 0.0151 0.0151 2.7182 0.7431

3.0055 0.0079 0.0079 0.6900 0.2918 3.0031 0.0042 0.0042 0.5810 0.2521 3.0022 0.0021 0.0021 2.8831 0.7634

0.9859 0.0080 0.0082 0.4913 0.2178 0.9872 0.0047 0.0048 0.9665 0.3832 0.9927 0.0021 0.0021 0.4287 0.1929

21 Donald and Paarsch (1993) introduce a technique, piecewise pseudo-maximum likelihood, that leads to asymptotically normal estimators for auction models.

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In experiment two, I consider first-price auctions with independent private values drawn from a uniform distribution with mean D 3 and standard deviation D 1.22 As above, n is uniform between 2 and 6, sample size is 50, 100 or 200 and the number of replications is 1000. Results are reported in Table V. As above, least-squares estimators are unbiased but less efficient than maximum likelihood, with the efficiency loss being more pronounced in the estimator of . The proportional loss of efficiency between the OLS and the ML estimators becomes larger as the sample size grows; this may be due to ML being super-consistent in this context, converging p at rate L rather than L; see Donald and Paarsch (1996). Results suggest that the ML estimator for the mean is biased downward. As predicted by Donald and Paarsch (1996), I find that the Jarque–Bera test strongly rejects normality of either maximum likelihood estimator, while the least-squares estimators are not rejected to be normally distributed for a sample size of 50. In the third experiment, I illustrate the application of Method 2 in the context of asymmetric auctions. I consider second-price independent private value auctions with three different types of bidders: bidders of the high type have values drawn independently from distribution N 12 C X ˇ, 3 , bidders of the medium type from N 11 C X ˇ, 2 , and bidders of the low type from N 10 C X ˇ, 1 . X is a product characteristic that has the same effect ˇ on all bidders. X is drawn i.i.d. U[0, 5]. The number of bidders is exogenous and random, with one or two of each bidder type per auction. The number of bidders of each type is observed, as it is necessary to apply the method. I seek to estimate ˇ D 1. I run Method 2 in 1000 replications of samples of size 100. I added a dummy for each O configuration of participation; in the table υijk is the coefficient for participation of i bidders of low type, j bidders of medium type and k bidders of high type. Results are shown in Table VI. I
Table V. Monte Carlo experiment: least squares v. maximum likelihood, first-price auctions, valuations uniformly distributed
True value O LS 3 O LS 1 O ML 3 O ML 1

Sample size D 50 Mean Variance Mean squared error Jarque–Bera p-value Sample size D 100 Mean Variance Mean squared error Jarque–Bera p-value Sample size D 200 Mean Variance Mean squared error Jarque–Bera p-value

2.9984 0.0043 0.0043 1.4223 0.5089 3.0011 0.0021004 0.0021017 9.4598 0.99117 2.9978 0.0010318 0.0010365 4.4603 0.89249

0.9999 0.0149 0.0149 1.2927 0.4760 0.999 0.0076885 0.0076895 4.1793 0.87627 1.0016 0.00363 0.0036326 1.3113 0.48089

2.9745 0.0022 0.0028 452.8485 1.0000 2.9872 0.00064033 0.00080441 5858.9 1 2.9959 0.00010281 0.00011948 2351.4 1

1.0067 0.0023 0.0023 194.9273 1.0000 1.0041 0.00065485 0.00067165 3324.6 1 0.99959 0.00011701 0.00011718 3014.3 1

22 We

used the uniform distribution to speed up the computation of equilibrium bids. J. Appl. Econ. 23: 925–948 (2008) DOI: 10.1002/jae

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Table VI. Monte Carlo experiment with an asymmetric model
O ˇ Mean Variance Jarque–Bera p-value 1.0025 0.0088 0.9957 0.3922 O υ111 10.7906 0.1992 0.3386 0.1557 O υ122 Mean Variance Jarque–Bera p-value 11.0366 0.1613 3.6535 0.8391 O υ112 11.8132 0.2603 6.6543 0.9641 O υ212 11.9127 0.2346 11.5586 0.9969 O υ121 11.4697 0.2069 5.1656 0.9244 O υ221 11.5863 0.1802 3.6440 0.8383 O υ211 12.2569 0.2412 14.4529 0.9993 O υ222 12.3214 0.2095 3.1769 0.7958

find that the method can readily estimate ˇ. The estimated patterns for the control coefficients does O O O not lend itself very easily to interpretation (for example, on average we obtain υ112 > υ121 < υ211 ).

APPENDIX B: PROOFS Proof of Lemma 1 Since ω k : n D k D n! 1!n k! vF v n k

1

Fv

k 1

dF v ,

nω k : n

1

n! vFn 1 k 1 F k 1 dF v k 1!n 1 k! n! D vFn 1 k 1 F k 1 [F C 1 k 1!n 1 k! n! vFn k 1 F k 1 dF v D k 1!n 1 k! n! vFn 1 k 1 F k dF v C k 1!n 1 k! D n k ω k : n C kω kC1:n

F ]dF v

so the recurrence relation holds.

Proof of Corollary 1 By induction, since it is immediate that with all ω k : n and a n C 1 , one can directly compute all remaining ω k : nC1 .

Proof of Lemma 2 The argument roughly follows Hoeffding (1953). I show that the distribution of ε kn ,n converges to the constant F 1 ˛ .
Copyright  2008 John Wiley & Sons, Ltd. J. Appl. Econ. 23: 925–948 (2008) DOI: 10.1002/jae

946

L. REZENDE

Given a quantile u, Pr[ε kn ,n < F

1

u ] can be written as u 1
0 1

t t

kn n kn

t t

dt dt

1
0

kn n kn

I must show that this goes to 0 for u < ˛ and to 1 for u > ˛. Take u < ˛. Fix v 2 u, ˛ . For a sufficiently high n, ˛n D 1 t˛n 1 t 1 ˛n is increasing for t < ˛n ; so u u

kn /n > v. The function

[t˛n 1
0 1

t t u v 1 ˛n n

] dt Ä ] dt v [t˛n 1
0 ˛n

t t

1 ˛n n

] dt Ä ] dt
1 ˛n 1 ˛n n

[t
0 u

˛n

1

1 ˛n n

[t˛n 1

1 ˛n n

[u˛n 1
0 ˛n v 1 ˛n n

] dt D ] dt ˛n

u

[v

˛n

1

1 ˛n n

u ˛n 1 v v˛n 1

u v

!0

The argument for u > ˛ is analogous, since the function t˛n 1 t > ˛n .

t

1 ˛n

is decreasing for

Proof of proposition 1 Suppose there was such distribution. Let a n D ω 2:n D c C bn for some constants c and b. It must necessarily be b ½ 0. If b D 0, by the mapping from Theorem 2 it is easy to verify that the distribution is degenerate. So b > 0. Now, successively apply the recursion formula to the case where k D 1. We obtain ω 1:n D D n n n n 1 ω 1:n
1

c C nb n 1 c C nb n 1 2 cC n 1 b CÐÐÐ

n 1 n

D nω 1:1 D n ω 1:1 b ! 1 n 1

cC n 1 b 1 ω 1:n 2 2 n 2 n n c C nb nn 1 n 1 n 1 1 C c nn 1 n 1 n 2 1 C 1 n 2 CÐÐÐ

C ÐÐÐ

as n ! 1, since the sum that multiplies b diverges (while the one that multiplies c does not). This contradicts the fact that ω 1:n should be increasing in n.
Copyright  2008 John Wiley & Sons, Ltd. J. Appl. Econ. 23: 925–948 (2008) DOI: 10.1002/jae

ECONOMETRICS OF AUCTIONS BY LEAST SQUARES

947

ACKNOWLEDGEMENTS

Parts of this research are based on chapters 4 and 5 of my PhD dissertation at Stanford University. I gratefully acknowledge the support of a John M. Olin Dissertation Fellowship and a Melvin and Joan Lane Stanford Graduate Fellowship. Thanks to Pat Bajari, Lanier Benkard, Tim Bresnahan, George Deltas, Phil Haile, Ali Horta¸ su, Thierry Magnac, Paul Milgrom, Harry Paarsch and two c anonymous referees for valuable suggestions. All errors are mine.

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Copyright  2008 John Wiley & Sons, Ltd.

J. Appl. Econ. 23: 925–948 (2008) DOI: 10.1002/jae