Identification and Estimation of Risk Aversion in First Price Auctions With Unobserved Auction Heterogeneity

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. Identification and Estimation of Risk Aversion in First Price Auctions With Unobserved Auction Heterogeneity Serafin J. Grundl and Yu Zhu 2015-089 Please cite this paper as: Grundl, Serafin J., and Yu Zhu (2015). “Identification and Estimation of Risk Aversion in First Price Auctions With Unobserved Auction Heterogeneity,” Finance and Economics DiscussionSeries2015-089. Washington: BoardofGovernorsoftheFederalReserveSystem, http://dx.doi.org/10.17016/FEDS.2015.089. NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

Identification and Estimation of Risk Aversion in First-Price Auctions with Unobserved Auction Heterogeneity Serafin Grundl and Yu Zhu ∗ September 29, 2015 Abstract We extent the point-identification result in Guerre, Perrigne, and Vuong (2009) to environments with one-dimensional unobserved auction heterogeneity. In addition, we also show a robustness result for the case where the exclusion restriction used for point identification is violated: We provide conditions to ensure that the primitives recovered under the violated exclusion restriction still bound the true primitives in this case. We proposeanewSieveMaximumLikelihoodEstimator,showitsconsistencyandillustrate its finite sample performance in a Monte Carlo experiment. We investigate the bias in risk aversion estimates if unobserved auction heterogeneity is ignored and explain why the sign of the bias depends on the correlation between the number of bidders and the unobserved auction heterogeneity. In an application to USFS timber auctions we find thatthebiddersareriskneutral, butwewouldrejectriskneutralitywithoutaccounting for unobserved auction heterogeneity. ∗We are very grateful to Amit Gandhi and Jack Porter for their advise and many helpful suggestions. We would also like to thank Andr´es Aradillas-L´opez, Bruce Hansen and Xiaoxia Shi for their helpful comments. Serafin Grundl: Federal Reserve Board of Governors, serafin.j.grundl@frb.gov. Yu Zhu: Department of Economics, University of Leicester, yz317@leicester.ac.uk. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the staff, by the Board of Governors, or by the Federal Reserve Banks. 1

1 Introduction Riskaversionplaysanimportantroleinauctiontheory. Riskaversionleadstomoreaggressive bidding in first-price auctions with independent private values whereas bidding in English auctions is not affected. Therefore first-price auctions generate higher revenues than English auctions if the bidders are risk averse (Holt (1980)).1 Matthews (1987) compares auction formats from the perspective of risk averse bidders and Maskin and Riley (1984) study the optimalauctionmechanismunderriskaversion. Infirstpriceauctionsriskaversionalsotends to reduce the optimal reserve price because aggressive bidding does not have to be induced with the help of a high reserve price (Riley and Samuelson (1981), Hu, Matthews, and Zou (2010)). Knowing bidders’ risk aversion is therefore crucial for auction design. Guerre, Perrigne, and Vuong (2009) showed that risk aversion can be nonparametrically identified from bid data in first-price auctions under an exclusion restriction, by exploiting variation in the number of bidders. This paper tries to bridge the gap between this identification result and applications to field data. We begin by showing identification in environments with one-dimensional unobserved auction heterogeneity, i.e. one of the auction characteristics is observed by the bidders but not by the econometrician. Identification proceeds in two steps. In the first step multiple bids from the same auction are used to identify the bid distribution conditional on the unobserved characteristic. This step builds on results of Krasnokutskaya (2011), Hu, McAdams, and Shum (2013) and d’Haultfoeuille and F´evrier (2010b), who apply techniques from the measurement error literature. Intuitively, the bid distributions conditional on the unobserved characteristic can be identified using the dependence among bids from the same auction created by the unobserved characteristic. Applying the techniques from the measurement error literature to first-price auctions with risk averse bidders creates a technical challenge, because it requires the highest bid to be strictly increasing in the unobserved characteristic. 1Thisresultholdsforagivennumberofriskaversebidders. Therevenuerankingispreservedintheentry modelofLevinandSmith(1994a)butSmithandLevin(1996)showthatitcanbereversedwithendogenous entry and decreasing absolute risk aversion. 2

We provide new comparative statics results for auctions with risk averse bidders to establish this monotonicity condition. In the second step we apply the results of Guerre, Perrigne, and Vuong (2009) to the bid distributions conditional on the unobserved characteristic to identify the primitives. As we condition on the unobserved characteristic we can exploit variation in the number of bidders for identification even if the entry decision depends on the unobserved characteristic. The exclusion restriction required for point identification is that the distribution of valuations conditional on the unobserved characteristic does not depend on the number of bidders. We also discuss the case where the exclusion restriction is violated such that the (conditional) valuation distribution in an auction with more bidders first-order stochastically dominates thevaluationdistributionwithfewerbidders. Weprovideaconditionforthebiddistributions thatguaranteesrobustnesswithrespecttothisviolationinthefollowingsense: Theprimitives recovered under the violated exclusion restriction still bound the true primitives in this case and risk neutrality remains testable. Next, we turn to estimation and inference. In light of the typical sample size available in applications we consider a semi-parametric specification with constant relative risk aversion and multiplicative unobserved auction heterogeneity. We propose a new Sieve Maximum Likelihood Estimator and show its consistency under low level conditions.2 Monte Carlo experiments show that the estimator performs well with sample sizes commonly found in applications. The Monte Carlo study also presents estimates if the unobserved characteristic is ignored, which has two opposing effects on risk aversion estimates. First, if auctions with a better unobservablecharacteristicattractmorebiddersthisincreasestheshiftofthe(unconditional) bid distributions as the number of bidders increases. Risk aversion tends to decrease the 2Derivingtheasymptoticdistributionoftheestimatorisdifficultduetothenon-regularlikelihoodfunction and the semi-parametric specification. Ackerberg, Chen, and Hahn (2012) show that for a regular likelihood function, treating the problem as parametric is numerically identical to using the asymptotic formula for semi-parametric estimation. Unfortunately this result does not apply here because the likelihood function is non-regular. Simulation results suggest however that treating the problem as parametric works very well in practice. 3

shift of the bid distribution as the number of bidders increases, because bids are close to valuations even without fierce competition. Therefore this effects leads us to underestimate risk aversion. Second, the unobserved characteristic increases the dispersion of bids. Risk aversion also increases the dispersion of bids because the bid function at the lower bound of the valuation distribution not affected by risk aversion while bidders with higher values bid more aggressively. Hence, this effect leads us to to overestimate risk aversion. Which of the two effects dominates, and therefore the bias of the risk aversion estimates, depends on how strongly the number of bidders is correlated with the unobserved characteristic. In an illustrative application we study US Forest Service timber auctions. We find that thebiddingfirmsareclosetoriskneutral, butwewouldrejectriskneutralitywithoutallowing for unobserved auction heterogeneity. This paper connects two separate strands of the structural auction literature - unobserved auction heterogeneity and risk averse bidders. Krasnokutskaya(2011)andKrasnokutskaya(2012)considersidentificationandestimation with separable unobserved auction heterogeneity in first-price auctions while Hu, McAdams, and Shum (2013) consider identification in the non-separable case. Several papers have documented unobserved auction heterogeneity in US Forest Service timber auctions (e.g. Aradillas-Lo´pez, Gandhi, and Quint (2013a), Aradillas-Lo´pez, Gandhi, and Quint (2013b), Roberts and Sweeting (2010), Roberts and Sweeting (2013) and Athey, Levin, and Seira (2011)). The empirical literature on risk aversion in first price auctions started with laboratory experiments where risk aversion has been proposed as an explanation of the overbidding puzzle.3 Bajari and Hortacsu (2005) apply structural auction methods to experimental data and conclude that the canonical auction model with risk averse bidders fits experimental data better than some other models which give up the assumption of Bayesian Nash Equilibrium. 3The overbidding puzzle refers to the common finding in laboratory experiments that bidders bid more aggressivelythanpredictedbytheriskneutralBayesianNashEquilibrium. SeeforexampleCox, Smith, and Walker (1988). For more references see the excellent surveys Kagel (1995) and Kagel and Levin (2010)). 4

Several papers found the bidders in US Forest Service timber auctions to be risk averse. Their results cannot be directly compared to this paper, because they rely on different restrictions for the identification of risk aversion.4 Kim (2015b) and Zincenko (2014) propose nonparametric estimators to implement the main identification result in Guerre, Perrigne, and Vuong (2009). As we are interested to applications to field data we also consider identification with unobserved auction heterogeneity and partial identification if the exclusion restriction is relaxed. In an extension Guerre, Perrigne, and Vuong (2009) also consider the case where an unobserved auction characteristic affectsthenumberofbiddersandthedistributionofvaluations. Theyprovideconditionssuch that the model can still be identified if an instrument is available, which affects the number of bidders but not the distribution of valuations. They consider two alternative conditions to achieve identification: Under the first condition there is a monotone mapping between the number of bidders and the unobserved characteristic. Under the second condition, there is a monotone mapping between the instrument and the unobserved characteristic. These monotonicity assumptions allow the econometrician to back out the unoberved characteristic in a first step and then proceed as if the unobserved characteristic is observed, to identify the distribution of valuations and the utility function. In a complementary paper to our’s Gentry, Li, and Lu (2015) also consider identification and estimation of risk aversion in first-price auctions. In contrast to this paper, they consider a model where the bidders do not know the number of entrants when they submit a bid. ThereforetheresultofGuerre,Perrigne,andVuong(2009)doesnolongerapplyintheirmodel and identification is more challenging. They show that a parametric restriction on the copula governing entry usually restores point identification, while a parametric restriction of the utility function leads to partial identification. Allowing for unobserved auction heterogeneity in their framework would be an interesting avenue for future research. 4ForexampleLuandPerrigne(2008)usevariationintheauctionformat,whileCampo,Guerre,Perrigne, and Vuong (2011) impose mild parametric restrictions to identify risk aversion. For risk aversion in timber auction see also Baldwin (1995) and Athey and Levin (2001). Campo (2012) finds evidence of risk aversion in construction procurement auctions. 5

This rest of the paper is organized as follows. Section 2 presents the identification results. In section 3, we propose a semi-parametric Sieve Maximum Likelihood estimator. Section 4 conducts Monte Carlo experiments to evaluate the finite sample performance of the estimator. The Monte Carlo study also illustrates that the bias of risk aversion estimates if the unobserved characteristic is ignored. We explain why the sign of the bias depends the correlation between the unobserved characteristic and the number of bidders. Section 5 is an application to USFS timber auctions. 2 Identification There are n ≥ 2 active bidders with independent private values. Their values v are independent draws from the distribution F (·|u,n) with a continuous density f (·|u,n) supported on [v(u),v(u)] where 0 ≤ v(u) < v(u) ≤ ∞. The econometrician does not observe the one-dimensional auction characteristic u which follows the distribution Fu(·|n). The bidders share a common utility function U with U(cid:48)(·) ≥ 0, U(cid:48)(cid:48)(·) ≤ 0 and U(cid:48)(cid:48) is continuous. The utility function is normalized such that U (0) = 0 and U (1) = 1.5 Define λ(·) = U (·)/U(cid:48)(·). The equilibrium bidding strategy s (·,u) is characterized by the following first-order condin tion ∂s(cid:48) (v,u) f (v|u,n) n = (n−1) λ(v −s (v,u)), n ∂v F (v|u,n) with the boundary condition s (v(u),u) = v(u). n Assumption 1. F (·|u,n) = F (·|u). Guerre, Perrigne, and Vuong (2009, Proposition 3) showed that if u is observed, F and U are point identified from bid data under this assumption. This assumption holds in many commonentrymodelsunderfairlygeneralconditionsasshowninAppendixB.1.6 InTheorem 5As we normalize the utility function such that U(1) = 1 we implicitely assume that max v − v∈[v(u),v(u)] s (v,u) ≥ 1. If this condition is violated identification of λ and F is not affected, but we would have to n choose a smaller point for the normalization to solve the differential equation λ(·)=U(·)/U(cid:48)(·) for U. 6Noticethat,likemostoftheliterature,weassumethatthebiddersknowhowmanyoftheirrivalsdecided toentertheauctionwhentheybid. Intuitively,Assumption1isnotveryrestrictiveinthiscase,becauseonce 6

3 we provide a robustness result for the case where the condition is violated. In applications to field data we have to confront the possibility that u is not observed. Previous work studying such environments assumes that bidders are risk neutral and focuses ontheidentificationofF (·|u)(Krasnokutskaya(2011)andHu, McAdams, andShum(2013)). The identification arguments exploit the fact that the data contain more than one bid for each auction. The unobserved characteristic creates dependence among bids from the same auction, which allows the researcher to separately identify the distribution of u and the bidders’ private information. We combine this strategy with Guerre, Perrigne, and Vuong (2009). The first result is an extension of Krasnokutskaya (2011) which considers cases where valuations consist of two independent and separable components. Theorem 1. Suppose that Assumption 1 holds and we observe at least two randomly selected bids from auctions with n ,n ≥ 2 bidders. Suppose one of the following conditions holds: 1 2 (1). F (v|u) = F∗(v −u) for all v and u, for some F∗ with density f∗. In addition Assumption 7(1) (Appendix A.1) holds. (2). F (v|u) = F∗(v/u) for all v and u, for some F∗with density f∗. Bidders have constant relative risk aversion (CRRA) with CRRA-coefficient σ ∈ [0,1). In addition Assumption 7(2) (Appendix A.1) holds. Normalize the lower bound of the support of f∗ to 1. Then U, F∗, Fu(·|n ) and Fu(·|n ) 1 2 are identified. One insight from this result is that there is an important distinction between additive and multiplicative auction heterogeneity if the bidders are risk averse. If the unobserved characteristic enters valuations additively, it also enters the equilibrium bid function additively - regardless of the utility function. If the unobserved characteristic enters valuations we condition on all the variables observed by the potential bidders - including the unobserved characteristic and if it varies also the number of potential bidders - variation in the number of entrants is“exogenous”. See Gentry, Li, and Lu (2015) for identification of risk aversion in first-price auctions if the bidders do not know how many of their rivals decided to enter the auction when they bid. 7

multiplicatively and the bidders have constant relative risk aversion, it also enters the bid function multiplicatively. If the utility function is not of the CRRA form however, the bidding strategy is not separable in u and the deconvolution techniques in Kotlarski (1967) can therefore no longer be applied.7 The result requires a location normalization. To see why consider the additive case 1(1). If F∗ is shifted to the right by 1 while Fu(·|n ) and Fu(·|n ) are shifted to the left by 1, 1 2 the distribution of v and therefore the bid data remains unchanged. Hence this shifted set of primitives is observationally equivalent to the original set of primitives. An analogous argument can be made for the multiplicative case in Theorem 1(1). Besides allowing for risk aversion, Theorem 1 also generalizes Krasnokutskaya (2011) to accommodate an unbounded unobserved characteristic and unbounded private values. This is achieved by building on an extension of Kotlarski (1967) by Evdokimov and White (2012). If the unobserved characteristic does not enter in a separable way establishing identification is more involved. Hu, McAdams, and Shum (2013) show how to achieve identification if bidders are risk neutral and u takes on a finite number different values under the following monotonicity restriction on F. Assumption 2. F (v|u ,n) ≤ F (v|u ,n) for all v, u > u and n and there exists v such 1 2 1 2 that F (v|u ,n) < F (v|u ,n). 1 2 Proposition1. SupposeAssumption2holdsandv¯(u) < ∞foreveryu, thens (v¯(u ),u ) > n 1 1 s (v¯(u ),u ). n 2 2 This result says that the highest bid in an auction with unobserved characteristic u is 1 strictly higher than with u . Hu, McAdams, and Shum (2013) establish this property by 2 exploiting the closed form of bidding strategy if the bidders are risk neutral. If bidders are risk averse the bidding strategy does typically not have a closed form and establishing strict monotonicity of the highest bid is therefore more involved. 8 7This case will be covered in Theorem 2. 8Tothebestofourknowledgethisisanewcomparativestaticresultforauctionswithriskaversebidders. 8

Theorem 2. Suppose that Assumptions 1 and 2 hold and we observe three randomly selected bids from each auction with n ,n ≥ 3 bidders. Then U and F are identified if one of the 1 2 following two conditions is satisfied: (1). Discrete u: The support of u is 1,2,...K with K < ∞ for n and n . 1 2 (2). Continuous u: (a) [u(n ),u(n )]∩[u(n ),u(n )] (cid:54)= ∅. 1 1 2 2 (b) v(u) is strictly increasing in u. (c) Assumption 8 (Appendix A.1) holds. (d) u = v(u). Here [u(n),u(n)] is the support of the unobserved characteristic in an n bidder auction. Theorem 2(1) extends the result of Hu, McAdams, and Shum (2013) for discrete u. Theorem 2(2) builds on d’Haultfoeuille and F´evrier (2010a) and applies to cases where u is continuous. The condition for Theorem 2(1) can be broken up into three parts. First, the support of u has finitely many points. Second, the support is the same for n and n . Third, the 1 2 support is normalized to 1,2,...K. Next we turn to the condition for Theorem 2(2). First, we require that for some u we observe n and n bidder auctions - otherwise we could not 1 2 exploit variation in the number of bidders conditional on u for identification. Second, we assume that v(u) is strictly increasing in u. Together with Proposition 1 this implies that the lowest and the highest bid are both strictly increasing in u. The third assumption is a smoothness condition. The fourth assumption is a normalization of u.9 ItisimportantthatTheorems1and2allowthedistributionofutodependonthenumber of bidders. Intuitively, if the bidders observe u before they make their entry decision, then To show identification we only need to establish strict monotonicity of the highest bid in u, but the proof in Appendix A.3 shows that the whole bid distribution is (weakly) shifted to the right as u increases. 9To see why the the normalizations of u are required in Theorem 2 consider u=h(u) for some increasing (cid:101) function h and F(cid:101)(·|u,·)=F (cid:0) ·|h−1(u),· (cid:1) , which leads to the same distribution of valuations and bids as the true primitives. 9

auctions with a better unobserved characteristic should attract more bidders. In Appendix B.3 we confirm this intution for the case we consider in our application where the unobserved characteristic enters valuations multiplicatively and bidders have constant relative risk aversion. Formally, we show that the distribution of the unobserved characteristic is increasing in n in the sense of first-order stochastic dominance. Next, we relax Assumption 1 such that valuations are increasing in n in the sense of first-order stochastic dominance. Assumption 3. F (v|u,n ) ≥ F (v|u,n ) for all v, u and n < n . 1 2 1 2 Define 1 α R (α,u) = , i n −1g(b (α,u)|u,n ) i ni i where i = 1,2, α ∈ [0,1], g(·|u,n) is the bid density and b (α,u) is the α-th quantile of the n bid distribution. Condition 1. Let n < n . There is u∗ such that 1 2 (1). b (0,u∗) = b (0,u∗). n1 n2 (2). R (α,u∗) > R (α,u∗) for all α > 0 . 1 2 Thisisnotanassumptiononprimitivesbutaconditionforthebiddistribution. Therefore, it can be checked once the bid distribution conditional on u has been recovered. The first part of this condition states that the lowest bid in n and n bidder auctions is the same. To 1 2 interpret the second part note that the first-order condition for an i bidder auction can be written as R (α,u) = λ(v(α,u)−b (α,u)). Therefore, the condition says that bid shading i ni is larger at the αth quantile in an n bidder auction than in the more competitive n bidder 1 2 auction. Let λ(cid:101) with λ(cid:101)(0) = 0 be consistent with the bid distributions given u∗ if we (incor- (cid:18) (cid:19) rectly) impose Assumption 1 for n and n . Let x = λ(cid:101)−1 max R (α,u∗) . Let U(cid:101)(x) = 1 2 1 ´ ´ α∈[0,1] (cid:16) (cid:16) (cid:17)(cid:17) (cid:16) (cid:16) (cid:17)(cid:17) 1 x exp log λ(cid:101)(t) dt for x ∈ [0,1] and U(cid:101)(x) = exp − log λ(cid:101)(t) dt for x ∈ [1,x). x 1 10

Theorem 3. Suppose that that u is observed and that Assumption 3 and Condition 1 hold, then (1). λ(x) ≥ λ(cid:101)(x) for x ∈ [0,x), U (x) ≥ U(cid:101)(x) for x ∈ [0,1], and U (x) ≤ U(cid:101)(x) for x ∈ [1,x). (2). b (α,u∗) ≤ F−1(α|u∗,n ) ≤ λ(cid:101)−1(R (α,u∗))+b (α,u∗) for i = 1,2. ni i i ni To shorten the statement of the result it is assumed that u is observed, but the extension to unobserved u along the lines of Theorems 1 and 2 is straightforward. The first part of the result shows that λ(cid:101) bounds the true λ from below. By integrating up λ(·) = U (·)/U(cid:48)(·) with U (1) = 1, this bound can be translated into a bound on U. The second part shows that the valuations are bounded from below by the bids and from above by the inverse bid function consistent with λ(cid:101). This is a robustness result. It provides conditions to ensure that the primitives recovered under Assumption 1 remain meaningful as bounds even if the assumption is violated. For ˆ example suppose we estimate λ under Assumption 1 and conclude that the bidders are risk ˆ averse because λ(x) > x for some x. This conclusion remains valid if Assumption 1 is violated, but Assumption 3 and Condition 1 are satisfied. The primitives can be partially identified under Assumption 3, even if Condition 1 does not hold. In this case the bounds do however no longer coincide with the primitives recovered under Assumption 1. 3 Estimation In light of the typical sample size in applications we consider a semi-parametric specification with constant relative risk aversion and auction characteristics (observed and unobserved), which enter valuations multiplicatively. A bidder’s valuation is v = v∗uexp[log(X)γ]. The bidder’s private value v∗ follows the distribution F∗ with density f∗.10 To simplify the notation, let Fu denote the distribution of the unobservable characteristic Fu(·|u) and let n 10Notice that this notation implicitly imposes Assumption 1. 11

fu be its density. The private values v∗ and the unobserved characteristic u are independent n of each other.11 The p-dimensional vector X contains observable auction characteristics. We assume that X is independent of both v∗and u. Bidders share a CRRA utility function with coefficient σ. Following Proposition 1 in Krasnokutskaya (2011) it can be shown that uexp[log(X)γ] enters the bidding strategy multiplicatively. The data contain L auctions. Let L denote the number of auctions with n ≥ 2 acn tive bidders. Let N be the set of n such that L > 0. For the (cid:96)-th auction, we observe n Z = (b ,X ,n ). Here b is the vector of all bids, X is the vector of observed auction (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) (cid:96) characteristics and n is the number of active bidders. We also denote the i-th element of b (cid:96) (cid:96) (cid:0) (cid:1) as b . The primitives of the model are σ,γ,f∗,{fu} . This specification satisfies the i,(cid:96) n n∈N assumptions of Theorem 1(2) if N has at least two elements. To the best of our knowledge, estimation of bidders’ risk aversion in first-price auctions with unobserved auction heterogeneity has not been discussed in the literature. In this paper, we develop a Sieve Maximum Likelihood Estimator (Sieve MLE) based on the joint densities of all the bids from the same auction. We propose a computationally feasible method to compute the joint bid densities. We also show that the estimator is consistent.12 It is worth pointing out that there are several other methods in the literature that can be generalized to handle this estimation problem. One possibility is to combine the method suggested by Krasnokutskaya (2011) with Bajari and Hortacsu (2005) to form a two-step estimator.13 Another candidate is the Simulated Method of Moments (SMM) proposed by 11It is worth noting that this assumption is imposed on bidders who decided to enter the auction. In AppendixB.2,weimposethesameassumptiononpotentialbiddersandaskforwhichentrymodelsseparability andindependenceofuandv∗ carriesovertoentrants. Weshowthatifthepotentialbiddersobserveasignal for v∗ or u independence carries over to entrants, but not if they observe both. 12Formally deriving the asymptotic distribution of the estimator is beyond the scope of this paper. The major difficulty is that our likelihood is non-regular because the support of the bid densities depends on the parameters. Therefore, the results from Ackerberg, Chen, and Hahn (2012), which are valid for regular models,donotdirectlyapplyhere. TheMonteCarloexperimentsshowthattreatingthemodelasparametric and using the asymptotic results from Smith (1985) performs well in practice. It is worth noting that here the bid density does not jump at the boundary of its support, so the results in Donald and Paarsch (1993), Chernozhukov and Hong (2004) and Hirano and Porter (2003) do not apply. 13Inthefirststep,useKrasnokutskaya(2011)toseparateouttheeffectofunobservedauctionheterogeneity. In the second-step, use Bajari and Hortacsu (2005) to estimate the CRRA coefficient and the distribution of private values. 12

Bierens and Song (2011) which is extended to the case with unobserved auction heterogeneity by Grundl and Zhu (2015). A third possibility is to extend a Bayesian estimator proposed by Kim (2015a). The Sieve MLE method is our preferred choice for the following reasons. First, compared to the first two alternatives, MLE is more efficient. Second, the Bayesian method considered in Kim (2015a) does not have efficiency advantages in our problem and at the same time involves choosing several tuning parameters for discretization and priors.14 3.1 Parameter Space The support of the densities of unobserved heterogeneity and private values are [µ,u+µ] and [1,v∗ +1] with u > 0, v∗ > 0 known.15 Here u and v∗ are the length of the support which may be infinity. µ is the unknown lower bound of the support of u which has to be estimated. It lies in some known closed interval I ⊂ R with lower bound greater than 0. Without loss of generality, the lower bound of v∗ is normalized to be 1. Instead of working directly with primitives, we transform them into the parameter θ = (cid:0) (cid:1) σ,γ,µ,ψ∗,{ψu} where µ is the lower bound of unobserved heterogeneity and the ψs n n∈N are functions supported on [0,1] which take on values no less than −1 and integrate up to 0. f∗,{fu} can be expressed in terms of ψ functions. To do so, first choose some base n n∈N density functions hu and h∗ supported on [0,u] and[1,v∗ +1], respectively. Let H∗ and Hu be their corresponding distributions. With some abuse of notation, let the densities given θ be f∗(x;θ) = [Tψ∗](H∗(x))h∗(x) and fu(x;θ) = [Tψu](Hu(x))hu(x) where n n [1+ψ(x)]2 [Tψ](x) = ´ . 1+ ψ(x)2dx It is easy to show that for any primitives f∗ and fu, we can find θ such that f∗(·) = n 14UnlikeinKim(2015a),thejointbiddensitiesarecontinuousduetotheunobservedauctionheterogeneity. In a parametric model where the densities are continuous at the boundary of the support, the maximum likelihood estimator is efficient even if the support of the densities depends on the parameters, see Smith (1985). Therefore, we choose MLE to avoid picking several additional tuning parameters which is required in Kim (2015a). 15Alternatively, we could assume that u and v∗ are unknown but finite and treat them as parameters. 13

f∗(·;θ) and fu(·) = fu(·−µ;θ). This transformation allows us to work only with functions n n supported on [0,1].16 (cid:16) (cid:17) (cid:8) (cid:9) Let θ = σ ,γ ,µ ,ψ∗, ψu be the true parameter under h∗ and hu which lives 0 0 0 0 0 0,n n∈N in a known space Θ = Σ×Kp ×I ×A . Σ = [0,1−η], K ⊂ R is a compact set and I is a closed interval with lower bound greater than 0. A = Ψ(B)n+1 where  ˆ ˆ ˆ   (cid:88)  Ψ(B) = ψ ∈ Cq[0,1] : ψ(x)dx = 0, ψ2(x)dx < ∞,ψ+1 ≥ η, ψ(k)(x)2dx ≤ B ,   0≤k≤q where η is some arbitrarily small positive number. B is a known positive constant and q is a positive integer. Notice that Ψ(B) only contains functions that are smooth enough to guarantee that Ψ(B) is compact under the sup-norm. Therefore we avoid inconsistency problem due to a ill-posed inverse problem.17 We further define α = (cid:0) ψ∗,{ψu} (cid:1) , hence θ = (σ,γ,µ,α).With some abuse of notation, n n∈N let (cid:107)ψ(cid:107) = sup |ψ(x)| and ∞ x∈[0,1] (cid:26) (cid:27) (cid:107)α(cid:107) = max (cid:13) (cid:13)ψ∗(cid:13) (cid:13) ,max (cid:8)(cid:13) (cid:13)ψu(cid:13) (cid:13) (cid:9) , ∞ ∞ n∈N n ∞ where(cid:107)·(cid:107) is the standard Euclidean norm. One can show that Θ is a compact space under E (cid:107)·(cid:107) where s (cid:107)θ −θ (cid:107) = max{|σ −σ |,|µ −µ |,(cid:107)γ −γ (cid:107) ,(cid:107)α −α (cid:107) }. 1 2 s 1 2 1 2 1 2 E 1 2 ∞ 3.2 Sieve Maximum Likelihood Estimator One difficulty in constructing the Sieve MLE is to compute the joint bid densities. These possibly high dimensional objects are complicated functions of θ and have no closed forms. We compute the bid densities numerically by exploiting the separable form of the bidding 16This transformation follows Bierens and Song (2012a). 17This regularization follows Santos (2012). 14

function. Let g (·;θ) be the joint density of bids given θ in n-bidder auctions if logX = 0. n ˆ n 1 (cid:89) g (b;θ) = g∗(b /u;θ)fu(u−µ;θ)du. (1) n un n i n i=1 Here g∗ is the marginal bid distribution in an auction with n-bidders whose value density n is f∗(·;θ). g∗(b∗;θ) can be obtained by exploiting the first-order condition of the bidding n strategy. Notice that     1−σF∗(s∗ n −1(b∗;θ);θ) if 1 < b∗ ≤ s∗ (v¯∗;θ) g∗(b∗;θ) = n−1 s∗ n −1(b∗;θ)−b∗ n n   0 Otherwise where s∗−1(·;θ) is the inverse of the bidding strategy n ˆ v(cid:20) F∗(x;θ) (cid:21)n 1− − σ 1 s∗ (v;θ) = v − dx. n F∗(v;θ) 1 The likelihood function can be written as (cid:88) l(Z ;θ) = l(Z ;(σ,γ,µ,α)) = 1 logg (exp(logb −logX γ);θ). (cid:96) (cid:96) {n =n} n (cid:96) (cid:96) (cid:96) n∈N The Sieve Maximum Likelihood Estimator is defined as 1 (cid:88) θ(cid:98) = arg max l(Z ;θ). (2) L (cid:96) θ∈Θ L kL Θ = Σ × Kp × I × A is the sieve space where A is a sequence of finite dimensional kL kL kL spaces that grows with the sample size. The estimator of the CRRA coefficient σ is the first (cid:98)L element of θ(cid:98) . Let E be the expectation under the true primitives. L 0 Assumption 4. (1). hu and h∗ are bounded and strictly bigger than 0 in the interior of their support and they have bounded continuous derivatives. 15

(2). lim h∗(v)/(v −1)(cid:15) = C as v ↓ 1 for some (cid:15) ≥ 0 and C > 0. v↓1 (3). limsup h∗(v)v2+δ < C and limsup hu(v)v2+δ < C for some C,δ > 0. v→∞ v→∞ (4). Under h∗ and hu, θ lives in the interior of Θ. 0 Assumption 5. The sieve space satisfies (1). {A }∞ is an increasing sequence of closed subsets of A. kL L=1 (2). sup (cid:107)α−A(cid:107) = o(1). α∈A kL ∞ (cid:2) (cid:3) Assumption 6. E logXT logX has eigenvalues bounded away from 0 and ∞. 0 Assumption 4(1)-(3) are requirements for the choice of h∗ and hu. Many commonly used density functions satisfy these requirements. Assumption 4(4) rules out the possibility that θ is a boundary point of Θ. It implies that the densities in the primitives are their cor- 0 responding base densities multiplied by functions bounded from above and bounded away from 0.18 Under this assumption, identification is guaranteed by Theorem 1 if there is no co-variate. Assumption 5(1) requires that the sieve space is closed and increasing so that the maximization problem in (2) is well-defined. Assumption (5)(2) requires that A apkL proximates A well enough. In Assumption 6 XT is the transpose of X. This assumption guarantees γ is identified. 0 p Proposition 2 (Consistency). If Assumptions 4, 5 and 6 hold, θ(cid:98) −→ θ as L → ∞ under L 0 p (cid:107)·(cid:107) . In particular, σ −→ σ . s (cid:98)L 0 The proof is based on Theorem 5.14 in van der Vaart and Wellner (2000) and generalizes Wald’sconsistencyprooftotheSieveMLE.Thecomplicationinthiscaseisthattheexpected log likelihood function can take on the value −∞ for some θ. Bierens (2014) considers a similar case but he requires the parameters at which the expected log likelihood is greater 18Therefore, we rule out densities with unconnected support, unbounded first moment and unbounded values. 16

than −∞ to be dense in the parameter space. One can show that in the case considered here the set of θ such that E l(Z ,θ) = −∞ has interior points. It is worth noting that 0 (cid:96) Assumptions 4, 5 and 6 are low level conditions. A key step to prove consistency is to show that under these low level conditions, the likelihood function and the sieve spaces satisfy certain regularity conditions. In particular, we need to show that l(Z;θ) is upper semicontinuous in θ, Z-a.e. and that there exists θ ∈ Θ such that (cid:107)θ −θ (cid:107) → 0 and 0,kL kL 0,kL 0 s E l(Z ,θ ) → E l(Z ,θ ). Lemmas that establish these regularity conditions are collected 0 (cid:96) 0,kL 0 (cid:96) 0 in Appendix D. 4 Monte Carlo Experiments 4.1 Setup Each generated sample has 900 auctions and the number of bidders n ranges from 2 to 5.19 We consider three different data generating processes (DGPs). In all DGPs v = v∗uXγ0 with logX i ∼ id N (0,1)andγ = 0.9. Theunobservedcharacteristicisdrawnfromaχ2 distribution. 0 In DGP 1, there is no selection on u and the χ2 parameter is 2 for all n. In DGP 2, there is weak selection on u and the χ2 parameter increases from 2 for n = 2 to 2.6 for n = 5. In DGP 3 there is strong selection on u and the χ2 parameter increases from 2 for n = 2 to 6.5 for n = 5. In all DGPs, bidders’ private values v∗ are drawn from a χ2-distribution with parameter 3. We consider the CRRA coefficients σ = 0, 0.1, 0.2, 0.3, to assess how well 0 the estimation method can distinguish risk neutrality and moderate levels of risk aversion. We repeat the Monte Carlo experiment 1000 times. 4.2 Estimators Results for two estimators are reported. First, the Sieve MLE estimator proposed in section 3.2. H∗ and Hu are both exponential with parameter 8. ψ∗ and ψ∗ are both 4-th n 19The shares are 36% for n=2, 27% for n=3, 21% for n=4 and 16% for n=5. 17

order Legendre polynomials. We compute bidding strategies at 3000 points and interpolate linearly.20 Second, as a benchmark, we also estimate the CRRA coefficient without taking unobserved heterogeneity into account, following the method used in Bajari and Hortacsu (2005) (BH). This estimator is computationlly light and therefore a natural choice for a specification without unobserved auction heterogeneity. First, we estimate the following equation by OLS: logb = c+γlogX +(cid:15) i,(cid:96) (cid:96) i,(cid:96) Let γˆ be the OLS estimate. Then we construct the residual bids ˆ b∗ = exp(b −γˆlogX ). i,(cid:96) i,(cid:96) (cid:96) Next we estimate the following equation by OLS:   q q ˆ b∗ n1 (q)− ˆ b∗ n2 (q) = (1−σ) gˆ (cid:16) ˆ b∗ (q) i (cid:17) (n −1) − gˆ (cid:16) ˆ b∗ (q) i (cid:17) (n −1)  (3) n2 n2 2 n1 n1 1 Here q ∈ [0,1] and ˆ b∗ (q) is the q-th quantile in the empirical distribution of ˆ b∗ given n = n n i,(cid:96) (cid:96) (cid:16) (cid:17) and gˆ ˆ b∗ (q) is the corresponding density. A Gaussian kernel with the rule-of-thumb n n bandwidth is used to estimate gˆ . Equation 3 is estimated at 100 equally spaced quantiles n ranging from 0.25 to 0.75.21 We restrict the estimates to be between 0 and 1. We report results for n = 2 and n = 4.22 1 2 4.3 Results The discussion focuses on the results for the CRRA coefficient shown in Table 1. 23 20The grid points are chosen such that the grid is finer for low values, because the bidding strategy there can be very nonlinear. 21Toavoidboundaryeffectsweexcludequantilescloseto0and1. Weexperimentedwithdifferentquantiles ranges and found similar results (available upon request). 22The two-step estimator does not allow us to combine more than two n in an efficient manner. Results for other pairs of n are similar (available upon request). 23The Online Appendix shows the results for the value distribution and the distribution of the unobserved characteristicifunobservedheterogeneityistakenintoaccount(SieveMLE).Thesefindingssuggestthatthe Sieve MLE estimator is might be preferable to two-step estimation procedures (Krasnokutskaya (2011)) or Simulated Method of Moments estimators (Bierens and Song (2012b) and Grundl and Zhu (2015)) even if 18

First, consider the results if unobserved auction heterogeneity is taken into account using the Sieve MLE estimator shown in the upper half of Table 1. The estimator works well for all three DGPs. The bias is very small (at most 0.014) if σ (cid:54)= 0. , but if the parameter is 0 on the boundary of the parameter space (σ = 0) it is somewhat larger (up to 0.048). The 0 standard deviation is at most 0.102.24 Now consider the result if we ignore unobserved heterogeneity, using the two-step BH estimator shown in the lower half of Table 1. Interestingly, the sign of the bias depends on the DGP.25 The CRRA coefficient is significantly over-estimated under DGPs 1 (no selection) and 2 (moderate selection), but under-estimated under DGP 3 (strong selection). Section 4.4 provides some intution to understand why the sign of the bias depends on the correlation between the number of bidders and the unobserved characteristic. We also test risk neutrality using the sieve MLE estimator H : σ = 0, H : σ > 0. To 0 0 1 0 constructthetestwetreatthemodelasparametricandusetheasymptoticdistributionofthe estimator.26 Notice that under the null hypothesis, σ is on the boundary of the parameter 0 space. Following the insight from Andrews (1999), σˆ is asymptotically truncated normal. Therefore it is still valid for the one sided test to reject the null hypothesis if σˆ divided by the standard error exceeds the corresponding quantiles of a standard normal random variable. Table 2 shows the results for testing risk neutrality. We consider significance levels of 5% and 10%. The test has good size control. For all three DGPs the rejection probability is close to the significance level if σ = 0. The test also performs well in terms of power. The 0 rejection probability for a 5% significance level increases from about 20% if σ = 0.1, to about 0 55% if σ = 0.2 and about 85% σ = 0.3. In light of the sample size and the flexibility of the 0 0 model, it is not surprising that it is difficult to distinguish σ = 0.1 from risk neutrality. 0 the researcher assumes that the bidders are risk neutral. 24We found that the standard error of the estimator is about 60% smaller than for a Simulated Method of Moments Estimator (Bierens and Song (2012b) and Grundl and Zhu (2015)). 25As expected, the bias gets smaller as the variance of the unobserved characteristic is reduced. These Monte Carlo results are available upon request. 26To estimate the asymptotic distribution, we need the joint bid densities to vanish smoothly at the boundaryof theirsupports. Hence, werequire that atleastthatthe fu vanish smoothlyattheirboundaries. n 19

σ = 0 σ = 0.1 σ = 0.2 σ = 0.3 0 0 0 0 Allowing for Unobserved Heterogeneity Mean 0.046 0.114 0.202 0.292 DGP 1: No Selection Std 0.069 0.086 0.102 0.094 Mean 0.041 0.108 0.188 0.284 DGP 2: Weak Selection Std 0.063 0.090 0.096 0.097 Mean 0.048 0.109 0.198 0.288 DGP 3: Strong Selection Std 0.074 0.093 0.105 0.102 Ignoring Unobserved Heterogeneity Mean 0.698 0.714 0.737 0.754 DGP 1: No Selection Std 0.205 0.160 0.146 0.138 Mean 0.540 0.554 0.578 0.606 DGP 2: Weak Selection Std 0.232 0.193 0.174 0.156 Mean 0.019 0.007 0.001 0.000 DGP 3: Strong Selection Std 0.134 0.083 0.032 0.000 Table 1: This table shows results of the Monte Carlo study for two estimators of the CRRA coefficient σ. The upper half of the table shows results if unobserved auction heterogeneity is taken into account using the Sieve MLE described in section 3. The lower half of the table shows results if unobserved heterogeneity is ignored using the two-step estimator proposed by Bajari and Hortacsu (2005). 20

Sig. Level σ = 0 σ = 0.1 σ = 0.2 σ = 0.3 0 0 0 0 DGP 1: No Selection 10 8.8 34.3 72.5 93.8 5 5.7 24.4 57.6 88.2 DGP 2: Weak Selection 10 7.6 27.7 69.5 91.9 5 4.9 19.8 53.1 85.9 DGP 3: Strong Selection 10 10.1 30.1 71.7 92.2 5 7.1 21.2 58.3 84.7 Table 2: This table shows the probability (in %) that risk neutrality (σ = 0) is rejected if 0 the unobserved characteristic is taken into account (Sieve MLE). 4.4 Understanding the Bias if Unobserved Heterogeneity is Ignored Here we provide some intution for the bias in risk aversion estimates if unobserve auction heterogeneity is ignored. Figure I(a) shows bid functions of risk neutral and risk averse bidders in two and four bidder auctions. Private values are on the horizontal axis and the corresponding bids on the vertical axis. The solid blue line and the solid red line depict a risk neutral bidder’s strategies in two and four-bidder auctions, respectively. The dashed lines depict a risk averse bidder’s strategies. Figure I(b) shows the corresponding bid distributions. Consider risk neutral bidders first. Their bid shading depends only on the distribution of valuations. Intuitively, the bidders shade their bids more if the values are more dispersed and the bidders have more private information and thereby more market power. If the number of competitors increases, market power declines and the bidders shade their bids less. This shift in the bid function is smaller if the values are are not very dispersed, because then the bids are close to values even for a small number of competitors. Hence, the bid distribution tends 21

10 8 6 4 2 0 0 2 4 6 8 10 v b 1 0.8 0.6 0.4 0.2 0 0 2 4 6 b (a) Bid functions )b(G (b) Bid distributions Figure I: This graph illustrates how risk aversion is identified by variation in the number of bidders. The left panel shows bid functions and the right panel the corresponding bid distributions. Solidlinesdepictriskneutralbiddersanddashedlinesshowriskaversebidders. Blue lines show two bidder auctions and red lines show four bidder auctions. to respond more to changes in n if the values (and therefore the bids) are more dispersed. Now consider risk averse bidders who bid more aggressively, which affects how much the bid distribution responds to changes in n and the dispersion of bids. Risk averse bidders respond less to changes in n because the bids are close to values even for a small number of competitors. Second, the dispersion of their bids is larger, because risk aversion has no effect for bidders at the lower bound of the valuation distribution but increases the bids of bidders with higher values. Therefore we conclude that the bidders are risk averse if the bid distribution does not respond much to increases in n relative to the dispersion of the bid distributions. If unobserved auction heterogeneity is ignored the (unconditional) bid distributions appear very dispersed, as variation in bids due to the unobservable characteristic is attributed to bidders’ private information. In addition, if auctions with a higher unobserved characteristic attract more bidders, this increases the shift of the (unconditional) bid distribution as n increases. The first effect increases the dispersion of bids and therefore leads to over- 22

estimation of risk aversion. The second effect increases the shift of the bid distribution as n increases and therefore leads to under-estimation of risk aversion. Which of these two effects dominates, and therefore the sign of the bias, depends on how strongly the number of bidders is correlated with the unobserved characteristic. 5 Empirical Application 5.1 Data Description We estimate the risk aversion of bidders in USFS timber auctions.27 The data can be downloaded from Phil Haile’s website.28 Lu and Perrigne (2008) and Campo, Guerre, Perrigne, and Vuong (2011) found the bidding firms to be risk averse.29 Other work documented unobserved heterogeneity in these auctions (e.g. Aradillas-Lo´pez, Gandhi, and Quint (2013a), Aradillas-Lo´pez, Gandhi, and Quint (2013b), Roberts and Sweeting (2010), Roberts and Sweeting (2013) and Athey, Levin, and Seira (2011)). FollowingHaileandTamer(2003), weconstructasub-sampleofscaledsaleswithcontract lengthoflessthanoneyearbetween1982and1990, forwhichtheassumptionofprivatevalues is plausible.30 Geographically, we focus on timber tracts from the Southern Region, ranging from Texas and Oklahoma to Florida and Virginia, where most of the first-price auctions take place. To limit the number of parameters in the distributions of the unobserved characteristic, 27Baldwin, Marshall, and Richard (1997, Appendix A) contains a detailed description of the auction procedure and background on the timber industry. 28http://www.econ.yale.edu/˜pah29/timber/timber.htm 29The findings in these papers cannot be directly compared to the findings in this paper, because they do not rely on variation in the number of bidders for the identification of risk aversion. Lu and Perrigne (2008) use variation in the auction format, while Campo, Guerre, Perrigne, and Vuong (2011) impose mild parametric restrictions. For risk aversion in timber auctions see also Baldwin (1995) and Athey and Levin (2001). 30Inscaledsales,bidderspayonlyforthetimberthatisactuallyharvested;thisinsuresthebiddersagainst the risk of overestimating the volume of timber and reduces the common value component in the valuations. Shorttermcontractswithacontractlengthoflessthanoneyearlimitresaleopportunitiesandtherebyreduce the common value component generated by the resale market. In 1981 the Forest Service introduced new policies designed to limit subcontracting and speculative bidding (Haile (2001)). Therefore, only auctions after 1981 are included. The data does not contain sales after 1990 for this region. 23

we further restrict the sample to auctions with 2 to 5 bidders. Intuitively, auctions with few competiors contain most information about risk preferences. As the number of competitors increases the effect of risk aversion on bids becomes small, because competition drives bids close to the values even for risk neutral bidders. To reduce the influence of the extreme bids, we also discard 8 auctions with bids more than 8 times the appraisal value.31 The final sample includes 370 2-bidder, 263 3-bidder, 172 4-bidder, and 105 5-bidder auctions. OurestimatesconditiononlyontheappraisalvalueprovidedbytheUSForestService,but not on other observed timber tract characteristics such as the timber volume, acreage or the species composition. As the appraisal value is meant to capture all the relevant information it can plausibly be treated as a sufficient statistic for the various timber tract characteristics. If some observed characteristics contain important information, which is not captured by the appraisal value, this affects only the precision of the estimator but not its consistency, as we allow for unobserved auction heterogeneity. 5.2 Results and Discussion The point estimate for the CRRA coefficient is 0.0018. The p-value for testing risk neutrality is 0.4914 and the 95% confidence interval for σ is [0,0.163]. Hence, we reject high levels of 0 risk aversion. For comparison table 3 shows results if risk aversion is ignored using the estimator in Bajari and Hortacsu (2005) as described in section 4. We report results for different pairs of auction sizes. To assess the robustness of the results, we report estimates based on three choices of quantiles. The bandwidth for the bid density estimators are chosen to be std(b)L−1/4.32 n The point estimates for the CRRA coefficient range from 0.547 to 0.708. The estimated confidence intervals do not cover any values below 0.324. Hence we find that the bidders are close to risk neutral if we allow for an unobserved 31Theremainingbidsarealllessthan4timestheappraisalvalue. Thereforewebelievethatthebidsabove 8 times the appraisal value can plausibly be considered outliers. 32The results are robust to different bandwidth choices we tried. 24

auction characteristic, but reject risk neutrality in a specification without unobserved auction heterogeneity. This pattern is consistent with a low correlation between the unobserved characteristic and the number of bidders as explained in section 4.4. Indeed we find that the distribution of the unobserved characteristic for different numbers of bidders is fairly similar. A possible explanation is that the unobserved characteristic is observed by the bidder only after they decided to enter the auction. For example, characteristics that are only observable for entrants who typically cruise the auctioned tract, but not for potential bidders. We follow most of the structural auction literature in assuming that the bidders know the number of their opponents, who also decided to enter, when they submit their bid.33 Intuitively, a violation of this assumption would bias our risk aversion estimates upwards. To see this consider the case where the number of potential bidders is the same for all auctions.34 In this case the bid distribution would not vary with the number of entrants. Through the lense of our model this is consistent with extreme levels of risk aversion, such that the bids are very close to valuation regardless of the number of bidders. 2 and 3 bidders 2 and 4 bidders 2 and 5 bidders Quantiles σˆ 95% CI σˆ 95% CI σˆ 95% CI [0.20,0.80] 0.708 [0.501,1.000] 0.666 [0.480,0.898] 0.694 [0.552,0.912] [0.25,0.75] 0.652 [0.406,1.000] 0.606 [0.398,0.913] 0.635 [0.450,0.913] [0.30,0.70] 0.615 [0.333,1.000] 0.547 [0.324,0.891] 0.568 [0.357,0.870] Table 3: Estimates of the CRRA coefficient σ in a specification without unobserved auction heterogeneity. 33Athey, Levin, and Seira (2011) argues that this is a reasonable assumption for timber auctions as the bids are highly correlated with the number of active bidders even after controlling for a variety of variables including the number of potential bidders. See Gentry, Li, and Lu (2015) for identification of risk aversion in first-price auctions if this assumption does not hold. 34Alternatively, we could condition number of potential bidders if it is observed. 25

6 Conclusion This paper extends the point identification result in Guerre, Perrigne, and Vuong (2009) to environments with unobserved auction heterogeneity and provides conditions to ensure that the primitives recovered under the exclusion restriction for the number of bidders remain meaningful as bounds of the true primitives even if the exclusion restriction is violated. We propose a Sieve Maximum Likelihood Estimator and show its consitency under low level conditions. Weexplainwhythebiasinriskaversionestimatesifunobservedauctionheterogentiy is ignored depends on the correlation between the number of bidders and the unobserved auction heterogeneity. The application underscores the importance of accounting for unobserved heterogeneity as we find the bidders to be risk neutral, but would reject risk neutrality if unobserved heterogeneity is ignored. We see two avenues for future research. First, relaxing the assumptions of symmetric, independent and private values are important extensions for many applications. Relaxing the assumption of independent values is perhaps most pertinent, because this creates an additional source of correlation among bids from the same auction. The researcher then faces the challenging task disentangle which part of this correlation can be attributed to the unobserved characteristic and which part to the correlation of values conditional on the unobserved characteristic. Second, allowing for unobserved heterogeneity in the framework of Gentry, Li, and Lu (2015) where bidders do not know the number of entrants. For this extension we would have to confirm that the conditions to apply the techniques from the measurment error literature are still satisfied. 26

A Identification A.1 Technical Assumptions Assumption 7. Technical Assumptions for Theorem 1. (1). Additive Case: (a) The density f∗has non-negative interval support and f∗(x) < a exp(−a |x|) for 1 2 ´ some constants a ,a > 0. In addition, |u|dFu(u|n) < ∞ for all n. 1 2 (b) λ(x) < exp(a x) for some a > 0. In addition, either ∃a > 0 such that 3 3 4 liminfλ(x)/exp(a x) > 0 or a < a . 4 3 2 x→∞ ´ (2). Multiplicative Case: The density f∗ has positive interval support and |v|dF∗(v) < ∞. ´ In addition, |logu|dFu(u|n) < ∞ for all n. Assumption 8. (1). Fu(·|n) has a continuous density fu(·|n) supported on [u(n),u(n)]. (2). F (·|·,n) is continuously differentiable on {(v,u) : v ∈ [v(u),v(u)],u ∈ [u(n),u(n)]}. A.2 Proof of Theorem 1 In Theorem 1 (1) bids are additive in u and in Theorem 1 (2) log bids are additive in log(u). This follows from a slight generalization of Proposition 1 in Krasnokutskaya (2011) presented in section A.2.1. The main identification proof is presented in section A.2.2. A.2.1 Bidding Strategy Lemma 1. Let s (v,u) be the bidding strategy for a bidder with value v in an auction with n unobserved heterogeneity u and s∗ be the bidding strategy under F∗. n (1). If F (v|u) = F∗(v −u), then s (v,u) = s∗ (v −u)+u for all u ≥ 0 and v ≥ u+1. n n 27

(2). IfF (v|u) = F∗(v/u)andthebiddershaveconstantrelativeriskaversion, thens (v,u) = n s∗ (v/u)u for all u > 0 and v ≥ u . n Proof. The bidding strategy under F∗ is given by the boundary condition s∗ (1) = 1 and the n first-order condtion ds∗ (v) f∗(v) n = (n−1) λ(v −s∗ (v)). dv F∗(v) n IfF (v|u) = F∗(v −u),thens (v,u) = s∗ (v −u)+usatisfiestheinitialconditions (v(u),u) = n n n v(u) and the first order condition holds: ∂s (v,u) ds∗ (v−u) f∗(v−u) f∗(v|u) n = n =(n−1) λ(v−u−s∗ (v−u))=(n−1) λ(v−s (v,u)). ∂v dv F∗(v−u) n F∗(v|u) n If F (v|u) = F∗(v/u) , then s (v,u) = s∗ (v/u)u satisfies the initial condition s (v(u),u) = n n n v(u) and: ∂s (v,u) ds∗ (cid:0)v(cid:1) f∗(cid:0)v(cid:1) (cid:16)v (cid:16)v(cid:17)(cid:17) f∗(v|u) (cid:18) v−s (v,u) (cid:19) n = n u = (n−1) u λ −s∗ = (n−1) λ n u ∂v dv F∗ (cid:0)v(cid:1) u n u F∗(v|u) u u Notice that λ(·/u)u = λ(·) if and only if the utility function is of the CRRA form. Therefore s (v,u) = s∗ (v/u)u satisfies the first-order condidtion with CRRA in this case. n n A.2.2 Proof of Theorem 1 Proof. Let G∗ and g∗ be the bid distribution and the corresponding bid density in an n n n bidder auction if u = 0 in Theorem 1 (1) or if u = 1 in Theorem 1 (2). The proof proceeds in two steps. First, we identify g∗ and g∗ building on Lemma 2 in n1 n2 Evdokimov and White (2012). Second, we identify the model primitives from g∗ and g∗ n1 n2 building on Proposition 3 in Guerre, Perrigne, and Vuong (2009). Please refer to Evdokimov and White (2012) and Guerre, Perrigne, and Vuong (2009) for these results. Here, we only show that the joint bid distributions satisfy the conditions in Lemma 2 of Evdokimov and White (2012) for both cases in Theorem 1. For Theorem 1 (1), we can rewrite the model as v = v∗ + u with v∗ independent of u. 28

The bidding strategy in an auction with n bidders u is u+s∗ (v∗) where s∗ (1) = 1 and for n n v∗ > 1 ds∗ (v∗) f∗(v∗) n = (n−1) λ(v∗ −s∗ (v∗)). dv∗ F∗(v∗) n ToapplyLemma2fromEvdokimovandWhite(2012),weneedtoshowthat(a)E[|u|+|s∗ (v∗)|] < n ∞ and (b) g∗ has a tail bounded by an exponential function. Condition (a) is guaranteed n ´ by the fact that E|s∗ (v∗)| < E|v∗| < ∞ and the assumption |u|dFu(u|n) < ∞. For n condition (b), notice that by assumption ∃C > 0 such that for v∗ > C > 0 we have ds∗ (v∗) f∗(v∗) n = (n−1) λ(v∗−s∗ (v∗)) < (n−1)2a exp(−a v∗)exp(a (v∗−s∗ (v∗))). dv∗ F∗(v∗) n 1 2 3 n The inequality uses the exponential bound for f∗ and λ. Let s1 (v∗) be a function that solves n ds1 (v∗) n = (n−1)2a exp(−a v∗)exp (cid:0) a (cid:0) v∗ −s1 (v∗) (cid:1)(cid:1) (4) dv∗ 1 2 3 n with s1 (C) = s∗ (C). Then s1 (v∗) > s∗ (v∗) if v∗ > C.35 n n n n If a < a , it is easy to see that s1 is bounded, so g∗ has bounded support and is bounded 3 2 n n by an exponential tail. (cid:16) (cid:17) If a < a , (4) has the solution exp(s1 (v∗)) = c exp a3−a2v∗ +c where c > 0 and c 2 3 n 1 a3 2 1 2 are constants. As a3−a2 < 1, s∗ (v∗) < s1 (v∗) < c v∗ with 0 < c < 1 for v∗ large enough. a3 n n 2 2 Then from the first order condition, the density of s∗ (v∗) satisfies n f∗(v∗) F∗(v∗) g∗(s∗ (v∗)) = = n n ds∗(v∗) (n−1)λ(v∗ −s∗ (v∗)) d n v∗ n F∗(v∗) 1 < < (cid:16) (cid:17) (n−1)exp(a 4 (1−c 2 )v∗) (n−1)exp a4(1−c2)s∗ (v∗) c2 n The first inequality follows from the assumption that λ(x) > exp(a x) for x large enough. 4 Hence, g∗ has an exponential bound. n For Theorem 1 (2), we can rewrite the model as v = v∗u with v∗ independent of u . The 35This follows from a standard contradiction argument. The proof is available upon request. 29

bidding strategy is us∗ (v∗) with n ds∗ (v∗) f∗(v∗) f∗(v∗) n = (n−1) (1−σ)(v∗ −s∗ (v∗)) < (n−1) (1−σ)v∗ dv∗ F∗(v∗) n F∗(v∗) Now we need to show that (a) E[|logu|+|logs∗ (v∗)|] < ∞ and that (b) logs∗ (v∗) has a n n density with a tail bounded by an exponential function. First, let v∗ be the lower bound of ´ v∗. Then s∗ (v∗) ≤ (n−1) v∗ f∗(v) (1−σ)vdv is bounded from above by the assumption n v∗ F∗(v) ´ that vf∗(v)dv < ∞. In addition, the bidding function is bounded away from 0. Hence, the density of logs∗ (v∗) has a bounded support. Hence, the density satisfies (b) which also n suggests E|logs∗ (v∗)| < ∞. In addition, E|logu| < ∞ by assumption which implies (a) is n satisfied. We normalize the lower bound of the support of f∗and thereby the lower bounds of the supports of g∗ for all n to one. It follows from Lemma 2 in Evdokimov and White (2012) n that g∗and fu(·|n) are identified for n = n ,n . n 1 2 Next we apply Proposition 3 in Guerre, Perrigne, and Vuong (2009) to g∗ and g∗ . This n1 n2 allows us to identify f∗ and U. A.3 Proof of Proposition 1 To simplify the notation, let F (·) = F (·|u ) , v (α) = F−1(α) and si (·) be the bidding i i i i n strategy under F for i = 1,2. In addition, bi (α) = si (v (α)) is the αth quantile of the bid i n n i distribution. As v(cid:48)(α)f (v(α)) = 1 we can rewrite the first order condition as follows:       (n−1) α 1λ(v i (α)−bi n (α)) if α > 0  dbi (α)  n = (5) dα        (n−1)λ (cid:48) (0) 1 if α = 0 (n−1)λ(cid:48)(0)+1fi(vi(0)) Before we prove Proposition 1, we illustrate the main idea of the proof by showing that 30

the stronger assumption v (α) > v (α) for all α implies that b1 (α) > b2 (α) for all α. 1 2 n n To see this notice that b1 (0) > b2 (0). Now suppose towards contradiction that for some n n α > 0 we have b1 (α) ≤ b2 (α). By continuity of the bid functions there exists α = n n 1 min{α : b1 (α) = b2 (α)} > 0. Notice that b1 (α) > b2 (α) for α < α by construction. n n n n 1 At the same time we have ∂ b1 (α ) > ∂ b2 (α ) because v (α ) > v (α ). Therefore there ∂α n 1 ∂α n 1 1 1 2 1 exists some α slightly smaller than α such that b1 (α) < b2 (α) which is a contradiction. The 1 n n proof of Proposition 1 follows a similar idea but is more involved. Lemma 2. Under Assumption 2, b1 (α) ≥ b2 (α) for all α ∈ [0,1]. n n Proof. First, noticethatb1 (0) ≥ b2 (0)asv (0) ≥ v (0). Nowsupposetowardscontradiction n n 1 2 thatthereisα > 0suchthatb1 (α ) < b2 (α ). Defineα = max{α : b1 (α) ≥ b2 (α),α ≤ α }. 2 n 2 n 2 1 n n 2 By construction, b1 (α) < b2 (α) for α ∈ (α ,α ). As v (α) ≥ v (α) for all α we have n n 1 2 1 2 ´ db1 n (α) > db2 n (α) for all α ∈ (α ,α ). This implies that b1 (α ) = b1 (α )+ α2 db1 n (α) > b2 (α ) = dα ´ dα 1 2 n 2 n 1 α1 dα n 2 b2 (α )+ α2 db2 n (α) which is a contradiction. n 1 α1 dα Lemma 3. Under Assumption 2 , if v (α) > v (α) then b1 (α) > b2 (α), for α ∈ [0,1]. 1 2 n n Proof. For α = 0 this holds because bi (0) = v (0) for i = 1,2. Now suppose towards n i contradiction that b1(α) ≤ b2(α) for some α ∈ (0,1] such that v (α) > v (α). This implies n n 1 2 that db1 n (α) > db2 n (α). Therefore we can find α slightly smaller than α such that b1(α ) < dα dα 1 n 1 b2(α ) which contradicts Lemma 2. n 1 Proof of Proposition 1. Suppose towards contradiction that b1 (1) = b2 (1) and v (1) = n n 1 v (1). This is the only case left to be ruled out, because the remaining cases where b1 (1) ≤ 2 n b2 (1)arecoveredbyLemmas2and3. Define∆b(α) = b1 (α)−b2 (α),∆v(α) = v (α)−v (α) n n n 1 2 and let α = inf{α : v (α) = v (α) on [α,1]} > 0. Notice that ∆b(α) = 0 for all α ∈ [α,1].36 1 2 Take the difference of the first-order conditions for b1 and b2 and apply the mean value then n 36On this region both bid functions can be derived by solving the same differential equation given by equation 5 and the end condition b1 (1)=b2 (1). n n 31

orem twice to obtain (cid:0) (cid:1) (cid:0) (cid:1) α∆b(cid:48)(α) = (n−1)λ v (α)−b1(α) −(n−1)λ v (α)−b2(α) 1 n 2 n = (n−1)λ(cid:48)(r¯(α))(∆v(α)−∆b(α)) = (n−1)λ(cid:48)(cid:0) v (α)−b1(α) (cid:1) (∆v(α)−∆b(α)) 1 n +(n−1)λ(cid:48)(cid:48)(r˜(α)) (cid:0) r¯(α)− (cid:0) v (α)−b1(α) (cid:1)(cid:1) (∆v(α)−∆b(α)) (6) 1 n = (n−1)(c+δ(α))(∆v(α)−∆b(α)) Here r¯(α) is some value between v (α)−b1(α) and v (α)−b2(α), r˜(α) is some value between 1 n 2 n r¯(α)andv (α)−b1(α),c = λ(cid:48)(v (α)−b1(α)) ≥ 1andδ(α) = λ(cid:48)(cid:48)(r˜(α))(r¯(α)−(v (α)−b1(α))).If 1 n 1 n 1 n α → α then v (α)−bi(α) → v (α)−b1(α) for i = 1,2. Consequently, r¯(α) → v (α)−b1(α) i n 1 n 1 n and δ(α) → 0 as α → α. As c ≥ 1, we can find an (cid:15) > 0 such that α−(cid:15) > 0 and c+δ(α) > 0 for all α ∈ [α−(cid:15),α]. Suppose we know δ then we can solve the differential equation (6) for ∆b on [α−(cid:15),α] with the end condition ∆b(α) = 0 . The closed form solution is ´ ´ α [c+δ(w)]∆v(w)exp w c+δ(z)dzdw ∆b(α) = − α ´ α−(cid:15) z < 0. exp α c+δ(z)dz α−(cid:15) z This contradicts Lemma 2. Therefore, b1 (1) > b2 (1). n n A.4 Proof of Theorem 2 Proof. The bid distribution given u in an n-bidder auction is denoted by G (·|u) and the n corresponding density by g (·|u). First, consider the case where u is discrete. As the support n of u does not depend on the number of bidders, we can normalize u such that it takes values on 1,2,3,··· ,K for n and n . Hu, McAdams, and Shum (2013) shows that G (·|u) is 1 2 n identified if the highest bid is strictly increasing in u. This is satisfied by Proposition 1. We then pair G (·|u) and G (·|u) to identify U and F by applying Proposition 3 in Guerre, n1 n2 32

Perrigne, and Vuong (2009). Now consider the case where where u is continuous. We show that the relevant conditions for Steps 1 and 2 in the proof of Theorem 2.1 in d’Haultfoeuille and F´evrier (2010b) are satisfied: First,thehighestbidgivenuisstrictlyincreasinginubyProposition1. Second,thelowest bid is assumed to be strictly increasing in u. Third, G (·|·) , s (v¯(u),u) and s (v(u),u) n n n are continuously differentiable. To see this, notice that F (·|·) and the utility function U are both continuously differentiable by Assumption 8. By Theorem 1 in Campo, Guerre, Perrigne, and Vuong (2011), the bidding strategy s (v,u) is continuously differentiable on the n support of F (·|·) . Hence the highest bid s (v¯(u),u) is continuously differentiable with n respect to u and G (·|·) is continuously differentiable. Therefore, s (v(u),u) = v(u) is n n continuously differentiable. We normalize u = v(u). Now we can apply Theorem 2.1 from d’Haultfoeuille and F´evrier (2010b) to show that G (·|v(u)) is identified for n and n . As n 1 2 the supports of fu and fu overlap, we can find some v(u) such that we observe G (·|v(u)) n1 n2 n for n = n ,n . We then invoke Proposition 3 in Guerre, Perrigne, and Vuong (2009) to 1 2 identify U and F. A.5 Proof of Theorem 3 Proof. Let G and G be the bid distributions from n and n bidder auctions. We first 1 2 1 2 prove that the bid distribution G first order stochastically dominates G . To simplify the 2 1 notation we suppress u∗ from now on. Suppose towards contradiction that G does not first 2 order stochastically dominate G . Let v denote the common lower bound of the support of 1 both bid distributions (Condition 1(1)). Guerre, Perrigne, and Vuong (2009, Theorem 1) establish that the slope of the bid function at v is strictly higher in the n bidder auction. 2 Moreover Assumption 2 implies that the density of valuations in the n bidder auction is 2 weakly lower at v. Therefore g (v) < g (v) and at the smallest point (cid:101)b > v such that 2 1 (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) (cid:16) (cid:17) G (cid:101)b = G (cid:101)b = α < 1 we must have g (cid:101)b ≥ g (cid:101)b . The first order condition of the 1 2 2 1 33

bidding strategy can be written as (cid:16) (cid:17) 1 α ˜ g b = for i = 1,2. i (cid:16) (cid:17) n −1 ˜ i λ v (α)−b i , where v (α) is the α-th quantile of the valuation distribution. As n > n and v (α) ≥ i 2 1 2 (cid:16) (cid:17) (cid:16) (cid:17) v (α), we must have g (cid:101)b < g (cid:101)b which is a contradiction. Therefore, G must first order 1 2 1 2 stochastically dominate G . 1 As in Guerre, Perrigne, and Vuong (2009), construct a decreasing sequence of αs such that R (α ) = R (α ) with R (α ) = x. As R (α ) > R (α ), R (0) = 0 < R (α ) 1 t 2 t−1 1 0 1 t−1 2 t−1 1 2 t−1 and R is continuous, there exits an α ∈ (0,α ) such that R (α ) = R (α ) by the 1 t t−1 1 t 2 t−1 intermediate value theorem. Therefore, such a decreasing sequence of αs exists. In addition, this sequence converges to 0. This can be shown by contradiction. First, notice that the sequence is decreasing and bounded from below by 0. Hence, it must converge to some non-negative number c. Suppose towards contradiction that c > 0. As R and R are both 1 2 continuous, (cid:16) (cid:17) (cid:16) (cid:17) R (c) = R lim α = lim R (α ) = lim R (α ) = R lim α = R (c) 1 1 t 1 t 2 t−1 2 t−1 2 t→∞ t→∞ t→∞ t→∞ This violates the condition that R (α) > R (α) for α > 0 . 1 2 We want to bound λ−1(x). We define λ(cid:101) as the strictly increasing function satisfying λ(cid:101)−1(R (α)) − λ(cid:101)−1(R (α)) = b (α) − b (α) for α ∈ [0,1] with λ(cid:101)(0) = 0. Notice that if 1 2 2 1 Assumption 1 is violated the existence of this function is no longer guaranteed. We assume that it exists henceforth. Using the α sequence and recursive subsitution λ(cid:101) can be expressed as follows: ∞ ∞ ∞ (cid:88) (cid:88) (cid:88) λ(cid:101) −1(x) = [b (α )−b (α )] = b (α )−b (α ) = ∆b(α ) 2 t 1 t 2 t 1 t t t=0 t=0 t=0 34

(cid:20) (cid:19) with x ∈ 0, max R (α) . This infinite sum exists because for any finite T, (cid:80)T ∆b(α ) ≤ 1 t=0 t α∈[0,1] λ(cid:101)−1(x) and ∆b(α ) ≥ 0 by the first order stochastic dominance of bid distributions shown t above. The true λ satisfies the first-order condition R (α) = λ(v (α)−b (α)) for i = 1,2 so i i i ∞ ∞ (cid:88) (cid:88) b (α )−b (α ) ≥ [b (α )−b (α )+v (α )−v (α )] 2 t 1 t 2 t 1 t 1 t 2 t t=0 t=0 ∞ (cid:88)(cid:2) (cid:3) = λ−1(R (α ))−λ−1(R (α )) 1 t 2 t t=0 = λ−1(R (α ))− lim λ−1(R (α )) = λ−1(x) 1 0 2 t t→∞ TheinequalityfollowsfromAssumption2. Thelastequalityusesthefactthat lim λ−1(R (α )) = 2 t t→∞ 0becausetheαsequenceconvergestozero. Henceλ(cid:101)−1(·) ≥ λ−1(·)andthereforeλ(cid:101)(·) ≤ λ(·). Theboundsfortheutilityfunctionareobtainedbysolvingthedifferentialequationλ(x) = U(x) with the boundary condition U (1) = 1. U(cid:48)(x) The underlying valuations recovered under Assumption 1 bound the actual valuations from above: F−1(α|u,n ) = λ−1(R (α,u))+b (α,u) ≤ λ(cid:101)−1(R (α,u))+b (α,u) for i = 1,2. i i i i i Moreover, the valuations are bounded from below by the bids: F−1(α|u,n ) ≥ b (α,u) for i i i = 1,2. B Entry B.1 Entry and Assumption 1 In this section we show that Assumption 1 is satisfied in a fairly general entry framework with (conditionally) independent signals. There are N potential entrants. Prior to bidding, a potential bidder i observes a vector of auction characteristics X including the number of potential entrants and a private signal ξ (possibly multi-dimensional). Bidder i(cid:48)s entry i strategy is a function φ : (X,ξ ) → {0,1}. If φ takes the value 1, the potential bidder enters. i i i Most commonly considered entry models fit into this framework for some φ . To simplify the i 35

notation, we suppress the argument of φ from now on. i →− →− →− Let v = (v ,v ,··· ,v ) , ξ = (ξ ,ξ ,··· ,ξ ) and φ = (φ ,φ ,··· ,φ ). We use the 1 2 N 1 2 N 1 2 N subscript −i to denote the vector of variables from bidders other than i. Let F (·,·|·) A,B|C denote the joint distribution of A and B conditional on C. For instance, F −→−→ (·,·|·) is the v,ξ|X joint distribution of the value and signal for bidder i given X.37 The next lemma provides a sufficient condition to ensure that an entrant’s value distribution depends only on his identity and X. Therefore, if the bidders are symmetric, an active bidder’s value distribution does not vary with n and Assumption 1 holds. →− Lemma 4. If (v ,ξ ) ⊥ ξ for all i, then for every i, F (v|1,·,x) = F (v|1,x), i i X −i vi|φi,φ−i,X vi|φi,X x-a.e. →− →− Proof. Notice that (v ,ξ ) ⊥ ξ implies that (v ,φ ) ⊥ φ . By conditional independence i i −i i i −i X X F (v,1,·|x) F (v,1|x)P (φ = ·|x) F (v|1,·,x) = vi,φi,φ−i|,X = vi,φi,φ−i|,X −i = F (v|1,x) vi|φi,φ−i,X P (φ = 1,φ = ·|x) P (φ = 1|x)P (φ = ·|x) vi|φi,X i −i i −i B.2 Independence of v∗ and u with Entry Proposition 3. Suppose v∗ and u are independent for potential bidders, then v∗and u are independent conditional on n if one of the following conditions holds: (1). Potential bidders do not observe any information about v∗ or u. (2). Potential bidders observe only u. (3). Potential bidders observe a signal s of v∗, which is independent across bidders and i independent of u. 37For example, in the AS model studied in Li, Lu, and Zhao (2015), entry can be described by φ (X,ξ )= i i 1(ξ >c(X)) where c(X) be a cutoff determined by the model. i 36

Proof. We must allow for the possibility of mixed entry strategies. Bidder i uses a randomization device which generates a random variable (cid:15) independent of (u,v∗) to implement the i mixed strategy. For Proposition 3(1) the entry strategy can be described by some function g((cid:15) ) which i takes the value 0 or 1. Therefore, the joint distribution of (u,v∗) for entrants is Pr(u ≤ x,v∗ ≤ y|g((cid:15) ) = 1) = Pr(u ≤ x,v∗ ≤ y) = Pr(u ≤ x)P (v∗ ≤ y) i . For Proposition 3(2) the entry strategy can be described by g(u,(cid:15) ), the joint distribution i for entrants is Pr(u ≤ x,v∗ ≤ y|g(u,(cid:15) ) = 1) = E{E[1(u ≤ x,v∗ ≤ y)|u,(cid:15) ]|g(u,(cid:15) ) = 1}. i i i = E[1(u ≤ x)E[1(v∗ ≤ y)|u,(cid:15) ]|g(u,(cid:15) ) = 1] i i = Pr(v∗ ≤ y)Pr(u ≤ x|g(u,(cid:15) ) = 1) i The last step follows because v∗ is independent of (u,(cid:15) ). i Lastly, following an analogous argument one can show for Proposition 3(3) that P (u ≤ x,v∗ ≤ y|g(s ,(cid:15) ) = 1) = P (v∗ ≤ y|g(s ,(cid:15) ) = 1)P (u ≤ x). i i i i This result does not cover the case where bidders obtain some information about the private information v∗ and about the unobserved characteristic u before they enter. In this case v∗and u are generally no longer independent for entrants. 37

B.3 Selection on u With some abuse of notation, let F∗ and Fu denote the distributions of v∗ and u for potential bidders and F∗(·|n) and Fu(·|n) the distributions of v∗ and u for a given number of active bidders n. Let N be the number of potential bidders. For simplicity, assume that fu has a bounded interval support [u,u]. The entry cost is k. Potential bidders share a wealth level W > k and a utility function U. Define U (x) = U(x+W −k) − U(W −k) as the re-centered utility function. We assume that at the bidding stage the bidders know their own v∗, u and n. Proposition 4. Suppose that v = uv∗ where u and v∗ are independent, and U (x) = x1−σ. Potential bidders observe only u, but no signal for v∗. Then Fu(·|n) is first-order stochastically increasing in n. To prove this we use the following lemma. Lemma 5. Let p(u) ∈ [0,1] be a weakly increasing function on [u,u]. Let ˆ x Fu(x|n) = p(u)n[1−p(u)]N−nfu(u)du/C(n) u ´ with C(n) = u¯ p(u)n[1−p(u)]N−nfu(u)du. Then if n < n(cid:48), Fu (cid:0) x|n(cid:48)(cid:1) ≤ Fu(x|n) for u x ∈ [u,u]. Proof. This is proved by contradiction. As Fu (cid:0) u|n(cid:48)(cid:1) = Fu(u|n) = 0 any violation of first order stochastic dominance must be such that Fu (cid:0) x|n(cid:48)(cid:1) ≤ Fu(x|n) until a point x ∈ [u,u) and then Fu (cid:0) x|n(cid:48)(cid:1) > Fu(x|n) for x ∈ (x,x+(cid:15)) with (cid:15) > 0. If this is the case, we must have ∂ Fu (cid:0) x|n(cid:48)(cid:1) = fu (cid:0) x|n(cid:48)(cid:1) ≥ fu(x|n) = ∂ Fu(x|n). Now consider ∂x ∂x fu(x|n) C(n)p(x)n[1−p(x)]N−n C(n) (cid:20) 1−p(x) (cid:21)n (cid:48) −n = = . fu(x|n(cid:48)) C(n(cid:48))p(x)n(cid:48) [1−p(x)]N−n(cid:48) C(n(cid:48)) p(x) As n(cid:48) > n and p is weakly increasing, the ratio above is weakly decreasing in x. Consequently, fu (cid:0) x|n(cid:48)(cid:1) ≥ fu(x|n) for all x > x. As Fu (cid:0) x|n(cid:48)(cid:1) > Fu(x|n) for x ∈ (x,x+(cid:15)) this implies that 38

Fu (cid:0) u|n(cid:48)(cid:1) > Fu(u|n). This is a contradiction because by definition Fu (cid:0) u|n(cid:48)(cid:1) = Fu(u|n) = 1. Proof of Proposition 4. At the bidding stage bidder i solves the following problem: (cid:18) (cid:18) (cid:19) (cid:12) (cid:19)n−1 m b a i x[U(v i ∗u−b i +W −k)−U(W −k)]P S n m j(cid:54)= a i xv j ∗,u ≤b i (cid:12) (cid:12) (cid:12) u +U(W −k) = max(v i ∗u−b i )1−σP (cid:0) S n (cid:0) v j ∗,u (cid:1) ≤b i (cid:12) (cid:12)u (cid:1)n−1 +U(W −k) bi = u1−σ(v∗−s∗ (v∗))1−σF∗(v∗)n−1+U(W −k). (7) i n i i Here S is the bidding strategy and s∗ is the bidding strategy if u = 1. Let n n π (u,v∗) = u1−σ(v∗ −s∗ (v∗))1−σF∗(v∗)n−1 n i i n i i ´ Let Π (u) = π (u,v∗)dF∗(v∗) . Notice that Π (u) = u1−σΠ (1). n n i i n n Inequilibrium, potentialbiddersenterrandomlywithsomeprobabilitywhichisafunction of u. The symmetric entry probability p(u) satisfies 1. p(u) = 1 if u1−σ0Π (1) > U(W)−U(W −k) N 2. p(u) = 0 if u1−σ0Π (1) < U(W)−U(W −k). 1 3. Otherwise, N (cid:88) U(W)−U(W −k) [1−p(u)]N−np(u)n−1Π (1) = . (8) n u1−σ n=1 We only need to show that p(u) is weakly increasing in u then Lemma 5 implies that Fu(·|n) isincreasinginninthesenseoffirst-orderstochasticdominance. Thistakestwosteps. First, notice that Π (1) is decreasing in n. To see this, recall that n ˆ Π (1) = (v∗ −s∗ (v∗))1−σF∗(v∗)n−1dF∗(v∗). n i n i i i Herev∗−s∗ (v∗)andF∗(v∗)n−1 arebothdecreasinginn. HenceΠ (1)isdecreasinginnasthe n n integrandisalwaysdecreasinginn. Second,noticethatahigherureduces U(W)−U(W−k) which u1−σ 39

corresponds to the entry cost in Lemma 1 from Levin and Smith (1994b). Apply this lemma toconcludethatp(u)isincreasinginuifp(u)solves(8). Ifu1−σΠ (1) > U(W)−U(W −k) N or u1−σΠ (1) < U(W)−U(W −k) , d p(u) = 0. Then p(u) is weakly increasing in u. 1 du Proposition 4 considers the specification used in the application with constant relative risk aversion and multiplicative unobserved heterogeneity. The result can be extended to additive unobserved heterogeneity and in this case the restriction to constant relative risk aversion is no longer needed. C Proof of Proposition 2 Proof. Assumption6andtheidentificationresultimplythatE l(Z ;θ)isuniquelymaximized 0 (cid:96) at θ . l(Z ;θ) is bounded from above by a constant because by Lemma 9, g∗ is bounded. 0 (cid:96) n Then E l(Z ;θ) is upper semi-continuous by Lemma 11 and the Reverse Fatou Lemma, 0 (cid:96) limsupE l(Z ;θ ) ≤ E limsupl(Z ;θ ) ≤ E l(Z ;θ). 0 (cid:96) k 0 (cid:96) k 0 (cid:96) k→∞ k→∞ As Θ is compact, for any (cid:15) > 0, there exists a δ > 0 such that E l(Z ,θ )− sup E l(Z ,θ) > δ. 0 (cid:96) 0 0 (cid:96) (cid:107)θ−θ0(cid:107) s ≥(cid:15) Define Θ((cid:15)) = {θ ∈ Θ : (cid:107)θ−θ (cid:107) ≥ (cid:15)}. For any θ ∈ Θ((cid:15)), let N (θ) be a closed ball 0 s around it. Let l (Z ) = sup l (cid:0) Z ,θ(cid:48)(cid:1) . By Lemma 11, l(Z ,θ) is continuous in N(θ) (cid:96) θ(cid:48)∈N(θ) (cid:96) (cid:96) θ Z -a.s. Hence if N (θ) ↓ θ, l (Z ) ↓ l(Z ,θ) a.s. By the monotone convergence the- (cid:96) N(θ) (cid:96) (cid:96) orem, E l (Z ) ↓ E l(Z ,θ). Therefore, for any θ ∈ Θ((cid:15)), there exists N (θ) such that 0 N(θ) (cid:96) 0 (cid:96) E l(Z ,θ )−E l (Z ) > δ/2. Then Θ((cid:15)) ⊆ ∪ N (θ). Because Θ((cid:15)) is a closed subset 0 (cid:96) 0 0 N(θ) (cid:96) θ∈Θ((cid:15)) of a compact space Θ, it is also compact. Hence, there exist N = N (θ ), j = 1,2,··· ,J j j that cover Θ((cid:15)). By Lemma 6 and Lemma 9, g (b,θ) is bounded from above, so is l(Z ,θ) n (cid:96) and l (Z ). Therefore, we can still apply law of large numbers even if the expectation may Nj (cid:96) 40

be −∞. 1 (cid:88) 1 (cid:88) sup l(Z ,θ) ≤ sup l (Z ) →a.s. supE l (Z ) < E l(Z ,θ )−δ/2. (9) L (cid:96) L Nj (cid:96) 0 Nj (cid:96) 0 (cid:96) 0 θ∈Θ((cid:15)) j j There exists sequence {θ }∞ , θ ∈ Θ such that lim (cid:107)θ −θ (cid:107) = 0 , and 0,kL kL=1 0,kL kL kL→∞ 0,kL 0 s E l(Z ,θ )−E l(Z ,θ ) → 0 by Lemma 12. Therefore, we can find K large enough such 0 (cid:96) 0,kL 0 (cid:96) 0 that |E l(Z ,θ )−E l(Z ,θ )| < δ/4 and θ ∈ Θ((cid:15))c ∩ Θ for all k ≥ K. By this 0 (cid:96) 0,kL 0 (cid:96) 0 K kL L definition and (9), E l(Z ,θ )−supE l (Z ) > δ/4 (10) 0 (cid:96) K 0 Nj (cid:96) j For L large enough (cid:40) (cid:41) (cid:110) (cid:111) 1 (cid:88) 1 (cid:88) θ(cid:98) ∈ Θ((cid:15)) ⊆ sup l(Z ,θ) ≥ sup l(Z ,θ ) L (cid:96) (cid:96) L L L θ∈Θ((cid:15))∩Θ kL θ∈Θ((cid:15))c∩Θ kL (cid:40) (cid:41) 1 (cid:88) 1 (cid:88) ⊆ sup l(Z ,θ) ≥ sup l(Z ,θ ) (cid:96) (cid:96) L L L θ∈Θ((cid:15)) θ∈Θ((cid:15))c∩Θ kL (cid:40) (cid:41) 1 (cid:88) 1 (cid:88) ⊆ sup l(Z ,θ) ≥ l(Z ,θ ) (cid:96) (cid:96) K L L θ∈Θ((cid:15)) The probability of (cid:8) sup 1 (cid:80) l(Z ,θ) ≥ 1 (cid:80) l(Z ,θ ) (cid:9) converges to 0 because θ∈Θ((cid:15)) L (cid:96) L (cid:96) K (cid:32) (cid:33) (cid:18) (cid:19) 1 (cid:88) 1 (cid:88) 1 (cid:88) 1 (cid:88) limsupP sup l(Z ,θ)≥ l(Z ,θ ) ≤limsupP sup l (Z )≥ l(Z ,θ ) L (cid:96) L (cid:96) K L Nj (cid:96) L (cid:96) K L→∞ θ∈Θ((cid:15)) L→∞ j (cid:18) (cid:19) (cid:18) (cid:19) 1 (cid:88) ≤ P supE l (Z )+δ/8≥E l(Z ,θ )−δ/8 +limsupP l(Z ,θ )−E l(Z ,θ )<−δ/8 0 Nj (cid:96) 0 (cid:96) K L (cid:96) K 0 (cid:96) K j L→∞ (cid:18) (cid:19) 1 (cid:88) +limsupP sup l (Z )−supE l (Z )>δ/8 L Nj (cid:96) 0 Nj (cid:96) L→∞ j j (cid:18) (cid:19) (cid:18) (cid:19) 1 (cid:88) ≤ P E l(Z ,θ )−supE l (Z )≤δ/4 +limsupP sup l (Z )−supE l (Z )>δ/8 0 (cid:96) K 0 Nj (cid:96) L Nj (cid:96) 0 Nj (cid:96) j L→∞ j j (cid:18) (cid:19) 1 (cid:88) +limsupP l(Z ,θ )−E l(Z ,θ )<−δ/8 =0 L (cid:96) K 0 (cid:96) K L→∞ 41

The last step follows from (10) and the Law of Large Numbers. Hence, (cid:32) (cid:33) (cid:16) (cid:17) 1 (cid:88) 1 (cid:88) P θ(cid:98) ∈ Θ((cid:15)) ≤ P sup l(Z ,θ) ≥ l(Z ,θ ) → 0. L (cid:96) (cid:96) K L L θ∈Θ((cid:15)) 42

D Lemmas for the Proof of Proposition 2 (not for publication) Throughout this section, C and any C with a subscript are generic finite positive constants which may take different values at different places. Lemma 6. f∗(·;θ) and fu(·;θ) satisfies the following n (cid:16) √ (cid:17)2 (cid:16) √ (cid:17)2 (1). η2 h∗(·) ≤ f∗(·;θ) ≤ 1+3 B h∗(·), η2 hu(·) ≤ fu(·;θ) ≤ 1+3 B hu(·). 1+B 1+B n Hence, f∗(·,θ) and fu(·,θ) are uniformly bounded from above. n (cid:16) √ (cid:17)2 (cid:16) √ (cid:17)2 (2). η2 H∗(·) ≤ F∗(·;θ) ≤ 1+3 B H∗(·), η2 Hu(·) ≤ Fu(·;θ) ≤ 1+3 B Hu(·). 1+B 1+B n (3). sup (cid:107)f∗(x;θ )−f∗(x;θ )(cid:107) ≤ C(cid:107)θ −θ (cid:107) ∀θ ,θ ∈ Θ for some C < ∞. The same x 1 2 ∞ 1 2 s 1 2 holds for fu(·;θ) ∀n ∈ N. n (4). sup (cid:107)F∗(x;θ )−F∗(x;θ)(cid:107) ≤ C(cid:107)θ −θ (cid:107) ∀θ ,θ ∈ Θ for some C < ∞.The same x k ∞ 1 2 s 1 2 holds for Fu(·;θ) ∀n ∈ N. n √ Proof. We start with the first claim. ψ ∈ Ψ(B) implies |ψ(x)| ≤ 3 B. To see this, notice √ √ |ψ(0)| ≤ 2 B for any ψ ∈ Ψ(B). If not, we can find a ψ ∈ Ψ(B) such that |ψ(0)| > 2 B. √ Without loss of generality, we can assume that ψ(0) > 2 B. Then ˆ ˆ (cid:115)ˆ x x 1 √ ψ(x) = ψ(0)+ ψ(cid:48)(y)dy ≥ ψ(0)− (cid:12) (cid:12)ψ(cid:48)(y) (cid:12) (cid:12)dy ≥ ψ(0)− |ψ(cid:48)(y)|2dy > B 0 0 0 √ which suggestsψ ∈/ Ψ(B). This is a contradiction. Therefore, ψ(0) ≤ 2 B and ˆ ˆ (cid:115)ˆ (cid:12) x (cid:12) x 1 √ |ψ(x)| = (cid:12) (cid:12)ψ(0)+ ψ(cid:48)(y)dy (cid:12) (cid:12) ≤ |ψ(0)|+ (cid:12) (cid:12)ψ(cid:48)(y) (cid:12) (cid:12)dy ≤ |ψ(0)|+ |ψ(cid:48)(y)|2dy ≤ 3 B (11) (cid:12) (cid:12) 0 0 0 √ Because |ψ(x)| ≤ 3 B, η2 (1+ψ(x))2 (cid:16) √ (cid:17)2 ≤ ´ ≤ 1+3 B (12) 1+B 1+ ψ(x)2dx 43

Then we have (cid:16) √ (cid:17)2 η2 1+3 B hu(x) ≥ fu(x;θ) = (Tψu)[Hu(x)]hu(x) ≥ hu(x), n n 1+B The same inequalities hold for f∗(x;θ). The second claim holds by integrating the above inequalities. Next, we prove the third claim. We only need to show that the mapping Tψ = [1´+ψ(·)]2 1+ ψ2(x)dx is continuous in ψ under (cid:107)·(cid:107) on Ψ(B). If (cid:107)ψ −ψ(cid:107) → 0 as k → ∞, ∞ k ∞ (cid:12) (cid:12) (cid:12) [1+ψ (x)]2 [1+ψ (x)]2 (cid:12) (cid:107)Tψ −Tψ (cid:107) = sup (cid:12) ´ 1 − ´ 2 (cid:12) 1 2 ∞ (cid:12)1+ ψ2(x)dx 1+ ψ2(x)dx(cid:12) x∈[0,1](cid:12) 1 2 (cid:12) (cid:12) (cid:12) (cid:12) [1+ψ (x)]2 [1+ψ (x)]2 (cid:12) ≤ sup (cid:12) ´ 1 − ´ 2 (cid:12) (cid:12)1+ ψ2(x)dx 1+ ψ2(x)dx(cid:12) x∈[0,1](cid:12) 1 1 (cid:12) (cid:12) (cid:12) (cid:12) [1+ψ (x)]2 [1+ψ (x)]2 (cid:12) + sup (cid:12) ´ 2 − ´ 2 (cid:12) (cid:12)1+ ψ2(x)dx 1+ ψ2(x)dx(cid:12) x∈[0,1](cid:12) 1 2 (cid:12) ≤ (cid:107)ψ −ψ (cid:107) (cid:0) 2(cid:107)ψ (cid:107) +2(cid:107)ψ (cid:107) +4(1+(cid:107)ψ (cid:107) )2(cid:1) . 1 2 ∞ 1 ∞ 2 ∞ 2 ∞ (cid:18) √ (cid:16) √ (cid:17)2 (cid:19) By (11), (cid:107)Tψ −Tψ (cid:107) ≤ 6 B +4 1+3 B (cid:107)ψ −ψ (cid:107) ≡ C (cid:107)ψ −ψ (cid:107) which is 1 2 ∞ 1 2 ∞ 2 1 2 ∞ H¨older continuous in ψ. Trivially (cid:12)(cid:0) (cid:1) (cid:0) (cid:1) (cid:12) |fu(x;θ )−fu(x;θ )| = (cid:12) Tψu [Hu(x)]h (x)− Tψu [Hu(x)]h (x)(cid:12) n 1 n 2 n,1 u n,2 u ≤ C (cid:107)hu(cid:107) (cid:107)ψ −ψ (cid:107) ≤ C (cid:107)hu(cid:107) (cid:107)θ −θ (cid:107) 2 ∞ 1 2 ∞ 2 ∞ 1 2 s which is H¨older continuous in θ. Notice that the bound of the above expression does not dependonx. Thesameholdsforf∗(x,θ). Forthelastclaim,justnoticethat|Fu(x;θ )−Fu(x;θ )| ≤ n 1 n 2 ´ |fu(x;θ )−fu(x;θ )|dx ≤ C(cid:107)θ −θ (cid:107) . The same inequality holds for F∗(·;θ) n 1 n 2 1 2 ∞ Lemma 7. sup |s∗ (v;θ )−s∗ (v;θ)| → 0 for if (cid:107)θ −θ(cid:107) → 0. v∈[1,v¯∗+1] n k n k s Proof. First, suppose v∗ < ∞. We first show that that s∗ (·;θ ) converges to s∗ (·;θ) unin k n 44

formly on [1+(cid:15),v∗ +1] for any (cid:15) > 0. |s∗ (x;θ )−s∗ (x;θ)| = (cid:12) (cid:12) (cid:12) ˆ x(cid:20) F∗(v;θ k ) (cid:21) 1 n − − σ 1 k dv− ˆ x(cid:20) F∗(v;θ) (cid:21)n 1− − σ 1 dv (cid:12) (cid:12) (cid:12) n k n (cid:12) F∗(x;θ ) F∗(x;θ) (cid:12) (cid:12) 1 k 1 (cid:12) ≤ (cid:12) (cid:12) (cid:12) ˆ x(cid:20) F∗(v;θ k ) (cid:21) 1 n − − σ 1 k dv− ˆ x(cid:20) F∗(v;θ) (cid:21) 1 n − − σ 1 k dv (cid:12) (cid:12) (cid:12) (cid:12) F∗(x;θ ) F∗(x;θ ) (cid:12) (cid:12) 1 k 1 k (cid:12) + (cid:12) (cid:12) (cid:12) ˆ x(cid:20) F∗(v;θ) (cid:21) 1 n − − σ 1 k dv− ˆ x(cid:20) F∗(v;θ) (cid:21)n 1− − σ 1 dv (cid:12) (cid:12) (cid:12) (cid:12) F∗(x;θ ) F∗(x;θ ) (cid:12) (cid:12) 1 k 1 k (cid:12) (cid:12) (cid:12) ˆ x(cid:20) F∗(v;θ) (cid:21)n 1− − σ 1 ˆ x(cid:20) F∗(v;θ) (cid:21)n 1− − σ 1 (cid:12) (cid:12) +(cid:12) dv− dv(cid:12) (cid:12) F∗(x;θ ) F∗(x;θ) (cid:12) (cid:12) 1 k 1 (cid:12) = A (x)+A (x)+A (x). 1 2 3 A (x),A (x) and A (x) have bounds independent of x. 1 2 3 ˆ 1 x(cid:12) (cid:12) n−1 n−1 A 1 (x) ≤ n−1 (cid:12) (cid:12) F∗(v;θ k )1−σk −F∗(v;θ)1−σk (cid:12) (cid:12) dv F∗(x;θ k )1−σk ˆ 1 1 v¯∗+1(cid:12) (cid:12) n−1 n−1 ≤ n−1 (cid:12) (cid:12) F∗(v;θ k )1−σk −F∗(v;θ)1−σk (cid:12) (cid:12) dv F∗(x;θ k )1−σk 1 Because x > (cid:15), F∗(x) > 2δ for some δ > 0. F∗(v;θ ) → F∗(v;θ) uniformly by Lemma 6. k Hence, there exists an K such that for all k > K, F∗(x) > δ for all x > 1+(cid:15). Therefore, for k k large enough, ˆ 1 v¯∗+1(cid:12) (cid:12) n−1 n−1 A 1 (x) ≤ n−1 (cid:12) (cid:12) F∗(v;θ k )1−σk −F∗(v;θ)1−σk (cid:12) (cid:12) dv. δ1−σk 1 n−1 n−1 In addition, notice σ k → σ, δ1−σk → δ1−σ. Therefore, for k large enough ˆ 2 v∗+1(cid:12) (cid:12) n−1 n−1 A 1 (x) ≤ n−1 (cid:12) (cid:12) F∗(v;θ k )1−σk −F∗(v;θ)1−σk (cid:12) (cid:12) dv. δ1−σ 0 (cid:12) (cid:12) n−1 n−1 Because (cid:12) (cid:12) F∗(v;θ k )1−σk −F∗(v;θ)1−σk (cid:12) (cid:12) → 0 uniformly, A 1 (x) → 0 uniformly in x. We can apply a similar argument to A (x) and A (x) to conclude they converges to 0 uniformly 2 3 in x. Therefore, |s∗ (x;θ )−s∗ (x;θ)| converges to 0 uniformly on [1+(cid:15),1+v∗]. Because n k n 45

s∗ (1;θ ) = s∗ (1;θ) = 1 and bid functions are continuous and increasing, for any δ > 0 we n k n can find an (cid:15) such that s∗ ((cid:15);θ) < δ/3. There exists a K such that for all k > K, n |s∗ (x;θ )−s∗ (x;θ)| < δ/3, ∀x ∈ [1+(cid:15),1+v∗]. n k n If x ∈ [1,1+(cid:15)], one can easily show that f k > K, |s∗ (x;θ )−s∗ (x;θ)| ≤ sup(|s∗ (1;θ )−s∗ ((cid:15);θ)|,|s∗ ((cid:15);θ )−s∗ (1;θ)|) < δ n k n n k n n k n Because the above inequality holds for all δ > 0 and it is independent of x, we can conclude s∗ (·;θ ) converges to s∗ (·;θ) uniformly on [1,1+v¯∗] if (cid:107)θ −θ(cid:107) → 0 if v¯∗ < ∞. n k n k s If v¯∗ = ∞, by the above argument, for any c < ∞, sup |s∗ (v;θ )−s∗ (v;θ)| → 0. v∈[1,c+1] n k n By the FOC, for any θ ∈ Θ ∂s∗ (v;θ) n−1f∗(v;θ) n−1h∗(v) n = (v −s∗ (v;θ)) ≤ C v ∂v 1−σ F (v;θ) n η H(v) for some constant C. The last inequality holds because of Lemma 6 and s∗ (v;θ) ≥ 0. By n Assumption 4, h∗(v) has a tail bounded by C/v2+δ for some δ > 0, for v > v large enough, 1 2 ˆ v1 h∗(v) C |s∗ (v ;θ)−s∗ (v ;θ)| ≤ C vdv ≤ v−δ. n 2 n 1 H(v ) H(v ) 2 v2 2 2 Hence,forany(cid:15) > 0,thereexistsav((cid:15)) < ∞suchthatforv > v((cid:15)),|s∗ (v((cid:15));θ)−s∗ (v;θ)| ≤ n n (cid:15). Therefore, sup |s∗ (v;θ )−s∗ (v;θ)| n k n v≥v((cid:15)) ≤ sup |s∗ (v;θ )−s∗ (v((cid:15));θ )|+ sup |s∗ (v;θ)−s∗ (v((cid:15));θ)| n k n k n n v≥v((cid:15)) v≥v((cid:15)) +|s∗ (v((cid:15));θ )−s∗ (v((cid:15));θ)| ≤ 2(cid:15)+|s∗ (v((cid:15));θ )−s∗ (v((cid:15));θ)| n k n n k n Consequently, sup |s∗ (v;θ )−s∗ (v;θ)| → 0, because the following holds for any v∈[1,v∗+1] n k n 46

(cid:15) > 0 sup |s∗ (v;θ )−s∗ (v;θ)| ≤ sup |s∗ (v;θ )−s∗ (v;θ)|+ sup |s∗ (v;θ )−s∗ (v;θ)| n k n n k n n k n v∈[1,∞] v∈[1,v((cid:15))] v≥v((cid:15)) ≤ 2(cid:15)+2 sup |s∗ (v;θ )−s∗ (v;θ)| → 2(cid:15). n k n v∈[1,v((cid:15))] Lemma 8. For every n ∈ N and θ ,θ ∈ Θ, if (cid:107)θ −θ(cid:107) → 0 then g∗(x;θ ) → g∗(x;θ) k k s n k n x-a.e. Proof. Because s∗ (v;θ) is strictly increasing on [1,v∗ +1] for all θ ∈ Θ, by Lemma 7, for any n 1 < x < s∗ (v¯∗;θ), s∗−1(x;θ ) → s∗−1(x;θ). If not, we can find an v (cid:54)= s∗−1(x;θ) such that n n k n n s∗ (v;θ) = x which violates the strictly increasing property. For such an x, s∗−1(x;θ )−x → n n k s∗−1(x;θ)−x > 0, n (cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) (cid:12)F∗ s∗−1(x;θ );θ −F∗ s∗−1(x;θ);θ (cid:12) n k k n (cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) (cid:12) (cid:0) (cid:1) (cid:0) (cid:1)(cid:12) ≤ (cid:12)F∗ s∗−1(x;θ );θ −F∗ s∗−1(x;θ);θ (cid:12)+(cid:12)F∗ s∗−1(x;θ);θ −F∗ s∗−1(x;θ);θ (cid:12) n k k n k n k n (cid:12) (cid:12) ≤ (cid:107)f∗(·;θ )(cid:107) (cid:12)s∗−1(x;θ )−s∗−1(x;θ)(cid:12)+(cid:107)F∗(·;θ)−F∗(·;θ )(cid:107) k ∞ n k n k ∞ (cid:0)(cid:12) (cid:12) (cid:1) ≤ C (cid:12)s∗−1(x;θ )−s∗−1(x;θ)(cid:12)+(cid:107)θ −θ(cid:107) → 0 n k n k s In addition, 1−σ k → 1−σ n−1 n−1 1−σ F∗(s∗−1(x;θ );θ ) 1−σF∗(s∗−1(x;θ);θ) g∗(x;θ ) = k n k k → n = g∗(x;θ). n k n−1 s∗−1(x;θ )−x n−1 s∗−1(x;θ)−x n n k n It is easy to see for any x < 1, g (x;θ ) = g (x;θ) = 0. In addition, if x > s∗ (v∗ +1;θ), n k n n x > s∗ (v∗ +1;θ ) for k large enough by Lemma 7. Hence g (x;θ ) = g (x;θ) = 0 for all n k n k n k large enough. If x = 1, g (1;θ ) = f∗(1;θ k )(1−σ k ) → g (1;θ). Hence g∗(x;θ ) → g∗(x;θ) n k n−σ n n k n k x-a.e. Lemma 9. There exists a constant C > 0 such that, g∗(·;θ) ≤ C for all θ ∈ Θ. n 47

Proof. For any v ∈ [1,v∗ +1], the first order condition implies 1−σ F∗(v;θ) 1−σ F∗(v;θ) 1−σ F∗(v;θ) n 1− − σ σ g∗(s∗ (v;θ);θ) = = = ´ (13) n n n−1v−s∗ n (v;θ) n−1´ v (cid:104) F∗(s;θ) (cid:105)n 1− − σ 1 ds n−1 1 v F∗(s;θ) n 1− − σ 1 ds 1 F∗(v;θ) (cid:16) √ (cid:17)2 By Lemma 6, for C = η2 and C = 1+3 B , 1 1+B 2 1−σC n 1− − σ σ H∗(v) n 1− − σ σ H∗(v) n 1− − σ σ H∗(v) 1+n− η 1 g∗(s∗ (v;θ);θ) ≤ 2 ´ ≤ C ´ ≤ C ´ (14) n n n−1 C n 1− − σ σ v H∗(s) n 1− − σ 1 ds 3 v H∗(s) n 1− − σ 1 ds 3 v H∗(s) n− η 1 ds 1 0 1 1 Notice that H∗(v)1+n− η 1 (cid:18) n−1 (cid:19) lim ´ = +1 h∗(1) v→1 v H∗(s) n− η 1 ds η 1 and let v be H∗(v ) = 1/2, for any v > v 1 1 1 H∗(v)1+n− η 1 2 n− η 1 ´ < v n−1 v H∗(s) η ds 1 1 Therefore, (cid:40) (cid:41) (cid:18) n−1 (cid:19) H∗(v)1+n− η 1 g∗(s∗ (v;θ);θ) ≤ C max +1 h∗(1), max ´ = C n n 3 η v∈[1,v1] v H∗(s) n− η 1 ds 1 max H∗(v)1+n− η 1 is bounded because it is continuous in v on (1,v ] and smaller than v∈[1,v1] ´ vH∗(s) n− η 1 ds 1 (cid:16) (cid:17) 1 n−1 +1 h∗(1) if v approaches 1. η Lemma 10. g (logb/exp(logXγ);θ) is continuous in θ on Θ (b,X)-a.e.. n 48

Proof. ∀θ ∈ Θ and θ → θ under (cid:107)·(cid:107) , k s (cid:12) (cid:18) (cid:19) (cid:18) (cid:19)(cid:12) (cid:12) b b (cid:12) (cid:12) (cid:12) g n exp(logXγ ) ;θ k −g n exp(logXγ) ;θ (cid:12) (cid:12) k (cid:12)ˆ = (cid:12) (cid:12) 1 (cid:89) n g∗ (cid:18) b i ;θ (cid:19) fu(u;θ)du (cid:12) un n (u+µ)exp(logXγ) n (cid:12) i=1 ˆ (cid:12) − 1 (cid:89) n g∗ (cid:18) b i ;θ (cid:19) fu(u;θ )du (cid:12) (cid:12) un n (u+µ )exp(logXγ ) k n k (cid:12) k k (cid:12) i=1 ˆ n (cid:18) (cid:19) ≤ C (cid:89) g∗ b i ;θ |fu(u;θ )−fu(u;θ)|du+ n (u+µ )exp(logXγ ) k n k n k k i=1 ˆ (cid:12) (cid:12) C (cid:12) (cid:12) (cid:89) n g∗ (cid:18) b i ;θ (cid:19) − (cid:89) n g∗ (cid:18) b i ;θ (cid:19)(cid:12) (cid:12)fu(u;θ)du (cid:12) n (u+µ )exp(logXγ ) k n (u+µ)exp(logXγ) (cid:12) n (cid:12) k k (cid:12) i=1 i=1 = A +A . 1 2 The first inequality holds because µ has to be strictly greater than some positive number. Hence, u ≥ µ is bounded from below. Consequently, there exists C > 1/un for all u. Next, we show that A and A converge to 0. 1 2 ˆ n (cid:89) A ≤ C(cid:107)θ −θ(cid:107) g∗(b /(u+µ )exp(logXγ );θ )du ≤ C(cid:107)θ −θ(cid:107) → 0. 1 k s n i k k k k s i=1 The first inequality holds by Lemma 6 and the second inequality holds by the fact that g∗ is n bounded and it is a density function. ˆ (cid:12) (cid:12) A ≤ (cid:12) (cid:12) (cid:89) n g∗ (cid:18) b i ;θ (cid:19) − (cid:89) n g∗ (cid:18) b i ;θ (cid:19)(cid:12) (cid:12)fu(u;θ)du 2 (cid:12) n (u+µ )exp(logXγ ) k n (u+µ )exp(logXγ ) (cid:12) n (cid:12) k k k k (cid:12) i=1 i=1 ˆ (cid:12) (cid:12) + (cid:12) (cid:12) (cid:89) n g∗ (cid:18) b i ;θ (cid:19) − (cid:89) n g∗ (cid:18) b i ;θ (cid:19)(cid:12) (cid:12)fu(u;θ)du (cid:12) n (u+µ )exp(logXγ ) n (u+µ)exp(logXγ) (cid:12) n (cid:12) k k (cid:12) i=1 i=1 = B +B 1 2 First, use a change of variables to rewrite ˆ (cid:12) (cid:12) (cid:12)(cid:89) n (cid:89) n (cid:12) B = (cid:12) g∗(b /u;θ )− g∗(b /u;θ)(cid:12)fu(u/exp(logXγ )−µ ;θ)du. 1 (cid:12) n i k n i (cid:12) n k k (cid:12) (cid:12) i=1 i=1 | (cid:81)n g∗(b /u;θ )− (cid:81)n g∗(b /u;θ)| → 0 u-a.e. by Lemma 8. In addition, by Lemma 9 for i=1 n i k i=1 n i 49

some C < ∞ (cid:12) (cid:12) n n (cid:12)(cid:89) (cid:89) (cid:12) (cid:12) g∗(b /u;θ )− g∗(b /u;θ)(cid:12)fu(u/exp(logXγ )−µ ;θ) ≤ Cfu(u/exp(logXγ )−µ ;θ) (cid:12) n i k n i (cid:12) n k k n k k (cid:12) (cid:12) i=1 i=1 Notice that Cfu(u/exp(logXγ )−µ ;θ) → Cfu(u/exp(logXγ)−µ;θ) u-a.e. and n k k n ˆ ˆ Cfu(u/exp(logXγ )−µ ;θ)du = Cexp(Xγ ) → Cexp(Xγ) = Cfu(u/exp(logXγ)−µ;θ) n k k k n The generalized Dominated Convergence Theorem implies B → 0. Similarly, the Domi- 1 nated Convergence Theorem implies B → 0. Consequently, g (logb/exp(logXγ),θ) is 2 n continuous in θ (b,X)-a.e.. Lemma 11. l(Z,θ) is continuous in θ on Θ Z-a.e.. Proof. l(Z,θ) is the log of a continuous function g by Lemma 10. Therefore, it is continuous n in θ, Z-a.e. Lemma 12. Under Assumptions 4 and 5, there exists a sequence {θ }∞ such that θ ∈ 0,kL kL=1 0,kL Θ , lim (cid:107)θ −θ (cid:107) = 0 , and E l(Z ,θ )−E l(Z ,θ ) → 0. kL kL→∞ 0,kL 0 s 0 (cid:96) 0,kL 0 (cid:96) 0 Proof. By Assumption 5, there exists α ∈ A such that (cid:107)α −α (cid:107) → 0. Now we 0,kL kL 0,kL 0 ∞ find σ ↓ σ such that θ = (σ ,γ ,µ ,α ) which satisfies s∗ (·;θ ) ≤ s∗ (·;θ ) ≤ 0,kL 0 0,kL 0,kL 0 0 0,kL n 0 n 0,kL C(s∗ (·;θ )−1)+1 for some C > 1. Then we show that this sequence of θ is the desirable n 0 0,kL sequence. Recall the first order condition ∂s∗ (v;θ) n−1 f∗(v;θ) n = (v −s∗ (v;θ)). ∂v 1−σF∗(v;θ) n Therefores∗ (·;θ ) ≤ s∗ (·;θ )if n−1 f∗(v;θ0) ≤ n−1 f∗(v;θ 0,kL ) ,or 1−σ0 F∗(v;θ0) ≥ 1−σ 0,kL F∗(v;θ 0,kL ) n 0 n 0,kL 1−σ0F∗(v;θ0) 1−σ 0,kL F∗(v;θ 0,kL ) n−1 f∗(v;θ0) n−1 f∗(v;θ 0,kL ) which in turn can be written as ´ ´ (cid:16) (cid:17) 1−σ 0 1 v (Tψ 0 ∗)◦H∗(s)h∗(s)ds ≥ 1−σ 0,kL 1 v Tψ 0 ∗ ,kL ◦H∗(s)h∗(s)ds n−1 (Tψ 0 ∗)◦H∗(v)h∗(v) n−1 (cid:16) Tψ∗ (cid:17) ◦H∗(v)h∗(v) 0,kL 50

(cid:13) (cid:13) Because (cid:13)ψ∗ −ψ∗(cid:13) → 0, by Lemma 6, 0,kL 0 ∞ (cid:12) (cid:0) (cid:1) (cid:12) sup(cid:12)(Tψ∗)◦H∗(v)− Tψ∗ ◦H∗(v)(cid:12) = (cid:15) → 0 kL kL v (cid:16) √ (cid:17)2 By (12), η2 ≤ (Tψ∗)◦H∗(v) ≤ 1+3 B . Therefore, for k large enough 1+B L (cid:20) (cid:21) (cid:20) (cid:21) 1− (cid:15) kL (1+B) (Tψ∗)◦H∗(v)h∗(v)≤ (cid:0) Tψ∗ (cid:1) ◦H∗(v)h∗(v)≤ 1+ (cid:15) kL (1+B) (Tψ∗)◦H∗(v)h∗(v) η2 0 0,kL η2 0 which suggests (cid:20) (cid:21) (cid:20) (cid:21) (cid:15) (1+B) (cid:15) (1+B) 1− kL f∗(v;θ ) ≤ f∗(v;θ ) ≤ 1+ kL f∗(v;θ ) (15) η2 0 0,kL η2 0 (cid:20) (cid:21) (cid:20) (cid:21) (cid:15) (1+B) (cid:15) (1+B) 1− kL F∗(v;θ ) ≤ F∗(v;θ ) ≤ 1+ kL F∗(v;θ ) (16) η2 0 0,kL η2 0 (15) and (16) together imply 1−σ 0 F∗(v;θ 0 ) 1−σ 0 (cid:104) 1− (cid:15) kL ( η 1 2 +B) (cid:105) F∗(v;θ 0,kL ) 1−σ 0 F∗(v;θ 0 ) (cid:104) 1− (cid:15) kL ( η 1 2 +B) (cid:105)2 ≥ ≥ n−1 f∗(v;θ 0 ) n−1 (cid:104) 1+ (cid:15) kL ( η 1 2 +B) (cid:105) f∗(v;θ 0,kL ) n−1 f∗(v;θ 0 ) (cid:104) 1+ (cid:15) kL ( η 1 2 +B) (cid:105)2 (cid:104) (cid:105) (cid:104) (cid:105) (cid:15) (1+B) (cid:15) (1+B) Define σ = 1 − (1−σ ) 1− kL / 1+ kL ↓ σ as (cid:15) → 0. Then let C > kL 0 η2 η2 0 kL (cid:104) (cid:105)2 (cid:104) (cid:105)2 1+ (cid:15) kL (1+B) / 1− (cid:15) kL (1+B) for k large enough, then because s∗ (v;θ ) ≤ s∗ (v;θ ) η2 η2 L n 0 n 0,kL ∂s∗ (v;θ) n−1 f∗(v;θ ) n−1 f∗(v;θ ) ∂s∗ (v;θ ) C n =C 0 (v−s∗ (v;θ ))≥ 0,kL (v−s∗ (v;θ ))= n 0,kL ∂v 1−σ F∗(v;θ ) n 0 1−σ F∗(v;θ ) n 0,kL ∂v 0 0 0,kL 0,kL Therefore,s∗ (·;θ ) ≤ C(s∗ (·;θ )−1)+1where1comesfromtheinitialconditions∗(1;θ) = n kL n 0 1. Notice θ converges to θ under (cid:107)·(cid:107) . 0,kL 0 s It is left to show that E l(Z ;θ )−E l(Z ;θ ) → 0. First, notice l(Z;θ ) → l(Z;θ ) 0 (cid:96) 0,kL 0 (cid:96) 0 0,kL 0 Z-a.e. byLemma11. Thenif|l(Z;θ )|isboundedbyaintegrablefunction, theDominated 0,kL Convergence Theorem implies that E l(Z ;θ )−E l(Z ;θ ) → 0. First notice, Lemma 9 0 (cid:96) 0,kL 0 (cid:96) 0 ´ implies that g∗(·;θ) < C. Therefore g∗(b;θ) = 1 (cid:81)n g∗(b ;θ)fu(u;θ)du < C for some n n un i=1 n i n constant C . Because l(Z;θ ) is the log of g∗, it is bounded from above by a constant. 0,kL n 51

Then it suffices to show that l(Z;θ ) is bounded from below by an integrable function. 0,kL To this end we show g (b;θ ) > C g (b;θ ) for some constant C > 0, which implies n 0,kL 1 n 0 1 l(Z;θ ) ≥ l(Z;θ )+C . 0,kL 0 1 We first show b ∈ [1,s∗ (1+v∗;θ )], g∗(b;θ ) ≥ C g∗(b;θ ) for some constant C > 0. n 0 n 0,kL 2 n 0 2 Use (13) to obtain g n ∗(b;θ 0,kL ) = 1−σ kL F∗(cid:0) s∗ n −1(b;θ 0,kL );θ 0,kL (cid:1) n 1− − σ σ 0 0 , , k k L L ´ 1 s∗ n −1(b;θ0) F∗(s;θ 0 )1 n − − σ 1 0 ds ´ g n ∗(b;θ 0 ) 1−σ 0 F∗ (cid:0) s∗ n −1(b;θ 0 );θ 0 (cid:1)n 1− − σ σ 0 0 1 s∗ n −1(b;θ 0,kL ) F∗(s;θ 0,kL )1− n σ − 0, 1 kL ds By Lemma 6 and the fact σ ≥ σ , there exists a constant C such that 0,kL 0 3 ´ (cid:20) (cid:21) n−1 g n ∗(b;θ 0,kL ) ≥ C H∗(cid:0) s∗ n −1(b;θ kL ) (cid:1) 1 s∗ n −1(b;θ0) H∗(s H ∗ n − ∗ 1 (s (b ) ;θ0)) 1−σ0 ds g n ∗(b;θ 0 ) 3 H∗ (cid:0) s∗ n −1(b;θ 0 ) (cid:1) ´ s∗ n −1(b;θ kL ) (cid:20) H∗(s) (cid:21) 1− n σ − 0, 1 kL ds 1 H∗(s∗ n −1(b;θ kL )) ´ (cid:20) (cid:21) n−1 ≥ C H∗(cid:0) s∗ n −1(b;θ 0,kL ) (cid:1) 1 s∗ n −1(b;θ0) H∗(s H ∗ n − ∗ 1 (s (b ) ;θ0)) 1−σ0 ds 3 H∗ (cid:0) s∗−1(b;θ ) (cid:1) ´ (cid:20) (cid:21) n−1 n 0 s∗ n −1(b;θ 0,kL ) H∗(s) 1−σ0 ds 1 H∗(s∗ n −1(b;θ 0,kL )) ´ = C 3 H H ∗ ∗ (cid:0) s (cid:0) ∗ n s − ∗ n − 1 1 (b ( ; b θ ; 0 θ , 0 k ) L (cid:1) ) (cid:1) n 1− − n 1 σ − σ − 0 0 σ σ 0 0 ´ 1 s∗ n 1 s − ∗ n 1 − ( 1 b ( ; b θ ; 0 θ , 0 kL ) H ) H ∗( ∗ s ( ) s 1 ) n − − 1 n σ − 1 − 0 σ 1 0 ds ds First, because s∗−1(b;θ ) ≥ s∗−1(b;θ ), n 0 n 0,kL g n ∗(b;θ 0,kL ) ≥ C H∗(cid:0) s∗ n −1(b;θ 0,kL ) (cid:1)n 1− − σ σ 0 0 g n ∗(b;θ 0 ) 3 H∗ (cid:0) s∗ n −1(b;θ 0 ) (cid:1)n 1− − σ σ 0 0 If b = s∗ (v¯∗ +1;θ ), n 0 g n ∗ g∗ (b ( ; b θ ; 0 θ ,k ) L ) ≥ C 3 H∗(cid:0) s∗ n −1(b;θ kL n ) − (cid:1) σ n 0 1− − σ σ 0 0 ≥ C 3 /2 1+n− η 1 (17) n 0 H∗(v¯∗+1)1−σ0 for large k . This is because we can find an (cid:15) > 0 such that s∗−1(b−(cid:15);θ ) > v (1/2) where L n 0 H H∗(v (1/2)) = 1/2. Because s∗−1(b−(cid:15);θ ) → s∗−1(b−(cid:15);θ ) for any (cid:15) > 0 and H∗ is H n 0,kL n 0 continuous, H∗(s∗−1(b;θ )) > 1/2. Because n−σ0 ≤ 1 + n−1, the last inequality in (17) n 0,kL 1−σ0 η 52

holds. In addition, because s∗ (·;θ ) ≤ C(s∗ (·;θ )−1)+1 n kL n 0 g n ∗(b;θ kL ) ≥ C H∗(cid:0) s∗ n −1(b;θ kL ) (cid:1)n 1− − σ σ 0 0 ≥ C H∗(cid:0) s∗ n −1(cid:0)b− C 1 +1;θ 0 (cid:1)(cid:1)n 1− − σ σ 0 0 g n ∗(b;θ 0 ) 3 H∗ (cid:0) s∗ n −1(b;θ 0 ) (cid:1)n 1− − σ σ 0 0 3 H∗ (cid:0) s∗ n −1(b;θ 0 ) (cid:1)n 1− − σ σ 0 0 By Assumption 4, lim H∗(cid:0) s∗ n −1(cid:0)b− C 1 +1;θ 0 (cid:1)(cid:1)n 1− − σ σ 0 0 = lim (cid:0) s∗ n −1(cid:0)b− C 1 +1;θ 0 (cid:1) −1 (cid:1)n 1− − σ σ 0 0(1+(cid:15)) b↓1 H∗ (cid:0) s∗ n −1(b;θ 0 ) (cid:1)n 1− − σ σ 0 0 b↓1 (cid:0) s∗ n −1(b;θ 0 )−1 (cid:1)n 1− − σ σ 0 0(1+(cid:15))  n−σ0(1+(cid:15)) b−1 1 1−σ0 = l b i ↓ m 1  (b− C 1 s ) ∗ n (cid:48)(1;θ 1 0)  = 1/C n 1− − σ σ 0 0(1+(cid:15)) = C 4 s∗ n (cid:48)(1;θ0) g∗(b;θ ) Because n 0,kL > 0 for all b ∈ (1,s∗ (v¯∗ +1;θ )) and it is continuous, there exists an g n ∗(b;θ0) n 0 g∗(b;θ ) C > 0 such that n 0,kL ≥ C on [1,s∗ (v¯∗ +1;θ )] . In addition, g∗(b;θ) = 0 if b ∈/ 2 g n ∗(b;θ0) 2 n 0 n [1,s∗ (v¯∗ +1;θ )]. g∗(b;θ ) ≥ C g∗(b;θ ) for k large enough.By Lemma 6, fu(u;θ ) > n 0 n 0,kL 2 n 0 L n kL C fu(u;θ ) 5 n 0 ˆ ˆ n n 1 (cid:89) 1 (cid:89) g (b;θ ) = g∗(b /u;θ )fu(u;θ )du ≥ C g∗(b /u;θ )fu(u;θ )du n 0,kL un n i 0,kL n 0,kL 5 un n i k n 0 i=1 i=1 ˆ n 1 (cid:89) ≥ C C g∗(b /u;θ )fu(u;θ )du = C g (b;θ ) 5 2 un n i 0 n 0 1 n 0 i=1 Therefore, l(Z ;θ ) ≥ l(Z ;θ )+logC . Then by the Dominated Convergence Theorem, (cid:96) 0,kL (cid:96) 0 1 E l(Z ;θ ) → E l(Z ;θ ). 0 (cid:96) 0,kL 0 (cid:96) 0 53

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