Nonparametric Testing and Identification in Ascending Auctions with Unobserved Heterogeneity

Transcription

1 Nonparametric Testing and Identification in Ascending Auctions with Unobserved Heterogeneity Andrés Aradillas-ópez, Amit Gandhi, Daniel Quint June 3, 2010 Abstract When unobserved heterogeneity across auctions induces correlation in bidder valuations, the assumptions of the standard model are violated and ascending auctions are underidentified. We develop techniques to identify the relevant primitives when there is exogenous variation in the number of bidders in each auction. To justify this identifying assumption, we show that it implies a testable restriction on bid data which would likely be violated if it did not hold, and present an asymptotically-valid test statistic to see whether that restriction holds in the data. Applying our techniques to data from U.S. Forest Service timber auctions, we find the data to be consistent with our model; our identification techniques seem to rationalize the existing policy of using fairly low reserve prices, which has been criticized in the prior literature. 1 Introduction Applying the insights from auction theory to real world settings requires an empirical knowledge of the primitives that define the game being played by bidders. In the case of private value auctions, a key primitive of interest is the latent distribution of bidder valuations in the underlying population. In English, or ascending, auctions, this is in fact the only primitive of interest: the dominance solvability of the game renders the remaining characteristics of the bidders, such as their risk preferences or strategic sophistication, effectively moot. The distribution of private values represents the demand curve facing the seller; once demand is empirically known, a seller can determine the The authors thank Jeremy Fox, Phil Haile, Bruce Hansen, Ali Hortaçsu, Art Shneyerov, Chris Taber, seminar participants at Chicago Booth, Concordia, Duke, Michigan, Wisconsin and Yale, and conference attendees at IIOC 2009, SITE 2009, and IIOC 2010 for helpful comments. Aradillas-ópez acknowledges the support of the NSF through grant SES

2 optimal auction design parameters (such as the optimal reserve price), forecast expected revenue, and so on. If all auctions in the data are identical, all bidders independently bid their private values, and all bids are observable, then the identification of demand is trivial - the distribution of bids identifies the distribution of private values. Unfortunately, the actual data generating process represented by real-world auctions differs from this ideal in three important ways: (i) auctions in the data may be heterogenous, (ii) not all bidders can be presumed to bid their actual values, and (iii) not all bids are necessarily observed. The empirical literature on auctions has made remarkable progress on problems (ii) and (iii), but much less on problem (i). The main approach to dealing with auction heterogeneity is to assume that conditional on a vector of observed covariates Z, all auctions in a particular data set are identical, and bidder valuations can therefore be assumed to be identical and independent draws from a population distribution F V ( Z). Assuming that auction heterogeneity can be conditioned out in this fashion, it is possible to identify and estimate the underlying distribution of private values F V ( Z) in a way that solves problem (ii) (strategic bidding) and problem (iii) (censored data, such as having data on only the transaction price). Results of this form were first attained parametrically for the standard auction formats by affont and Vuong (1993) and Paarsch (1997), and nonparametrically by Guerre, Perrigne, and Vuong (2000), Athey and Haile (2002), and Haile and Tamer (2003). In this project, we seek to enrich the empirical approach to auctions by confronting issue (i), the heterogeneity of auctions. That is, we will dispense with the assumption that auction heterogeneity can be conditioned out, and instead allow for the presence of unobserved heterogeneity across the auctions in the data. What we aim to show is that such unobserved heterogeneity has significant effects for the analysis of auction data, and in particular for the inference of optimal reserve policy. Unobserved heterogeneity induces correlation among the private values of bidders in an auction, and reserve prices in a correlated environment entail (potentially significant) costs that would be underestimated under the standard assumption of independent private values. Indeed, our analysis of the United States Forest Service timber auction data reveals that this correlation suggests a much more cautious approach to reserve price policy, relative to the policy that would appear optimal under analysis based on the assumption that valuations in an auction were independent conditional on observables. Two recent papers by Krasnokutskaya (2009) and Hu, McAdams, and Shum (2009), building on work by i, Perrigne, and Vuong (2000), have addressed the issue of unobserved heterogeneity in firstprice auctions. By restricting unobserved heterogeneity to a scalar unobservable affecting private 2

3 values in a parametric or monotonic way, and by imposing regularity conditions on the distribution of the unobservable scalar, these authors have shown that it is possible to use the statistics of measurement error to recover the distribution of the unobservable, and hence by extension, the joint distribution of private values among bidders in a first price auction. However the key to this identification strategy is that there exist multiple informative bids at an auction that are related to the underlying valuations of bidders through the equilibrium of the game. Ascending auctions on the other hand are difficult to empirically analyze because only one bidder s valuation, the secondhighest, can be tightly pinned down by equilibrium bidding (Haile and Tamer (2003)). This makes it difficult to detect and measure the correlation among values; the issue of unobserved heterogeneity in ascending auctions thus remains very much an open question. In this paper we develop an empirical strategy for studying the implications of correlation among bidders at an ascending auction. Unlike the approach taken for first price auctions, our strategy does not place any restrictions on the dimensionality or the distribution of the unobserved heterogeneity that induces correlation at an auction. Not surprisingly, it will be impossible to identify the joint distribution of private values in an auction without any restrictions. However, despite the fact that the joint distribution of private values is fundamentally underidentified, we show there exist important features of the joint distribution that are identified, at least partially, using variation across auctions in the number of bidders. The features of demand that we do identify turn out to be sufficient for the purposes of the mechanism design. Our main identifying assumption is that the number of bidders participating in each auction is independent of the unobserved characteristics of that auction. (As we discuss below, this same assumption has been exploited in several recent papers on first-price auctions, to identify risk preferences and non-nash bidding, neither of which is a concern in ascending auctions.) Under this assumption, variation in transaction prices across auctions of different sizes can be used to pin down enough of the unobserved structure of demand in auctions of a fixed size to forecast the revenue implications of a change in reserve price. While exogeneity of the number of bidders can potentially be defended on a variety of grounds, it remains a strong identifying assumption; one of the major goals of our paper is to show that this assumption is testable. We develop an informative test of exogenous auction size, which takes the form of an inequality relating the distributions of winning bids across auctions of different sizes. (The test generalizes the nonparametric tests for independent private values in ascending auctions proposed by Athey and Haile (2002) and Haile and Tamer (2003).) We introduce a test statistic that is applicable to this set of inequalities, and show that it has a known asymptotic distribution. We thus construct (for the first time to our knowledge) an asymptotically valid nonparametric test of 3

4 two of the standard assumptions used in auction theory, private values and exogenous participation, which accounts for the presence of unobserved heterogeneity. Given exogenous variation in auction size, we construct upper and lower bounds on expected revenue and the profit-maximizing reserve price. We also show that partial identification results still obtain if our identifying assumption is violated in the natural direction: if valuations are stochastically higher in auctions with more bidders, the upper bounds we calculate for expected revenue remain valid, although the lower bounds do not. We then apply these methods to data from U.S. Forest Service timber auctions. We fail to reject the hypothesis of exogenous variation in auction size in our data. Our empirical implementation of the identification of seller s revenue suggests that accounting for unobserved heterogeneity rationalizes a much lower reserve price than analysis under the assumption of independence; this may answer the puzzle found in the existing literature of why reserve prices in timber auctions have historically been set so low, and also supports the change in policy by the USFS towards a transaction price method of setting appraisal prices for timber. 2 Correlated Demand in Auctions We use standard notation for private value auctions. N denotes the number of bidders in an auction, with n and n denoting generic values it can take. 1 Each bidder i has a valuation V i which is his or her private information, and gets payoff V i P from winning the auction (where P is the price paid) and 0 from losing. If there are n bidders at the auction, let the joint distribution of the valuations V =(V 1,...,V n ) be given by F V (v 1,...,v n Z), where Z is a vector of observables that characterize demand at the auction. Departing from the usual independent private value framework, we allow F V to be correlated even after controlling for Z. We assume that this correlation is induced by unobserved heterogeneity (from the perspective of the seller). That is, in addition to the observable covariates Z, thereexists a vector of unobserved characteristics θ Θ that affect demand at the auction. Bidder valuations for the object for sale in an auction with the complete vector of characteristics (Z,θ) are i.i.d. draws from a probability distribution F ( Z,θ). Since θ Θ is a demand shifter that is unobserved by the seller, the seller effectively faces the correlated demand environment given by n F V (v 1,...,v n Z) = F (v i θ, Z) dg(θ Z) θ Θ i=1 1 In principle, this should refer to the number of potential bidders, that is, the number of bidders who learned their valuations and considered participating. This is often assumed to be observed in the data; in our application, as we discuss later, there are specific institutional features that make this a reasonable assumption. 4

5 where G( Z) is the distribution of θ (conditional on Z). Thus, heterogeneity in the unobservable θ is the sole mechanism that induces correlation in bidders valuations, while any variation not caused by θ is idiosyncratic and independent across bidders. Since bidders valuations are assumed to be symmetrically distributed and independent conditional on (θ, Z), we will refer to this model as CIPV (conditionally independent private values), in contrast to the usual IPV assumption. For the remainder of this section, we will suppress the dependence on Z for ease of exposition, but everything is still implicitly being done conditional on Z. For an auction with n bidders, knowledge of the joint distribution F V (v 1,...,v n ) allows the seller to anticipate the revenue effects of changing auction design parameters. In the context of an ascending auction, the main auction design parameter of interest is the reserve price, a central question being whether there is any value in setting a strategic reserve price (i.e., a reserve price above the seller s own value), and if so what is the optimal amount of markup over opportunity cost that should be used to set the reserve. Under the assumption that the values V i at an auction are IPV, a key result in auction theory is the so called exclusion principle (that it is always in the seller s interest to use a strategic reserve), and the optimal markup for the seller is given by arg max r>0 (r v 0)(1 F V (r)) where v 0 is the seller s valuation for the unsold good. However if bidder values at an auction are correlated, neither result holds: it remains an empirical problem whether a strategic reserve price is profitable at all; and even if it is, the gains from using one, and the degree of the optimal markup, are likely much smaller in the correlated demand environment as compared to an IPV setting. 2 Intuitively, this is because the optimal reserve price trades off the decreased likelihood of a sale (due to the possibility that all bidders have valuations below the reserve) against the increased revenue in the event the reserve binds (just one bidder has a valuation above the reserve). Correlation among bidder valuations makes the former more likely when bidder values are correlated, they are more likely to all be low simultaneously while decreasing the value of the latter when bidder values are correlated, the top two are likely to be closer together, so the increase in revenue is smaller than under independence. In order to address the revenue implications of correlated demand, the auctioneer must have empirical knowledge of the joint distribution F V. et V k:n denote the k th lowest bidder valuation in auctions with n bidders, and F k:n its probability distribution. Haile and Tamer (2003) show that the assumptions that bidders have private values and play undominated strategies lead to sharp bounds on V n 1:n given the top two bids in an ascending 2 Quint (2008) shows that this is true when values are symmetric and affiliated: both the expected profit at any given reserve, and the profit-maximizing reserve price, are lower under affiliated values than under independent values consistent with the same observables. 5

6 auction. For expositional ease, we will make the stronger assumption that V n 1:n is exactly equal to transaction price, so that F n 1:n is point-identified from the data. This would be true to within the minimum bid increment under the assumptions of Haile and Tamer (2003) if there are no jump bids at the end of an auction; in our data, the winning bid is on average just 2% higher than the second-highest, so this is a reasonable approximation. (V n 1:n would be exactly equal to the transaction price in a so-called button auction, where bidders hold down a button to remain active and release it to drop out.) All of our results extend without difficulty to the more general bidding assumptions of Haile and Tamer (2003). In an IPV setting, the marginal distribution F n 1:n for any one n is sufficient to identify the parent distribution F V. In a correlated demand environment, however, the marginal distribution F n 1:n does not uniquely pin down the distribution F V, and is insufficient for mechanism design purposes. What we now show, however, is that the seller s profit problem in an auction of size n does not depend on the entire joint distribution F V, but only on the marginal distributions of the top two valuations, F n 1:n and F n:n. The fact that we do not even need to identify the joint distribution of these two order statistics, but only their marginals, to identify the seller s revenue curve is a key simplification that will enable us to empirically study the model presented above. 3 Theorem 1 In an ascending auction with n bidders and reserve price r, the seller s expected profit and each bidder s ex-ante expected surplus depend only on F n 1:n and F n:n. Proof. Again, we assume that transaction price equals V n 1:n,provideditexceedsthereserve price; if not, then the winner pays r. We show in Appendix A.1 that expected profit π n (r), and each bidder s ex-ante expected surplus u n (r), can be written as π n (r) = (F n 1:n (r) F n:n (r)) (r v 0 )+ + r (v v 0 )df n 1:n (v) (1) u n (r) = 1 n (E {max{v n:n,r}} E {max{v n 1:n,r}}) (2) where v 0 is again the seller s valuation for the unsold good. 4 3 In a different but analogous setting, Athey and Haile (2007) point out that identification of the joint distribution of (V n 1:n,V n:n) issufficientfor evaluationofrentextractionbytheseller,theeffectsofintroducingareserveprice, and the outcomes under a number of alternative selling mechanisms. Here, however, we do not rely on the joint distribution of V n 1:n and V n:n, onlythemarginal distribution of each. 4 Under two further assumptions that bidders observe θ in addition to their own valuations, and that they play Bayesian Nash equilibrium strategies (1) and (2) holds exactly for any standard auction format, that is, any mechanism satisfying the usual conditions for revenue equivalence. If bidders know θ, thenconditioningonany realization of θ, theyareinanipvsetting;soforagivenθ, anymechanismunderwhichtheobjectisalwaysallocated to the bidder with the highest valuation (provided it is greater than r), and a bidder with value less than r gets 6

7 Of course, once π n (r) is known, we can solve max r π n (r) to calculate the optimal reserve price for an auction with a fixed number n bidders. 5 Thus, if he is unable to observe θ, everythingthe seller could benefit from knowing is summarized in the distributions F n 1:n and F n:n. As we have already discussed, F n 1:n is directly revealed by the auction data; so what remains to be identified is F n:n. Of course V n:n is the key order statistic that is censored in an ascending auction - we never observe what the winning bidder would have been willing to pay since the auction ends before his reservation value is reached. Our main identification result is that variation in transaction prices across auctions of different sizes identify F n:n under the right assumptions. To see how this identification is achieved, consider the following thought experiment. Start with an n + 1-bidder auction, and randomly remove one of the bidders. With probability n n+1,thebidder with the highest valuation was not removed, and so the highest of the n remaining is the highest of the original n + 1. With probability 1 n+1, the bidder with the highest valuation was removed, in which case the the highest of the remaining n chosen is the second-highest of the original n + 1. If we let F m n be the joint CDF of n bidders chosen at random from an m-bidder auction, and F m k:n the marginal distribution of the k th lowest of the n, this tells us that Fn:n n+1 (v) = 1 n +1 F n:n+1 n+1 (v)+ n n +1 F n+1:n+1 n+1 (v) (3) (This relationship among order statistics is a special case of Equation 9 from Athey and Haile (2002). 6 ) However, it is not F n+1 n:n we want to identify, but F n n:n. Thus, in order to make useful inferences from (3), we need to be able to make comparisons between bidders in auctions of different sizes. The identifying assumption behind most of our results is that the number of bidders at an auction is independent of their valuations: Definition 1 N is exogenous (or there is exogenous variation in auction size) if for any (n, m, m ) with m, m n, F m n = F m n. That is, N is exogenous if the distribution of valuations of a set of bidders is independent of the size of the auction they were taken from the three bidders in a three-bidder auction are expected payoff 0, would be revenue-equivalent to the second-price sealed-bid auction, which satisfies (1) and (2) exactly. Averaging over θ, then,wouldcompletetheproof. 5 Under independent private values, the optimal r does not depend on n; butwithcorrelatedvalues,itdoes. Aseller who must select a reserve price without knowing n would solve max r n p(n)πn(r), where p(n) istheprobabilityof n bidders participating; as we discuss later, in our particular application, it is reasonable to assume the seller knows n when selecting a reserve price. 6 Note that (3), and the identification results that follow from it, require only exchangeability of the joint distribution F V,nottheadditionalstructureofconditionalindependence. However,bothourtestandthebounds we place on the trailing term of (4) below do require conditional independence. 7

8 indistinguishable (in expectation) from three bidders chosen at random from a four-bidder auction. Within the context of our model, this can be interpreted simply as N being independent of θ (and independent of the realizations of the draws of bidder values from the distribution F ( θ)). Exogenous N leads to the following result: Theorem 2 If N is exogenous, F n:n (v) = for any n >nand any v. n m=n+1 1 n 1 m 1 i=n i 1 F m 1:m (v) + n n i +1 F n: n(v) (4) Proof. A full proof is given in Appendix A.2. Exogenous N means F n n:n = F n+1 n:n, so (3) becomes F n:n (v) = 1 n+1 F n:n+1(v)+ n n+1 F n+1:n+1(v); iterating gives (4). In Sections 3 and 4, we will introduce an indirect test of the assumption of exogenous N, and, finding it to be supported in our data, use Theorem 2 as the basis for our identification and counterfactual analysis. As n grows, the coefficient on the trailing term in (4) vanishes, so in the limit where F m 1:m is learned for every m = n +1,n+2,..., exact identification of F n:n is achieved under exogenous N. In the case where n cannot be taken to infinity, we will place bounds on the trailing term F n: n (v) to get bounds on F n:n. (This is done in detail in Section 5.) In some applications, however, N may not be exogenous, leading (hopefully) to a rejection of exogenous N by our test. When our identifying assumption fails, it is likely to fail in a predictable direction: we would expect auctions with more bidders to also, on average, have higher-valued objects. When this is the case, partial identification still holds. To show this, we first relax Definition 1 to formalize what it means to have positive correlation between auction size and valuations. Definition 2 N is positively associated with valuations if for any n, F n+1 n:n FOSD F n n:n. That is, N is positively associated with valuations if auctions with more bidders also tend to have bidders with weakly higher valuations three bidders chosen at random from a four-bidder auction have a stochastically greater highest valuation than the three bidders in a three-bidder auction. When this is the case, we get partial identification: specifically, we can calculate a pointwise lower bound on F n:n, which leads to a pointwise upper bound on π n ( ). Theorem 3 If N is positively associated with valuations, n m 1 1 i 1 F n:n (v) F m 1:m (v) + n n n 1 i +1 F n: n(v) (5) for any n >nand any v. m=n+1 i=n 8

9 Proof. Again, see Appendix A.2 for a complete proof; by assumption, F n+1 n:n FOSD F n n:n, and so F n n:n(v) F n+1 n:n (v) = 1 n+1 F n:n+1(v)+ n n+1 F n+1:n+1(v); iterating gives (5). In this paper, we remain agnostic as to the actual process determining bidder participation; we simply test whether the assumption of exogenous N is supported by the data, and show how it leads to identification of the model. (Even if N is the endogenous outcome of an entry game such as the one described by evin and Smith (1994), this assumption would still be satisfied if potential bidders had the same information as the seller prior to entry that is, if bidders only observed θ after deciding whether to participate in an auction.) We do not explicitly test for positive association between N and valuations; we see it as the natural alternative hypothesis to exogenous N, and therefore view the partial results it allows as robustness checks for the conclusions under exogenous N. Exogenous N has been used as an identifying assumption in several papers on first-price auctions with independent signals: by Guerre, Perrigne, and Vuong (2009) to identify the coefficient of risk aversion; by Haile, Hong, and Shum (2003) to test between common and private values; and by Gillen (2009), to identify the distribution of behavioral types in a level-k model. We, on the other hand, work in a setting where private values are generally believed to hold, and consider ascending auctions, in which strategic sophistication and risk preferences play no role; thus, we are able to use this assumption to identify a complexity that these other papers must assume away: unobserved heterogeneity. 7 What Theorems 1 and 2 establish is that within our model of conditionally independent private values, exogenous variation in auction size allows identification of the primitives necessary to calculate expected revenue (as a function of reserve price), and therefore revenue-maximizing reserve price. In addition, while this assumption has been used several times, it has never been formally tested. In our context, exogenous N imposes nonparametric restrictions on bid distributions, which we argue would likely be violated if the assumption failed. We can therefore use these restrictions to design a nonparametric test of the assumption of exogenous N. The rest of the paper proceeds as follows. In section 3, we show that exogenous variation in N implies a non-parametrically testable restriction in the CIPV framework. The restriction is a generalization of an equality test proposed by Athey and Haile (2002) for IPV ascending auctions, to an inequality test applicable to auctions with unobserved heterogeneity. We show that this test has power against the most likely sources of endogenous auction size, positive correlation between higher N and auction primitives favoring high valuations. In section 4, we introduce nonparametric test 7 Both Guerre, Perrigne, and Vuong (2009) and Haile, Hong, and Shum (2003) make some attempt to control for unobserved heterogeneity in first-price auctions, but under much stronger assumptions than we require. 9

10 statistics for the inequality restrictions implied by the model, and derive their asymptotic properties. We apply this test to data from U.S. Forest Service timber auctions, and show that the data fails to reject exogenous variation in N. In section 5, we use Theorem 2 to calculate bounds on F n:n, and as a result π n and u n,whenn is exogenous, and show which bounds still hold when N is positively associated with valuations. We then apply this identification strategy to the same USFS timber data, and demonstrate the counterfactual expected revenue and optimal reserve implications. These implications are substantially different from the implications that would follow from standard analysis under the assumption of independent values, and support the cautious approach to reserve price policy that the timber service historically has taken. Section 6 concludes. Proofs and examples omitted from the text are in Appendix A; proofs of the asymptotic properties of the test statistics are in a separate technical appendix, which also includes a discussion of alternative formulations of the test statistics. 3 Testable Implications of Exogenous Auction Size 3.1 Benchmark: Testing under IPV Our test for exogenous variation in auction size within our model is most easily understood if we first introduce an analogous test within the independent private values framework. For n>1, define a function ψ n 1:n :[0, 1] [0, 1] by ψ n 1:n (s) = ns n 1 (n 1)s n (6) This function has the property that given n independent draws from a probability distribution F ( ), the CDF of the second-highest of the draws is ψ n 1:n (F ( )). In the symmetric IPV model, valuations of participating bidders are assumed to be independent draws from some probability distribution F V,soF n 1:n (v) =ψ n 1:n (F V (v)), or 8 ψ 1 n 1:n (F n 1:n(v)) = F V (v). Since we assume F n 1:n is identified from bid data for each n, a test of the symmetric IPV model with exogenous N is therefore whether ψ 1 n 1:n (F n 1:n(v)) = ψ 1 n 1:n (F n 1:n(v)) (7) for each pair (n,n). (This is the test considered in Theorem 1 of Athey and Haile (2002).) 3.2 Our Test under CIPV Returning to our model of conditionally independent private values, the assumption of exogenous N would imply that the valuations of bidders participating in an auction of size n are independent 8 From (6), it is straightforward to show that ψ n 1:n is strictly increasing and onto, and therefore invertible. 10

11 draws from a distribution F ( θ), where θ is independent of N. To see why the test above is not still valid, consider the extreme case of perfectly correlated values, where θ is one-dimensional and each bidder s valuation in an auction with characteristics θ is simply θ. In this case, F n 1:n (v) for n 2 is simply the ex-ante distribution of θ, and therefore does not vary with n; ψ 1 n 1:n (F n 1:n(v)), then, is strictly increasing in n, and (7) fails. However, under CIPV, the way in which the test fails is predictable, and leads us to a similarlymotivated inequality test. Under our model, F n 1:n (v) must still be weakly decreasing in n; butwe show below that under CIPV with exogenous N, it decreases more slowly than it would under IPV, and so ψ 1 n 1:n (F n 1:n(v)) must be increasing in n. We prove this result, then argue why this test should have power against the most likely source of endogeneity. Theorem 4 Under symmetric, conditionally independent private values with exogenous N, for any v, F n 1:n (v) is decreasing in n and ψ 1 n 1:n (F n 1:n(v)) is increasing in n. Proof. For the first result, note that ψ n 1:n (s) is decreasing in n. 9 At a given realization of θ, the distribution of V n 1:n is ψ n 1:n (F ( θ)); so the unconditional distribution of V n 1:n, is decreasing in n as well. F n 1:n (v) = E θ {ψ n 1:n (F (v θ))} For the second result, it suffices to show ψ 1 n:n+1 (F n:n+1(s)) ψ 1 n 1:n (F n 1:n(s)). The key step of the proof is the observation that ψ n 1:n ψn:n+1 1 :[0, 1] [0, 1] is concave. For any two differentiable functions f and g with g invertible, f g 1 (s) =f g 1 (s) g 1 f (s) = (g 1 (s)) g (g 1 (s)),yielding where t = ψ 1 n:n+1 (s). ψn 1:n ψn:n+1 1 ψ (s) = n 1:n (t) ψn:n+1 (t) = n(n 1)tn 2 (1 t) (n+1)nt n 1 (1 t) = n 1 n+1 1 t So ψ n 1:n ψn:n+1 1 (s) = establishing that ψ n 1:n ψn:n+1 1 is concave. Jensen s inequality then yields n 1 n+1 ψ 1 n:n+1 (s), which is decreasing in s, ψn 1:n ψn:n+1 1 (Fn:n+1 (v)) = ψ n 1:n ψn:n+1 1 Eθ {ψ n:n+1 (F (v θ))} E θ ψn 1:n ψ 1 n:n+1 = E θ {ψ n 1:n (F (v θ))} = F n 1:n (v) (ψn:n+1 (F (v θ))) so ψ 1 n:n+1 (F n:n+1(v)) ψ 1 n 1:n (F n 1:n(v)). Again, since we assume F n 1:n for each n can be inferred from bid data, Theorem 4 gives us two testable inequality restrictions implied by exogenous auction size under the assumption of symmetric, 9 For n>2, it s just algebra to show that ψ n 1:n (s) ψ n 2:n 1 (s) = (n 1)s n 2 (1 s)

12 conditionally independent private values: specifically, that for any n, n and any v. 10 n>n F n 1:n (v) F n 1:n(v) (8) n>n ψn 1:n (F n 1:n(v)) ψ 1 n 1:n (F n 1:n(v)) (9) 3.3 Power Against Endogenous N Since we cannot directly test for exogenous N, only for a consequence of it, it remains to argue why we expect this test to have power; that is, why endogeneity of N would likely lead to a violation of either (8) or (9). The logic is as follows. Under IPV with exogenous N, as n increases, the distribution of V n 1:n shifts to the right, so F n 1:n (v) decreases; but the function ψn 1:n ( ) increases, and the two effects exactly balance each other, so ψ 1 n 1:n (F n 1:n(v)) is unchanged. Under CIPV with exogenous N, the change in ψ 1 n 1:n ( ) is the same as before, but ψ 1 n 1:n (F n 1:n(v)) is now increasing rather than constant; thus, the second part of Theorem 4 therefore says that F n 1:n (v) must decrease in n more slowly under CIPV than it would under IPV. Anything that speeds up the dependence of F n 1:n (v) on n, beyond the rate of change under IPV, would lead to a violation of (9), and therefore a failure of the test. The most plausible source of endogeneity of auction size would be if bidders were more prone to enter auctions for more valuable objects either auctions with characteristics θ which make them likely to be more valuable, or auctions for objects that the bidder already knows he values more highly. Such positive selection would mean that auctions with larger n were more likely to have high prices, beyond the natural effect of drawing from a larger pool of bidders; this is exactly the behavior that the test (9) will detect. Of course, since selection increases the dependence of F n 1:n on n but correlation weakens it, the two effects could potentially cancel out and be undetected if selection were weak enough; but below we show that under fairly general conditions, it will not. To formalize this intuition, let E θ n denote an expectation taken over the distribution of θ among auctions with exactly n bidders. Exogenous N is the assumption that θ is independent of N, sothis expectation would be the same for every n. Below, we show that if θ N and auctions with more bidders tend to have characteristics which put more weight in the upper tail of the distribution of bidders values, this will always lead to a violation of (9). 10 Under the incomplete bidding model of Haile and Tamer (2003), with G n 1:n and G n:n the (observable) distributions of the second-highest and highest bids in n-bidder auctions, (8) and (9) would become G n:n(v δ) G n 1:n(v) andψ 1 n 1:n (G n 1:n(v)) ψ 1 n 1:n (G n :n(v δ)), where δ is the minimum bid increment. With some modification, the tests we describe below could be applied to these inequalities instead. 12

13 Theorem 5 Suppose that for each θ, F ( θ) is continuous and twice differentiable, with derivative f( θ), andhasthesameboundedsupport[v, v]. If E θ n (f(v θ)) 2 >E θ n (f(v θ)) 2 then ψn 1:n (F n 1:n(v)) <ψ 1 n 1:n (F n 1:n(v)) for v sufficiently close to v. Thus, if N is endogenous and higher n are associated with realizations of θ putting more weight f(v θ) at the top of the range of bidder values, we are guaranteed to have a violation of (9). For example: Corollary 6 In addition to the continuity and differentiability assumptions of Theorem 5, suppose Θ and f(v θ) is increasing in θ. et p n denote the distribution of θ conditional on N = n. If n>n and p n FOSD p n, then (9) will be violated for v sufficiently close to v. Proof. Since (f(v θ)) 2 is increasing in θ, p n FOSD p n implies E θ n (f(v θ)) 2 >E θ n (f(v θ)) 2, giving ψn 1:n (F n 1:n(v)) <ψ 1 n 1:n (F n 1:n(v)) for v close to v. In Appendix A.3, we prove Theorem 5. We then show general conditions under which the endogenous entry models of evin and Smith (1994) and Samuelson (1985) would both lead to violations of (9) under Corollary 6. We solve examples of these two entry models to show that the curves ψ 1 n 1:n (F n 1:n(v)) are indeed incorrectly ordered (relative to (9)) over a large part of their support. Negative selection bidders being more prone to participate in auctions for less-valuable objects would not lead to a violation of (9); but if it were a strong enough effect, it would lead to a violation of (8). So strong negative selection should also be detected. 11 however, might not be detected. 12 Weak negative selection, (While counterintuitive, negative selection could occur naturally if more valuable objects also tend to be valued more consistently by different buyers. This could be the case, for example, if the 11 In settings where exogenous N is taken for granted, a violation of (8) (a left-shift in the distribution of winning bids as n increases) is interpreted as an indication of common values, since bidders shade their bids more as n increases due to a stronger winner s curse. (See, for example, Hong and Shum (2002), and the discussion in Bajari and Hortacsu (2003).) Here, we take private values as given, and argue that such a left-shift must therefore indicate negative selection. 12 In theory, we could replace (8) with the stronger result that F n 1:n (v) F n 2:n 1 (v) (n 1)(n 3) (F n 3:n 2 (v) F n 2:n 1 (v)) 2 (n 2) 2 F n 4:n 3 (v) F n 3:n 2 (v) to give a stronger test against negative selection. In practice, however, due to the difference term in the denominator, this latter test is highly unstable when applied to a reasonably-sized dataset. 13

14 most valuable objects tended to be purchased by professional dealers, who were fairly unanimous in their assessments, while lower-value objects appealed to individual collectors, whose tastes were more idiosyncratic. Still, we feel intuitively that positive selection is the more likely source of endogeneity in N; thus, we expect (9) to be the more meaningful test.) 3.4 Description of Data We apply our test to data from timber auctions run by the Unites States Forest Service. These auctions have received a great deal of attention in the empirical auctions literature; but except for some efforts by Haile, Hong, and Shum (2003), the literature has thus far ignored the issue of unobserved heterogeneity. Thus we hope this application to be fruitful as it is a widely studied auction in which the role of heterogeneity has not been addressed. These auctions proceed as follows. The forestry service first conducts a cruise of the tract and publishes detailed information on the tract for potential bidders. It also announces an appraisal value for the tract, which serves as a reserve price for the qualifying round of bids: bidders must submit sealed bids of at least that reserve price to be eligible to participate. The oral round of the auction then begins at the highest of these sealed bids. Data on these auctions comes to us from Phil Haile, who has posted timber auction data from 1978 to We have data on the size of the tract, the estimated volumes of the various timber species, estimated costs of harvesting and delivering the timber to mills, and estimated revenue from such sales, in addition to other measures. Potential bidders can also conduct their own cruises (which is rarely performed for the subset of auctions we consider). It is commonly understood that the reserve prices set by the Forest Service are too low and do not bind. A particular empirical question of interest for us is whether the Forest Service should be setting a higher reserve to extract greater rents, which has been a subject of ongoing debate within the timber service. We use the cleaning conventions of Haile and Tamer (2003) to select auctions which are most likely to satisfy the assumption of private values. We focus on sales in Region 6 (which encompasses mostly Oregon) and select sales whose contracts expire within a year, to shut down the effect of resale possibilities on valuations. We focus on scaled sales, where common-value uncertainty about the total amount of timber should not affect valuations. What is left is a sample of auctions in which the private values assumption is thought best to hold. A unique feature of the data is that we have clean observations on the number of bidders in the room when the bidding begins, which is crucial to both our test and our identification strategy. A number of other papers have considered these Forest Service auctions. Baldwin, Marshall, and Richard (1997) provide much institutional background. Their focus is to test for collusion. 14

15 Haile (2001) considers the effects of resale on valuations. Haile, Hong, and Shum (2003) consider testing for common values against private values, assuming away (for the most part) unobserved heterogeneity. u and Perrigne (2008) use the USFS data to estimate risk aversion among bidders, using the fact that the service conducts both first price sealed bid auctions and open ascending auctions. Finally, Athey and evin (2001), Athey, evin, and Seira (2008), and Haile and Tamer (2003) analyze the data to empirically study mechanism design issues. 13 All papers besides a few elements of Haile, Hong, and Shum (2003) do not consider the effects of unobserved heterogeneity in drawing inferences from the data, which is our current concern. An interesting feature of the data is that the observable auction characteristics broadly appear to be independent of the number of bidders. As we discuss below, the government s appraisal value is the most significant covariate in explaining transaction price; but it has almost no relation to N. (If we regress N on appraisal value or its log, we get a slightly negative slope and an R 2 less than 0.01.) Other auction-specific covariates also do not appear significantly related to N. Sincebidders do not appear to be selectively participating in auctions based on their observable characteristics, it might be reasonable to think that they are also not selectively participating based on unobservable characteristics.we now move on to testing this assertion. 3.5 Visual Test Recall that Theorem 4 gives two strings of inequalities which must hold under conditionally independent private values with exogenous N: F 1:2 (v) F 2:3 (v) F 3:4 (v) ψ 1 1:2 (F 1:2(v)) ψ 1 2:3 (F 2:3(v)) ψ 1 3:4 (F 3:4(v)) (10) As discussed above, we assume that the transaction price in each auction in the data was exactly equal to the second-highest valuation V n 1:n. As a first check, then, we can simply construct empirical analogs of the distribution functions F n 1:n (v) directly from bid data, plot F n 1:n (v) and ψn 1:n Fn 1:n (v) against v for various n, and check whether these distributions satisfy (10). Even without formalizing this into a proper test, this should give some intuition for whether exogenous 13 One of the key insights of Athey, evin, and Seira (2008) is that there exist ex-ante asymmetries between two types of bidders: millers and loggers. This does not contradict our model. Suppose bidders of each type have private values drawn from the distributions F m(v i θ, Z) andf l (v i θ, Z), respectively, and that a fraction q(θ, Z) of bidders are expected to be millers. As long as each bidder is imagined to be randomly (independently) either a miller or a logger, each bidder s valuation can still be seen as an independent draw from F (v i θ, Z) =q(θ, Z)F m(v i θ, Z)+(1 q(θ, Z))F l (v i θ, Z). Since beliefs about other bidders valuations do not affect bidding in a private-value ascending auction, our model still applies, although our identifying assumption now requires that q be independent of N. 15

16 auction size is a plausible hypothesis given our data, or whether either positive or negative selection seems a likely problem. For a first pass, we let θ capture all auction heterogeneity, that is, we make no attempt to control even for observable covariates. Figure 1 shows F n 1:n (v) and ψn 1:n Fn 1:n (v) as n varies from 2 to 8. (For visual ease, colors go in rainbow order red, orange, yellow, green, blue, indigo, violet as n increases from 2 to 8.) Visual inspection shows that these curves do indeed shift nearly monotonically in the predicted direction as n changes. Thus, without controlling for any observed heterogeneity, the bid data seems to be consistent with our model and the assumption of exogenous auction size. Figure 1: The eyeball test for exogenous auction size (assuming V n 1:n = transaction price) F n 1:n (v) (should shift right as n increases) ψn 1:n Fn 1:n (v) (should shift left as n increases) Of course, in reality, auction-by-auction heterogeneity will typically be a mix of observed and unobserved variables. If X denotes the observed covariates and θ (as before) the unobserved, our test above was really a test of whether valuations are i.i.d. draws from a distribution F ( X, θ), with exogenous N meaning that N is independent of (X, θ). If this is true, then conditional on a realization of X, N is still independent of θ; and so if our unconditional model is valid, there is no need to worry about it becoming invalid when we condition on any subset of the covariates. However, conditioning on observable covariates is potentially appealing for several reasons. First, if N is independent of θ but not of X, then our hypothesis of exogenous N (and thus our identification strategy) is invalid when we do not condition, but valid when we condition on X. Second, we pointed 16

17 out above that positive correlation between N and valuations would speed up the rate at which F n 1:n shifts to the right as n increases, while correlation slows it down; thus, heterogeneity can in a sense conceal some degree of endogeneity of N. By conditioning on X, we remove some of the heterogeneity that could do this, giving us a more powerful test. So conditioning on available covariates makes our model more likely to be valid, and more likely to be rejected if it is not. And finally, we would like to know whether our model is in some sense necessary that is, whether there truly is unobservable heterogeneity inducing correlation, or whether the data could be well described by the standard (IPV) model when all observable covariates were controlled for. As we discuss further in section 5, we can mimic the analysis presented in u and Perrigne (2008) to show that aside from N, the most important observable covariate for explaining variation in the winning bid is the appraisal price, given in 1983 dollars per thousand board feet. (u and Perrigne (2008) find the volume of timber to also have explanatory power; however, we focus only on scaled sales, where we would not expect volume to play a significant role, as we confirm empirically.) As a first approximation, then, we capture the information contained in observable covariates through appraisal price. We discuss the role of appraisal price as a conditioning variable further in Section 5. We now replicate the visual test (Figure 1 above) with nonparametric estimates of the distributions of transaction prices for each N, conditional on appraisal value. etting (W i,n i,x i ) i=1 denote the transaction price, number of bidders, and appraisal value observed in the i th auction in the data, we construct a Nadaraya-Watson kernel estimate for each distribution F n 1:n (v X) as i=1 {N i = n}k b (X X i ) {W i v} F n 1:n (v X) = i=1 {N i = n}k b (X X i ) where K b is the Gaussian kernel with bandwidth b. We use a bandwidth based on the rule of thumb used by Haile and Tamer (2003) for the same data. We can then apply the same visual test as before, to these conditional CDFs. Figure 2 shows graphs of F n 1:n ( X) and ψn 1:n Fn 1:n ( X) for three representative appraisal values, 26, 56, and 92 the 25 th, 50 th, and 75 th percentiles of appraisal values in the data. Our test is again the claim that in each row, the curves in the first column shift to the right as n increases, while the curves in the second column shift to the left. Examining Figure (2), there are two main features to notice. The first is that by controlling for the observable auction heterogeneity through appraisal price, there is less heterogeneity in the residual θ to slow down the competition effect, and hence the curves in the right column are all much closer together than the unconditional case, which is exactly what the theory predicts. Second, the data seems to fit the hypothesis of exogenous N fairly well. All three plots of F n 1:n (v X) showa clear shift to the right as n increases, although there is some crossing of curves in a few spots. As 17

18 Figure 2: The same eyeball test, using estimates of F n 1:n conditional on appraisal value F n 1:n (v X), X = 26 ψn 1:n Fn 1:n (v X), X = 26 F n 1:n (v X), X = 56 ψn 1:n Fn 1:n (v X), X = 56 F n 1:n (v X), X = 92 ψn 1:n Fn 1:n (v X), X = 92 18

19 for the plots of ψn 1:n Fn 1:n (v X), for X = 56 and 92, these are also for the most part ordered correctly (shifting to the left as n increases); at X = 26, for v above about 75, the curves are virtually indistinguishable from each other. Thus, the data again appears to be broadly consistent with our predictions. While both of these eyeball tests seem to suggest the data is consistent with our model and exogenous N, we would still like to develop a more formal test of this. Visually, we can see the curves in the graphs of the conditional test cross in certain places, and hence there is the natural question of the actual statistical significance of these visual events. In the next section we move on to a formal econometric test of (8) and (9), to see whether the occasional apparent violations of these predictions are explained by the sample size or refute the model. In addition, the assumption of IPV implies the added restriction that (9) holds everywhere with equality, which appears not to hold in Figures 1 and 2; we naturally would like to ask whether the curves represent real econometric rejections of the equality test proposed in Athey and Haile (2002). 4 Econometric Test Next, we introduce formal econometric tests for the inequalities (8) and (9). We construct a test statistic for each inequality which is calculated from observed auction data; asymptotically, each one will diverge to + if the corresponding inequality is violated anywhere; converge to 0 in probability if it holds everywhere as a strict inequality; and converge to a normal distribution with mean 0 and bounded variance if it holds with equality on a positive measure of v. ee, inton, and Whang (2009) have recently described a test for stochastic monotonicity; their test follows a Kolmogorov-Smirnov (K-S) type of approach, based on the supremum of a properly normalized statistic. This approach could be used for one of our two tests, a comparison of two distributions functions; however, it is not designed to test the other, which involves a nonlinear transformation of one of the distributions. By its nature, it seems unavoidable that any test of (9) must rely on an explicit estimation of a distribution function. A K-S approach in this setting would have the drawback of leading to a nonpivotal asymptotic distribution (see Remark 2.1 in ee, inton, and Whang (2009)). Consequently, critical values for the resulting test would have to rely on simulation or resampling. Rather than relying on this approach, our test is based on the expected value of a properly-designed functional. The key advantage of our approach is that the critical values come from a known distribution. The asymptotic analysis of our test statistic is also simpler than the one that would follow from a K-S approach. 19

20 Our test is based on functionals whose expected values are zero if and only if (8) and (9) are satisfied with probability 1. Similar functionals were also used by Khan and Tamer (2009) in a setting of censored regression. As in their case, the sample analogs of the functionals we use take the form of U-statistics. The rejection rules that we recommend are based on the asymptotic properties of these U-statistics. For clarity, we first present the test without conditioning on any auction-specific covariates. After that, we will extend the test to be run conditional on observable covariates; we run the test both unconditionally and conditional on appraisal value. 4.1 Test Statistic Our unconditional test statistics are based on the realizations of N and V N 1:N in each auction in the data. We show that the inequalities (8) and (9) hold for every pair (n, n ) and almost every v if and only if a certain pair of functions have mean zero. We characterize these functions and develop a test for these moment restrictions. We continue to assume that the transaction price in each auction exactly reveals the second-highest valuation; the test could easily be modified to work under the weaker bidding assumptions of Haile and Tamer (2003). We assume we have observations from ascending auctions, indexed by i {1, 2,...,}. For each auction i, we observe the winning bid (transaction price), W i, and the number of bidders N i. We assume these observations Z i =(W i,n i ) are i.i.d. draws from some ex-ante distribution of auctions Z =(W, N). et p N (n) =Pr(N = n), F W N (w n) =Pr(W w N = n), and F W (w) =Pr(W w) be the unconditional distribution of N, the conditional distribution of W given N, and the unconditional distribution of W under the true data-generating process. et S Z denote the joint support of (W, N), and S W and S N the marginal supports of W and N, respectively. We assume that p N (n) > 0 for every n S N, and that F W N ( n) is continuous for each n. The function ψ n 1:n ( ) was defined in (6). For any pair n, n S N,define Ω(s, n, n ) = ψ n 1:n ψ 1 n 1:n (s) W N (w, n, n ) = F W N (w n) F W N (w n ) Φ W N (w, n, n ) = Ω F W N (w n ),n,n F W N (w n) so (8) and (9) are exactly the conditions that W N (w, n, n ) 0 and Φ W N (w, n, n ) 0 for n>n. et N S N be a compact, pre-specified set. For any n, n N with n = n,letw n,n S W be any compact subset on which both F W N ( n) and F W N ( n ) are strictly bounded away from 20

21 0 and 1. We will test whether W N (w, n, n ) 0 and Φ W N (w, n, n ) 0 for almost all w W n,n, for all (n, n )withn, n N and n>n. Throughout the following, this is what we will mean by almost everywhere. Take any distinct i, j {1,...,}. For any n, n N, consider the two functions T F W N n,n (Z i,z j ) = T Ω W N n,n (Z i,z j ) = {W i W j } F W N (W j n ) {N i = n} W N (W j,n,n ) 0 {W j W n,n } Ω F W N (W j n ),n,n {W i W j } {N i = n} ΦW N (W j,n,n ) 0 {W j W n,n } (11) and let µ F W N n,n E T F W N n,n (Z i,z j ) and µ Ω W N n,n E T Ω W N n,n (Z i,z j ) (12) Theorem 7 If Z i and Z j are i.i.d., then for any n, n N, µ F W N n,n 0 and µω W N n,n for n>n, 0. Further, (i) µ F W N n,n =0if and only if F W N(w n) F W N (w n ) for almost all w W n,n (ii) µ Ω W N n,n W n,n =0if and only if ψ 1 n 1:n FW N (w n) ψ 1 n 1:n FW N (w n ) for almost all w Proof. To prove part (ii), fix (N i,z j ), and define T Ω W N n,n (N i,z j) E Wi N i,z j T Ω W N n,n (Z i,z j) = E Wi N i Ω F W N (W j n ), n, n {W i W j} {N i = n} ΦW N (W j, n, n ) 0 {W j W n,n } = Ω F W N (W j n ), n, n F W N (W j N i) {N i = n} ΦW N (W j, n, n ) 0 {W j W n,n } = {N i = n} {W j W n,n } max 0, Φ W N (W j, n, n ) so T Ω W N n,n (N i,z j ) 0, and T Ω W N n,n (N i,z j ) > 0 if and only if N i = n, W j W n,n, and Φ W N (W j,n,n ) > 0. By iterated expectations, µ Ω W N n,n = p N (n) E max 0,Φ W N (W j,n,n ) Wj W n,n, which is therefore weakly positive, and strictly positive if and only if Φ W N (W j,n,n ) > 0occurs with positive probability on W n,n. The analogous result holds for µ F W N n,n and W N(W j,n,n ). Thus, T Ω W N n,n (Z i,z j ) is a function whose expected value is 0 if (9) holds almost everywhere in W n,n and is strictly positive if (9) is violated with positive probability on W n,n ; and likewise T F W N n,n (Z i,z j ) with (8). Therefore, summing over different pairs (n, n ), the sums n,n N,n>n T Ω W N n,n (Z i,z j ) and n,n N,n>n T F W N n,n (Z i,z j ) will similarly have expected value 21

22 0 if (8) and (9) hold almost everywhere in W n,n for each n>n in N, and each will have strictly positive expected value if the corresponding inequality is violated with positive probability. principle, then, we would like to calculate sample averages of these sums over the pairs (Z i,z j )in our data, and test whether the resulting averages are significantly different from 0. Sample analog estimators would take the form of (second order) U-statistics. U-statistics arise as generalizations of sample averages. They were introduced by Hoeffding (1948) and Halmos (1946); for a detailed overview of their general properties, see Chapter 5 in Serfling (1980). In many instances (but not always), statistics of this class have an asymptotically normal behavior. However, we cannot directly calculate T Ω W N n,n (Z i,z j ) and T F W N n,n (Z i,z j ), since the distributions F W N, and therefore the functions W N and Φ W N, are not known exactly; we therefore replace them with nonparametric estimates. In order to ensure the asymptotic properties of the test statistics are well-behaved, we exclude observations i and j from the estimates of F W N T Ω W N n,n (Z i,z j ) and T F W N n,n (Z i,z j ). That is, we define R i,j W N (w n) 1 2 =i,j {W w} {N = n} p i,j(n) 1 N 2 =i,j {N = n} In that are used in F i,j W N (w n) R i,j W N (w n) p i,j (n) if p i,j(n) = 0 N N 0 otherwise and calculate corresponding estimates for W N and Φ W N i,j W N (w, n, n ) = F i,j W N (w n) F i,j W N (w n ) Φ i,j W N (w, n, n ) = Ω F i,j W N (w n ),n,n F i,j W N (w n) In addition to replacing F W N, W N, and Φ W N with estimates, we make one other modification to (11). Our null hypothesis allows for both (8) and (9) to hold strictly or with equality. To deal with the possibility that either holds with equality with positive probability, we introduce a nonzero bandwidth b, which disappears at a rate slower than, so that we can characterize exponential bounds for the probability that i,j W N (W j,n,n ) < b and W N (W j,n,n ) 0 (or that Φ i,j W N (W j,n,n ) < b and Φ W N (W j,n,n ) 0). Thus, for actual calculation, we define T F W N n,n (Z i,z j ) = {W i W j } F i,j W N (W j n ) {N i = n} i,j W N (W j,n,n ) b {Wj W n,n } T Ω W N n,n (Z i,z j ) = Ω F i,j W N (W j n ),n,n {W i W j } {N i = n} Φ i,j W N (W j,n,n ) b {Wj W n,n } 22 (13)

23 Our test statistics are then the sample averages of T F W N n,n and T Ω W N n,n,summedoverthedifferent (n, n ) combinations: U T F W N (2) = U T Ω W N (2) = 1 ( 1) 1 ( 1) i,j {1,...,} i=j i,j {1,...,} i=j T F W N n,n (Z i,z j ) n,n N n>n T Ω W N n,n (Z i,z j ) n,n N n>n Appendix A.5 provides regularity conditions for the distributions F W N under which we can fully characterize the asymptotic distribution of these test statistics. 14 The results are given in Theorem 13; the proof is contained in a separate technical appendix. In brief, we define residual functions η F W N N (Z i ) and η Ω W N N (Z i ), and show that under appropriate regularity conditions, F U T W N = (2) n,n + 1 η F W N N (Z i )+o p (1) Ω U T W N = (2) n,n N,n>n µ FW N µ ΩW N n,n n,n N,n>n i=1 + 1 i=1 η Ω W N N (Z i )+o p (1) with E[η F W N N (Z i )] = E[η Ω W N N (Z i )] = 0. We then characterize the functions η F W N N (Z i ) and η Ω W N N (Z i ), utilizing the Hoeffding decomposition of U-statistics (see Hoeffding (1961) and Serfling (1980), Chapter 5). This leads us to the following asymptotic results: 15 Preview of Theorem 13. et Σ FW N (14) (15) be a consistent estimator of Var η F W N N (Z i ). 16 et c 1 > 0 be any positive number, and D FW N = Σ 1/2 F W N +c 1. Under appropriate regularity conditions (described in Appendix A.5), as, 1a. U T F W N (2) DFW N p + if (8) is violated with positive probability 1b. U T F W N (2) DFW N d N(0,σ 2 ) with σ 2 (0, 1) if (8) holds almost everywhere, and holds with equality with positive probability 14 The conditions required are reminiscent of conditions used in a number of nonparametric estimation and inferential problems. For example, while we allow for the distribution of W N (W i,n,n ) to have a probability mass at zero, we require that its density be uniformly bounded on some interval [ s, 0), and impose a similar restriction on Φ W N (W i,n,n ). Appendix A.5 describes the required assumptions in detail. 15 Note that positive probability refers to non-zero probability over W n,n for some n, n N with n>n,and almost everywhere refers to probability 1 over W n,n for every n, n N with n>n. 16 Analytic expressions for Σ FW N and Σ ΩW N are given in Appendix A.5; alternatively, they can be estimated via subsampling. 23

24 1c. T U F W N p DFW (2) N 0 if (8) holds strictly almost everywhere Similarly, let Σ ΩW N be a consistent estimator of Var η Ω W N N (Z i ), c 2 > 0 any positive number, and D ΩW N = Σ 1/2 Ω W N + c 2 ; under appropriate regularity conditions, as, 2a. U T Ω W N (2) DΩW N p + if (9) is violated with positive probability 2b. U T Ω W N (2) DΩW N d N(0,σ 2 ) with σ 2 (0, 1) if (9) holds almost everywhere, and holds with equality with positive probability 2c. U T Ω W N (2) DΩW N p 0 if (9) holds strictly almost everywhere Based on these results, we can apply rejection rules based on the standard normal distribution. Consider the null hypothesis H F W N 0 :(8) is satisfied at almost all v W n,n, for every n, n N with n>n. et α (0, 1) denote a pre-specified significance level and let z α where Z N(0, 1). Consider the rule satisfy Pr(Z z α )=α, Reject H F W N 0 if F U T W N (2) z α (16A) D FW N This decision rule has the following asymptotic properties: lim Pr H F W N 0 is rejected when it is true α lim Pr H F W N 0 is rejected when it is false = 1 (17) The equivalent result holds for H Ω W N 0 :(9) is satisfied at almost all v W n,n, for every n, n N with n>n, and a decision rule of the form Reject H Ω W N 0 if Ω U T W N (2) z α (16B) D ΩW N Our test statistics defined above take the form of second-order U-statistics. The presence of U-statistics in econometrics is extensive, both for estimation and inference. Examples of M- estimators where the objective function is written explicitly as a U-statistic include the maximum rank correlation estimator studied by Han (1987) and Sherman (1993), as well as the estimation procedures in Dominguez and obato (2004) and Khan and Tamer (2009). U-statistics have also been used to construct consistent specification tests. Some examples include Fan and i (1996), Zheng (1998), Chen and Fan (1999). Even though our goal is to devise a specification test (as opposed to the estimation of an unknown parameter), Khan and Tamer (2009) is perhaps the closest to the spirit of our test, which relies on transforming functional (e.g, conditional moment) 24

25 inequalities into moment equalities. The goal in Khan and Tamer (2009) is to estimate a finitedimensional parameter that is point-identified by conditional moment inequalities that must hold with probability one. They propose an M-estimation procedure based on a U-statistic objective function whose probability limit is uniquely minimized at the true parameter value. The general idea of transforming moment inequalities into equalities has also been recently studied by Andrews and Shi (2009) and Kim (2009) who propose inferential methods for finite-dimensional parameters partially identified by conditional moment inequalities. The approach in Andrews and Shi (2009) relies on the use of properly chosen instruments, while Kim (2009) employs U-statistics of the type used in Khan and Tamer (2009). 4.2 Monte Carlo Simulations To understand how much data is required to give our test power, we perform our test of (9) on simulated data. For three different data-generating processes, we generated 1000 data sets of sizes = 200, = 400 and = 800, and calculated the test-statistic for each simulated data set. The three data-generating processes were: Endogenous N: jointly generate N and W. we used an example of the entry game from evin and Smith (1994) to (This example was solved in Appendix A.3; theoretical graphs of ψ 1 n 1:n (F n 1:n(v)), showing a violation of (9), are shown in Figure 7.) Exogenous N (A): we used the same data-generating process for valuations as the last case, but chose N independently of θ; this led to (9) holding strictly everywhere. Exogenous N (B): we used a different data-generating process for valuations, chosen so that (9) would hold with equality with positive probability. The data-generating processes themselves, and the bandwidths b used, are described in Appendix A.4. Applying the test above to the simulated data, with a target significance level of 5%, led to the following rejection rates: Empirical Rejection Rates at target significance level of 5% = 200 = 400 = 800 Endogenous N 58.4% 82.4% 98.3% Exogenous N (A) 3.1% 1.2% 1.1% Exogenous N (B) 7.0% 4.7% 5.2% 25

26 These results exactly reflect the asymptotic predictions in Theorem 13. As grows, the probability of rejecting exogeneity of N when it is endogenous is converging to 1. This probability is vanishing when the true data-generating process has exogenous N and (9) holds as a strict inequality with probability one. Finally, if these inequalities are binding with nonzero probability, the empirical rejection probabilities are close to the target significance level of 5%. 4.3 Results of the Unconditional Test Next, we apply the test to our timber data, using N = {2, 3, 4,...,11} and W n,n F W N (w m) 0.98 for m = n, n ; that is, we used = w : F i,j W N (W j n) F i,j W N (W j n ) 0.98 to estimate {W j W n,n } in the calculation of T F W N n,n (Z i,z j ) and T Ω W N n,n (Z i,z j ). Rejecting (8) or (9) would imply the rejection not only of CIPV, but also of IPV. 17 On the other hand, failing to reject either (8) or (9) could be attributed to the data being consistent with IPV. To check whether this is the case, we can test whether the reverse inequality in (9) holds: namely, whether ψn 1:n (F n 1:n(v)) ψ 1 n 1:n (F n 1:n(v)) (9 ) for every n, n N with n>n and almost all v W n,n. If the data supports (9), then rejecting (9 ) implies that the inequalities in (9) are strict with positive probability, which rules out IPV as the true model. A test of (9 ) would replace T Ω W N n,n (Z i,z j )with T Ω W N n,n (Z i,z j ) = {W i W j } Ω F W N (W j n ),n,n {N i = n} (11 ) ΦW N (W j,n,n ) 0 {W j W n,n } The estimator T Ω W N n,n (Z i,z j ) is constructed accordingly. The resulting U-statistic will have asymptotic properties analogous to those of (14) under the same type of regularity conditions (see the technical appendix for details). The test statistics were constructed using the analytic expressions for variance given in Appendix A.5, and with c 1 = c 2 = 10 6.Weusedb =0.001 to compute both the statistics and the estimates of their asymptotic variance, as well as the trimming set W n,n. The same constants were used for the rejection rule of the IPV test. The results obtained are the following: 17 That is, it would reject the notion that N is exogenous and IPV holds. 26

27 Unconditional Test Results Eq. (8) Eq. (9) IPV ( ) Test-statisticfor(9 ),using(11 ). These results strongly suggest a data-generating process satisfying (8) and (9). (Their magnitude could be the result of both holding as strict inequalities with probability one, in which case the variance estimators used in the denominator might be quite small. Asymptotically, the variance estimators would be dominated by c 1 = c 2 = 10 6, causing both test statistics to converge to 0 in probability; but at our sample size, this may not yet be happening. Nonetheless, we see strong support for (8) and (9); since both test statistics are negative, we could not reject either (8) or (9) regardless of the normalizing variance.) Our results also reflect strong evidence against (9 ). Thus, if we do not condition on any covariates, our data are consistent with conditionally independent private values and exogenous N, but not consistent with independent private values, confirming the visual intuition in Section 3.5. All of our qualitative results were found to be robust to moderate changes in b. arger values of b led to reductions in the absolute value of all three statistics, but the qualitative rejection results remained intact for a 5% significance level for moderately large increases in this bandwidth. 4.4 Testing Conditional on Observable Covariates Next, we modify the test to condition on observable covariates. et X be a vector of covariates, and let Y (N,X) and Z (W, Y ). We maintain the assumption of having an i.i.d. sample (Z i ) i=1. Define F W Y (w n, x) Pr(W w N = n, X = x) W Y (w, n, n,x) F W Y (w n, x) F W Y (w n,x) Φ W Y (w, n, n,x) = Ω F W Y (w n,x),n,n ) F W Y (w n, x) We will test whether conditional analogs of (8) and (9) hold at each realization of X that is, whether n>n implies F W Y (w n, x) F W Y (w n,x) (18) and ψn 1:n FW Y (w n, x) ψ 1 n 1:n FW Y (w n,x) (19) (or, equivalently, W Y (w, n, n,x) 0andΦ W Y (w, n, n x) 0) almost everywhere. For clarity, we present the case where X is one-dimensional and continuously distributed, as is the case in our 27

28 application. ater we discuss extending the test to cases where X is multi-dimensional and includes both discrete and continuous covariates. As before, let N S N be a fixed subset with p N (n) > 0 for all n N, and let C n,n S X S W be a pre-specified range of values of (x, w) on which we will test (18) and (19). (Again, C n,n should be chosen such that F W Y ( n, ) and F W Y ( n, ) are both bounded away from 0 and 1 on C n,n.) et h 0 be a nonnegative bandwidth sequence converging to zero, and let K : be a bias-reducing kernel 18.Define S F W Y (Z n,n i,z j,z k ) = {W i W k } F W Y (W k n,x j ) {N i = n} W Y (W k,n,n,x j ) 0 (Xj,W k ) C n,n 1 h K S Ω W Y (Z n,n i,z j,z k ) = Ω F W Y (W k n,x j ),n,n {W i W k } {N i = n} ΦW Y (W k,n,n,x j ) 0 (X j,w k ) C n,n 1 h K Both functions are constructed such that only triples (Z i,z j,z k ) for which X i X j vanishing neighborhood around zero will matter asymptotically. X i X j h X i X j h (20) lies in a Weighting methods with these types of asymptotic properties are referred to as pairwise differencing and they have been studied, for example, in Honoré and Powell (1994), Honoré and Powell (2005), Aradillas-opez, Honoré, and Powell (2007) and Hong and Shum (2009). The following result describes a set of sufficient conditions under which lim E S F W Y (Z n,n i,z j,z k ) = 0 and lim E S Ω W Y (Z n,n i,z j,z k ) = 0 if and only if W Y and Φ W Y are negative almost everywhere within the test range. Theorem 8 et f X N ( n) denote the density of X conditional on N = n. Supposethatthesupport of f X N ( n) is the same for all n N. Further, suppose that for any n N, f X N (x n) and F W Y (w n, x) are both M times differentiable with respect to x, the latter with bounded derivatives, at almost all x for which (x, w) C n,n for some w. Then for i = j = k, E S F W Y n,n (Z i,z j,z k ) = γ F W Y n,n + O(hM ) E S Ω W Y n,n (Z i,z j,z k ) = γ Ω W Y n,n + O(hM ) where γ F W Y n,n = p N (n) E max 0, W Y (W k,n,n,x j ) f X N (X j n) γ Ω W Y n,n = p N (n) E max 0, Φ W Y (W k,n,n,x j ) f X N (X j n) (Xj,W k ) C n,n (Xj,W k ) C n,n 18 That is, K satisfies K(s) = K( s), K(s)ds = 1, sr K(s)ds = 0 for all r = 1,...,M 1, and sm K(s)ds <. 28

29 Proof. We focus on E S Ω W Y n,n (Z i,z j,z k ) ; the proof for E S F W Y n,n (Z i,z j,z k ) follows identical steps. As before, define S Ω W Y (N n,n i,z j,z k ) = E Wi,X i N i S Ω W Y (Z n,n i,z j,z k ) = E Xi N i E Wi N i,x i S Ω W Y (Z n,n i,z j,z k ) {N i = n} = E Xi N i Ω F W Y (W k n,x j ),n,n F W Y (W k N i,x i ) ΦW Y (W k,n,n,x j ) 0 (Xj,W k ) C n,n 1 h K h = E Xi N i Ω F W Y (W k n,x j ),n,n 1 F W Y (W k N i,x i ) h K {N i = n} ΦW Y (W k,n,n,x j ) 0 (Xj,W k ) C n,n X i X j X i X j h = Ω F W Y (W k n,x j ),n,n F W Y (W k n, X j )f X N (X j n)+o(h M ) {N i = n} ΦW Y (W k,n,n,x j ) 0 (Xj,W k ) C n,n = {N i = n} (Xj,W k ) C n,n max{0, Φ W Y (W k,n,n,x j )} f X N (X j n)+o(h M ) The second-to-last inequality comes from an M th -order Taylor approximation, which given our smoothness assumption yields h K E Xi N i Ω F W Y (W k n,x j ),n,n 1 F W Y (W k N i,x i ) Ω F W Y (W k n,x j ),n,n F W Y (W k N i,x) = = X i X j h 1 h K x X j h f X N (x N i )dx Ω F W Y (W k n,x j ),n,n F W Y (W k N i,x j ) f X N (X j N i )+O(h M ) along with the fact that the entire expression is zero unless N i = n. Taking the expectation of S Ω W Y n,n over (N i,z j,z k )thenprovestheresult. As with the unconditional test, we modify (20) by replacing F W Y with a nonparametric estimate calculated on all the observations but i, j, k and introducing a positive but vanishing bandwidth b. et K and h be an additional kernel function and bandwidth sequence, and define R i,j,k W Y (w n, x) = 1 ( 3) h =i,j,k {W w} {N = n} K X x h p i,j,k 1 (x, n) = Y ( 3) h =i,j,k {N = n} K X x h F i,j,k W Y (w n, x) = R i,j,k W Y (w n, x) p i,j,k (x, n) if p i,j,k (x, n) = 0 Y Y 0 otherwise 29

30 and i,j,k W Y (w, n, n,x) = F i,j,k W Y (w n, x) F i,j,k W Y (w n,x) Define Φ i,j,k S F W Y n,n (Z i,z j,z k ) = S Ω W Y n,n (Z i,z j,z k ) = W Y (w, n, n,x) = Ω F i,j,k W Y (w n,x),n,n ) F i,j,k W Y (w n, x) {W i W k } F i,j,k W Y (W k n,x j ) {N i = n} i,j,k W Y (W k,n,n,x j ) b (Xj,W k ) C n,n Ω F i,j,k W Y (W k n,x j ),n,n {W i W k } {N i = n} Φ i,j,k W Y (W k,n,n,x j ) b (Xj,W k ) C n,n 1 h K 1 h K X i X j h X i X j h Our conditional test statistics are S U F W Y = (3) S U Ω W Y = (3) 1 ( 1)( 2) S F W Y (Z n,n i,z j,z k ) i,j,k {1,...,} i=j=k n,n N n>n 1 ( 1)( 2) S Ω W Y (Z n,n i,z j,z k ) i,j,k {1,...,} i=j=k n,n N n>n (21) Appendix A.6 gives conditions under which S U F W Y (3) and S U Ω W Y (3) have asymptotic behavior analogous to the unconditional test statistics; the formal result is given as Theorem 14. A detailed proof can be found in the technical appendix. As we show in Appendix A.6, equipped with this result we can employ rejection rules analogous to the unconditional test case. (The added conditions to guarantee asymptotic normality of the conditional test statistics are M-times differentiability of certain functions of F W Y with respect to the covariate X, and conditions on the relative convergence rates of the bandwidths b, h and h. The smallest integer value of M for which all these restrictions can be satisfied is M = ) Our methodology can be extended to the case where X is multivariate, with both discrete and continuous covariates. Our nonparametric estimators would use indicator functions for the discrete components of X, and a multivariate (e.g, multiplicative) bias-reducing kernel for the continuous elements. The value of M needed to preserve -asymptotic normality of the test statistics would increase with the dimension of continuous components in X. 19 The entire set of bandwidth convergence restrictions would be satisfied for M =16,e.g,ifwechooseh 0.073, h and b A high degree of smoothness is required for our statistics to converge at the parametric rate of. Smaller values of M would yield slower rates of convergence, and in any such case our test would proceed by normalizing S U F W Y and U Ω W Y S by the relevant convergence rate, which would be (3) (3) slower than. 30

31 4.5 Results of the Conditional Test As discussed above, in our implementation, we condition only on appraisal value. As before, we apply the test with N = {2, 3,...,11} and C n,n = (x, w) :0.02 F W Y (w m, x) 0.98 for m = n, n for each (n, n ). In the construction of both S F W Y n,n (Z i,z j,z k ) and S Ω W Y n,n (Z i,z j,z k ), we used to estimate 0.02 F i,j,k W Y (W k n, X j ) F i,j,k W Y (W k n,x j ) 0.98 {(X j,w k ) C n,n }. The smallest value of M (the number of derivatives assumed to exist for the various functionals involved) that is consistent with the conditions of Theorem 14 is M = 16. This dictates the order of the bias-reducing kernels required. We employed polynomial kernels with bounded support of the following form, K(z) = K(z) = 7 a 0 + a j z 2j j=1 Ck z C k. The a j s are chosen to satisfy the bias-reducing requirements. In our results we used C k = 8, which yields a support large enough to include most of the probability mass of, say, a Standard Normal kernel. Using polynomial kernels helped simplify the computation of our test-statistics significantly. The bandwidths we used are of the type b = , h = d 1 σ X 0.073, and h = d 2 σ X σ X denotes the sample standard deviation of X (appraisal value). The expression for b was chosen so that b given our sample size. The exponents for the remaining bandwidths satisfy all our assumptions when M = 16. The constants of proportionality d 1 and d 2 were fixed at 2 in our implementation. As in the unconditional case, we also test if the reverse inequality in (19) holds everywhere in our trimming set. Once again, rejecting this would rule out independent private values conditional on X (appraisal value). Our results include such a test, labeled IPV below. Its asymptotic properties are described in the technical appendix. S As we can see in Equations (31A)-(31B) of Appendix A.6, estimating the variances of U F W Y (3) S and U Ω W Y from their analytic expressions requires the use of a fourth-order U-statistic. Given (3) our sample size, this would amount to a sum with approximately terms. In order to make the task of estimating this variance computationally feasible, we chose 20 random subsamples of size = 500 and computed the corresponding estimate of the variance for each subsample. Our final estimate was computed as the average of these 20 estimates. The same bandwidths and kernels described above were used in the estimation of these variances. As in the conditional case, we used the additive terms c 1 = c 2 = 10 6 in the normalization (see Equation (32), Appendix A.6). 31

32 Conditional Test Results Eq. (18) Eq. (19) IPV As in the unconditional case, our results indicate that, conditional on appraisal value, our data is consistent with conditionally independent private values and exogenous N, but not consistent with independent private values. These conclusions were robust to moderate changes in our bandwidths, and in the support of the kernels used. 5 Identification and Counterfactual Analysis 5.1 Theory Now that we have reason to believe our data are consistent with our identifying assumption, we return to the problem of identification. Recall from Section 2 that the main empirical question is to identify F n:n, and that F n:n (v) = n m=n+1 1 n 1 m 1 i=n i 1 F m 1:m (v) + i +1 n n F n: n(v) if N is exogenous (and F n:n (v) is bounded below by the right-hand side if N is positively associated with values). Since {F m 1:m } are identified from bidding data from auctions with m bidders, our next task is to place bounds on the trailing term. 20 Theorem 9 Fix n and v. et αm n 1 m 1 n 1 i=n i 1 i+1 for m>n.iff m 1:m(v) F m 1:m (v), F m 1:m (v) for every m {n, n +1,..., n}, then if N is exogenous, and F n:n (v) F n:n (v) n m=n+1 αn mf m 1:m (v)+ n n F n 1: n(v) (22) F n:n (v) F n:n (v) n m=n+1 αn mf m 1:m (v)+ n n ψ 1 n 1: n F n 1: n (v) n (23) if N is exogenous or positively associated with valuations. 20 The bounds placed on F n: n in Theorem 9 are the tightest available using only F n 1: n,butcouldpotentiallybe tightened using {F n 1:n } n= n ;thus,theboundsdefinedincorollary10arenotsharp. Asforthetests,(8)isnot sharp (as noted in footnote 12), but (9) is sharp under a data-generating process satisfying independent private values and exogenous N, (9)wouldholdwithequalityeverywhere. 32

33 Proof. Since V n: n V n 1: n, F n: n (v) F n 1: n (v) F n 1: n (v), so if N is exogenous, F n:n (v) = n m=n+1 αn mf m 1:m (v)+ n n F n: n(v) n m=n+1 αn mf m 1:m (v)+ n n F n 1: n(v) Next, we show that F n: n (v) ψ 1 n 1: n (F n 1: n(v)) n :letψ n: n (s) =s n, and note that ψ n: n ψ 1 ψ n 1: n (s) = n: n (t) nt n 1 t ψ n 1: n (t) = n( n 1)t n 2 (1 t) = ( n 1)(1 t) where t = ψ 1 n 1: n (s). t ( n 1)(1 t) is increasing in t, so ψ n: n ψ 1 n 1: n is increasing, or ψ n: n ψ 1 n 1: n is convex. For a given θ, the CDF of the greatest of n independent draws from F (v θ) is(f (v θ)) n ; so taking the expectation over θ implies F n: n (v) =E θ (F (v θ)) n, and applying Jensen s Inequality gives 21 F n: n(v) = E θ {ψ n: n (F (v θ))} = E θ ψ n: n ψ 1 n 1: n (ψ n 1: n (F (v θ))) ψ n: n ψ 1 n 1: n (Eθ {ψ n 1: n (F (v θ))}) = ψ n: n ψ 1 n 1: n (F n 1: n(v)) Then if N is either exogenous or positively associated with valuations, completing the proof. F n:n (v) n m=n+1 αn mf m 1:m (v)+ n m=n+1 αn mf m 1:m (v)+ n n n m=n+1 αn mf m 1:m (v)+ n n n n F n: n(v) ψ 1 n n 1: n (F n 1: n(v)) ψ 1 n 1: n F n 1: n (v) n Following from Theorem 1, bounds on F n:n lead to bounds on the relevant economic measures: Corollary 10 Fix n, and let F m 1:m, F m 1:m n m=n be integrable mappings from + to [0, 1]. Define F n:n and F n:n as in (22) and (23), and let π n (r) F n 1:n (r) F n:n (r) (r v 0 )+ r (v v 0)dF n 1:n (v) π n (r) F n 1:n (r) F n:n (r) (r v 0 )+ r (v v 0)dF n 1:n (v) r n min {r v 0 : π n (r) max r π n (r )} r n max {r v 0 : π n (r) max r π n (r )} r n max {r v 0 : π n (r) π n (v 0 )} u n (r) 1 n u n (r) 1 n 0 0 max{r, v}df n:n (v) 0 max{r, v}df n 1:n (v) max{r, v}df n:n (v) 0 max{r, v}df n 1:n (v) 21 Quint (2008) shows this same lower bound F n:n(v) ψ n:n ψ 1 n 1:n (F n 1:n(v)) for symmetric, affiliated private values, whether or not they are conditionally independent. 33

34 et rn = arg max r π n (r) be the profit-maximizing reserve price. If F m 1:m (v) F m 1:m (v), F m 1:m (v) for every m {n, n +1,..., n} and every v, then 1. If N is exogenous, π n (r) [π n (r), π n (r)], rn [r n, r n ],andu n (r) [u n (r), u n (r)] 2. If N is positively associated with valuations, π n (r) π n (r), rn r n,andu n (r) u n (r) Proof. Quint (2008) shows that (1) (the expression for π n (r)) is stochastically increasing in both V n 1:n and V n:n (in the first-order stochastic dominance sense); so when N is exogenous, plugging the upper bounds F n 1:n and F n:n into (1) gives the lower bound π n (r), and the lower bounds F n 1:n and F n:n give the upper bound π n (r). As in Haile and Tamer (2003), we can use π n and π n to bound the profit-maximizing reserve price for an auction of a given size: letting r = arg max r π n (r ), π n (r n) π n (r n) π n (r ) π n (r ) = max r π n (r ) and so r n {r : π n (r) max r π n (r )}. By (2), u n can be expressed as the expected value of a function which is increasing in V n:n and decreasing in V n 1:n. So expected bidder surplus is stochastically increasing in V n:n and decreasing in V n 1:n ; plugging F n:n and F n 1:n into (2) gives the lower bound on u n (r), and F n:n and F n 1:n yield the upper bound. When N is positively associated with valuations, the lower bound F n:n is still valid, so the upper bounds π n and u n hold. Since π n (v 0 ) depends only on F n 1:n, not F n:n, π n (v 0 ) π n (v 0 ); so π n (r n) π n (r n) π n (v 0 ) π n (v 0 ) gives the bound r n. It is worth noting that when N is exogenous, the bounds on F n:n, and therefore π n (r) and r n, will be tighter when n is lower exactly the cases where setting the correct reserve price is most important. 22 To see this, consider the case where the data set is huge but bounded, so that F m 1:m is learned very precisely for m up to an upper bound n. SinceF n:n (v) = n m=n+1 αn mf m 1:m (v)+ n n F n: n(v), so all uncertainty about F n:n (v) comes from uncertainty about F n: n (v); but the lower is n, theloweristhecoefficient n n on F n: n(v), and therefore the lower is the uncertainty in F n:n. 5.2 Empirical Application to USFS Timber Auctions The main goal of our study of the timber auction data is to examine the empirical consequences of unobserved heterogeneity in data for auction design. As discussed above, we control for the largest source of observable heterogeneity by conditioning on the appraisal price of each auction tract. Regressions of transaction price on auction-specific covariates show that appraisal price explains much of the variation in sales price across timber tracts. This is not coincidental since the appraisal 22 As n grows, the probabilities that a given reserve price r either binds or precludes a sale both go to 0. 34

35 prices in our data were constructed using the residual method, which takes the estimated revenue from the downstream sales of the lumber derived from the timber and subtracts off the estimated costs of cutting, transporting, and processing the timber. (The residual method also subtracts off an additional amount that is designed to serve as a reasonable profit for lumber supply.) Thus the appraisal price in our data is meant to contain information on all the auction-tract covariates that are directly related to profits, and it thus plausibly plays the role of a sufficient statistic for all observable auction heterogeneity. (We will later contrast this method for appraising timber with the transaction method, which appraises timber based on the predicted sales price of a timber tract based on a regression of transaction prices on observable auction characteristics.) By conditioning only on appraisal price, we allow the unobservable θ to capture the effects of any remaining observable heterogeneity, in addition to any heterogeneity that is not being captured through the observables. Since appraisal value is meant to capture the effect of all available information on expected profitability, however, conditioning on appraisal price makes the effects of observed heterogeneity left in θ arguably small, and hence most of what θ is capturing may be true unobserved heterogeneity (i.e., information outside of the seller s information set). Our main empirical question moving forward is to study the revenue implications of a strategic reserve policy when the demand facing the seller contains some amount of unobserved heterogeneity. In the counterfactuals we consider, we exploit the fact that the forest service holds timber auctions in two rounds. In the first round, bidders must submit bids of at least the reserve (equal to the appraisal value) in order to qualify for the oral auction round of bidding. The oral auction then starts at the highest of the sealed bids. The first round of bidding thus reveals to both the seller and the other bidders the number of potential bidders. It is generally recognized that the reserve price posted rarely if ever deters a potential bidders from submitting a sealed bid (see, e.g., Froeb and McAfee (1988) and Haile and Tamer (2003)). Thus the residual method for setting appraisal values is understood to underestimate potential bidder values for the tract. Furthermore, the number of bidders who submit during the first round provides a very tight measure of the number of potential bidders in the ensuing ascending auction. The seller can thus use the information on the revealed number of potential bidders to design alternative strategic reserve price policies for the oral round of bidding, which would be operationalized by starting the oral auction at some strategic level above the highest of the sealed bids. We begin by visually demonstrating the effect of correlation on the calculation of the distribution F n:n of the highest valuation. As an example, we consider a timber tract with the median appraisal value in the data (appraisal = 55.9) and N = 3 bidders. Figure 3 shows three different ways to calculate F n:n. Upper bound and lower bound refer to the bounds F n:n and F n:n defined in 35

36 (22) and (23). (These are calculated using nonparametric estimates of the conditional distributions F m 1:m as a function of appraisal value.) IPV is the standard calculation of F n:n under the assumption that, conditional on appraisal value, bidder valuations are independent; this calculation is F n:n (v x) = ψ 1 n 1:n (F n 1:n(v x)) n. Figure 3: F n:n (v x), for n = 3 and x = 56, calculated three ways CDF Upper Bound ower Bound IPV reserve We see immediately that, even though we do not achieve point-identification of F 3:3, our bounds are nonetheless informative. The implications of the standard model lie outside the bounds we calculate; and our upper and lower buonds are closer to each other than either one is to IPV. Further, Theorem 9 showed that the lower bound F n:n remained valid even when the assumption of exogenous N was replaced with the much weaker assumption of positive selection; so the observation that F 3:3 lies above the IPV calculation is robust to the most likely violation of our identifying assumption. Next, we examine the effect that these different estimates of F n:n have on matters of economic interest: the seller s expected revenue and expected profit at different reserve prices. Expected revenue is simply the expected value of transaction price, counted as 0 in the event that all n bidders have valuations below the reserve and therefore no sale occurs; this is calculated as (F n 1:n (r) F n:n (r)) r + vdf r n 1:n (v). Plugging the conditional estimates of F 2:3 and F 3:3 at the median appraisal value x = 56 into this expression gives us an estimate of expected revenue based on IPV, and upper and lower bounds based on our model, pictured in Figure 4. As we can see, IPV implies a small but noticeable increase in revenue from using a reserve price around 100; the bounds on expected revenue accounting for unobserved heterogeneity suggest there is no such increase, and that expected revenue begins to decrease in reserve much sooner. (Recall the intuition we gave earlier for why correlated values reduce the benefit of a reserve price. Reserve 36

37 Figure 4: Expected Revenue, n = 3 and x = 56 profit 100 ower Upper IPV reserve prices trade off increased revenue in the event just one bidder s value exceeds the reserve, against reduced revenue in the event none of them do and the reserve precludes a sale. When bidder values are positively correlated, then when n 1 bidders have values below r, the last bidder is more likely to have a low value as well; so the former event is less likely, while the latter event is more likely.) Next, we consider the seller s expected profit at various reserve prices. Expected profit is given by (1). However, to calculate it, we need to know the seller s cost for the object, or its value to the seller should it go unsold. Here, we make the natural assumption that the value to the seller is equal to the appraisal value. (Since appraisal value is meant as a conservative estimate of the profitability of logging the land, this seems appropriate.) Figure 5 shows expected profit under this assumption, calculated under IPV and under our method. We think of a nonstrategic reserve price as a reserve price equal to the seller s valuation, and a strategic reserve price as a reserve higher than that. Both IPV and our method show that there are gains to using a strategic reserve; but the magnitude of the gains, as well as the reserve that maximize them, are much smaller under our analysis than under IPV. Recall Corollary 10: when there is positive selection, the upper bounds on expected revenue and profit still hold; so in both Figures 4 and 5, the upper bounds are valid even when the assumption of exogenous N is violated. This means that the qualitative takeaway that both the optmial reserve price and the increase in profits are actually much lower than IPV would suggest still holds when our main identifying assumption fails. 37

38 Figure 5: Expected Profit, n = 3 and v 0 = x = 56 profit 60 ower Upper 50 IPV reserve Table 1 shows, under each of the three estimates above, the reserve price r that maximizes expected profit; the multiple of the seller s valuation that that represents; the expected profit under a nonstrategic reserve price; the expected profit at the optimal reserve price; and the increase in profits that this represents. Table 1: Optimal Reserve and Effect on Profits, Calculated Three Ways r r v 0 π n (r = v 0 ) π n (r = r ) % IPV % Upper % ower % As discussed above, our analysis shows that there is indeed a gain to using a strategic reserve price, but that this gain is much smaller than the gain suggested by IPV. From Corollary 10, bounds on the optimal reserve price are given by π n (r) max r π n (r ) = 49.2; this constrains r to lie within the interval [73.7, 122.8]. While this is a wide range, it is still informative, as the optimal reserve is sure to be below the value of implied by IPV. Further, at the IPV-implied optimal reserve of 131.4, the upper and lower bound on expected profit accounting for unobserved heterogeneity are 47.8 and 41.1, respectively, implying anything from a negligible gain to a non-negligible loss relative to a non-strategic reserve. While this is just for one particular auction, it is fairly representative. To better understand the implications of unobserved heterogeneity, however, we can consider the effect of different reserve price policies if they were applied to every auction in our data set. 38

39 First, we consider an alternative policy of setting each auction s reserve price such that the probability of no sale (i.e., no bidder having a value above the reserve) is 15%. (The optimal reserve price suggested by IPV is substantially higher than this for most auctions.) The 15% number comes from the mandate of the Forest Service to sell at least 85 percent of timber tracts. For each auction in the data, we calculate the reserve price that generates this probability under the assumption of IPV, and examine the revenue implications under both IPV and the upper and lower bounds from the correlated values model. As pointed out by Haile and Tamer (2003), the IPV model predicts that the government could nearly double the reserve price in most auctions without threatening the 15% no-sale benchmark. In fact, if all reserve prices in the data were raised to the level predicting a 15% chance of no-sale under IPV, the average ratio of reserve price to appraisal value would be 3.1, and the median would be 2. However, when unobserved heterogeneity is accounted for, the probability of no-sale is substantially higher; again averaging across the auctions in the data, the upper and lower bounds on the probability of no-sale at these reserve prices are 36.7% and 24.2%, respectively. That is, at the reserve price which IPV predicts gives a 15% chance of no-sale, we calculate the chance of no-sale to instead be between 24.2% and 36.7%. Table 2 shows these results, along with the impact this reserve price policy would have on expected profits (given the mix of auctions in the data, and continuing to assume that the government s value of unsold tracts is equal to their appraisal values). Bids and values are per unit of timber, but our data also contains the volume of timber in each tract, so we can calculate the gain or loss per auction in dollars as well. (Since upper bounds on expected profit are calculated from lower bounds on F n:n and vice versa, the upper and lower labels may seem slightly ill-defined; here, upper refers to π n and F n:n, the most optimistic case consistent with the data.) Table 2: Percent Change in Expected Profit from 15 Percent No Sale Policy ower Upper IPV Average probability of no sale 36.7% 24.2% 15.0% Average change in expected profit (relative to r = v 0 = x) 3.9% 7.4% 16.0% Change in expected profit, median auction 1.6% 6.9% 13.5% Average change in expected profit, dollars $9, 269 $3, 447 $7, 896 Change in expected profit, median auction $215 $1, 845 $4, 548 As a final policy exercise, we note that in the 1990 s, the USFS made an effort to move towards a market based method of appraising timber. The idea was that instead of using historical cost 39

40 and revenue accounting information to set advertised rates, timber could be appraised using price data on past sales of similar tracts. Given a predicted value of the high bid in such an auction, this value is then discounted by some factor to determine the actual appraisal as a way to encourage competition. In our data, we mimic this transactions-based approach by regressing transaction price on appraisal value in our data, and then using a discount factor of 20 percent. Table 3 shows features of the distribution (across the different auctions in our data set) of the change in expected profits from moving from a non-strategic reserve price to this transactions-based approach. Table 3: Percent and Dollar Change in Revenue from Transaction Price Reserve Policy ower Upper IPV ower Upper IPV Mean 2.5% 5.3% 9.8% $252 $2, 654 $4, 274 Median 0.8% 4.1% 6.4% $190 $1, 379 $2, th percentile 0.6% 2.4% 3.2% $289 $479 $804 75th percentile 4.9% 7.4% 13.9% $1, 023 $3, 346 $5, 397 As Table 3 shows, the transactions-based approach gives a reasonable upside and a muchmitigated downside relative to the more aggressive policies discussed above. (This is due to the fact that the implied reserve prices are substantially lower than those in the first experiment.) 6 Conclusion In this paper, we have introduced new empirical methods for identification and counterfactual analysis in ascending auctions, which are robust to correlation in values induced by unobserved heterogeneity. Applying them to timber data, we found that analysis accounting for unobserved heterogeneity rationalized a much more cautious reserve price policy than traditional analysis, at least when appraisal value was taken to be a sufficient statistic for observable covariates. Our identifying assumption is quite strong; however, it seems to fit well in the timber auction setting, and lends itself to formal testing. In light of Theorem 5, and the corollaries and examples in Appendix A.3, we expect our test to have power against the most likely forms of endogeneity of N; conditional on passing the test, then, exogeneity of N may be no more a leap of faith than independent values or any other standard modeling assumption. More generally, we believe the methods presented above to be useful in three types of situations: 1. Bidder valuations are correlated, but this correlation cannot be linked to any observable auction-specific covariates 40

41 2. Covariates known to shift bidders valuations are missing from the data 3. Conditioning nonparametrically on all relevant covariates is unappealing due to sample size and the curse of dimensionality The last of these may in fact be the most common. Even if bidder values are indeed independent conditional on all relevant observables, use of the IPV framework constrains the analyst to be fully nonparametric and to condition on everything, since any misspecification would lead to correlated residuals and invalidate the theoretical model. Our techniques, on the other hand, allow for more flexibility. To make this point more explicit, suppose N (X, θ) and V i iid F ( X, θ). et X denote the support of X, and g : X k be any function of X on which we choose to condition our analysis. 23 Conditional on a realization of g(x), V i are still i.i.d. draws from F ( X, θ), just with a different (posterior) distribution of (X, θ) for each value of g(x). 24 Regardless of the choice of g, our model (and identifying assumption) still hold, and we can get valid estimates of expected revenue and do valid counterfactuals from estimates F n 1:n ( g(x)) even if g does not accurately model the effect of X on V i in the data-generating process and is therefore misspecified. 25 This suggests a tradeoff we hope to explore in future work. The more covariates we control for, the less unmodelled heterogeneity remains across auctions, and so the more precise our results and counterfactuals will be in the limit. At the same time, the precision of our conclusions is limited by the precision of our estimates of the conditional distribution of transaction prices, F n 1:n ( g(x)). The more covariates we use, especially nonparametrically, the more data would be required to put reasonable error bands on these estimates, and therefore on our estimates of π n and π n and our policy conclusions. Conditioning on more variables (and with fewer parametric assumptions) thus involves a tradeoff between asymptotically sharper results and noisier estimates at a fixed sample size. At the very least, our techniques allow the analyst to select an interior optimum. 23 If g is a constant, we are conditioning on nothing, and letting all of (X, θ) bepickedupasunobservedheterogeneity. If k =dim(x) andg is the identity map, we are conditioning on all of X. Ifk<dim(X) andg(x) =(X 1,X 2,...,X k ), then (as in this paper) we are conditioning on a subset of X. Ifk =1,g is a one-dimensional index constructed from X. Andsoon. 24 Specifically, if ρ(x, θ) isthepriordistributionof(x, θ), then ρ(x, θ g(x) =g) = ρ(x,θ) {g(x)=g} X Θ ρ(x,θ ) {g(x )=g}dx dθ. 25 Another alternative would be to model valuations parametrically, for example, as V i = βx + i.evenifthetrue effect of X on V i is not linear, the residuals i = V i βx are still i.i.d. conditional on (X, θ); so our techniques would still pin down the distribution of n:n from the distributions of m 1:m,givingusthe average expectedrevenue across all sets of covariates X with the same value of βx. 41

42 A Appendix A.1 Proof of Theorem 1 In a second-price sealed-bid auction, bidders have a dominant strategy of bidding their valuations. Seller profits are therefore 0 if V n:n <r, r v 0 if V n:n r V n 1:n, and V n 1:n v 0 if V n:n V n 1:n >r(where v 0 is the seller s cost and V 0:1 is understood to be 0). We can therefore write expected profits as π n (r) = E Vn 1:n,V n:n 1Vn:n r V n 1:n (r v 0 )+1 Vn:n V n 1:n>r (V n 1:n v 0 ) Since the first event happens with probability F n 1:n (r) F n:n (r), and the second happens whenever V n 1:n >r, we can rewrite this as π n (r) = (F n 1:n (r) F n:n (r)) (r v 0 )+ + (v v r 0 )df n 1:n (v) As for bidder surplus, each bidder has ex-ante probability 1 n of having the highest value, which earns a surplus of 0 when V n:n r and V n:n max{v n 1:n,r} when V n:n >r(where V 0:1 is under stood to be 0). So u n (r) = 1 n E V n 1:n,V n:n {1 Vn:n>r (V n:n max{v n 1:n,r})} = 1 n E 1 r Vn:n V n 1:n (r r)+1 Vn:n>r V n 1:n (V n:n r)+1 Vn:n V n 1:n>r (V n:n V n 1:n ) = 1 n E 1 Vn:n rr + 1 Vn:n>rV n:n + 1 Vn 1:n rr + 1 Vn 1:n>rV n 1:n = 1 n (E {max{v n:n,r}} E {max{v n 1:n,r}}) As argued in the text, conditional on a realization of θ, bidder values are IPV, and so other standard auctions are revenue-equivalent to a sealed-bid second-price auction. A.2 Proof of Theorems 2 and 3 As noted in the text, exogenous N implies F n:n (v) = 1 n+1 F n:n+1(v)+ n n+1 F n+1:n+1(v). Fixing n, we will use induction on n to show (4), F n:n (v) = 1 n 1 n m=n+1 m 1 i=n i 1 F m 1:m (v) + i +1 n n F n: n(v) for any n >n. For the base case, n = n + 1, the right-hand side is 1 n+1 F n:n+1(v)+ n n+1 F n+1:n+1(v), which as noted is equal to F n:n (v). For the inductive step, if (4) holds for n = K, then 1 K+1 m 1 n 1 m=n+1 i=n i 1 i+1 F m 1:m (v)+ n K+1 F K+1:K+1(v) = 1 K m 1 K n 1 m=n+1 i=n i 1 i+1 F m 1:m (v)+ 1 n 1 i=n i 1 i+1 F K:K+1 (v)+ (n 1)n = F n:n (v) n K F K:K(v)+ 1 n 1 K(K+1) F K:K+1(v)+ n = F n:n (v)+ n K F K:K (v)+ 1 K+1 F K:K+1(v)+ K K+1 F K+1:K+1(v) K+1 F K+1:K+1(v) n K+1 F K+1:K+1(v) 42

43 which is again equal to F n:n (v) since by (4), the terms in parentheses sum to 0; so (4) holds for n = K + 1. To show (5) holds when N is positively associated with valuations, the steps are the same, only starting with the weak inequality F n:n (v) 1 n+1 F n:n+1(v)+ n n+1 F n+1:n+1(v). Under exogenous N, a more general version of (3) also holds: for any j < k, F j:k 1 (v) = k j k F j:k(v)+ j k F j+1:k(v), which we can rearrange to F k i:k (v) = k i F (k 1) (i 1):k 1(v) k i F k (i 1):k (v) i We can use this to identify F k i:k for i>1 by induction on i: {F n 2:n } are pinned down from {F n 1:n }, {F n 3:n } from {F n 2:n }, and so on. A.3 Power of the Test Proof of Theorem 5. The proof is basically the Taylor expansion of ψ 1 n 1:n F n 1:n around v, which gives ψn 1:n (F n 1:n(v)) = 1 (v v) E θ n (f(v θ)) 2 + O (v v) 2 (24) Calculation is complicated by the fact that ψn 1:n is infinite at 1. ψn 1:n F n 1:n differentiable at v, because F n 1:n we calculate things. Differentiating gives ψ 1 n 1:n F n 1:n (v) = ψ n 1:n (Fn 1:n (v)) Fn 1:n(v) so letting v v, is still is very flat near v; we just need to be very careful in how = 1 ψ n 1:n(ψ 1 n 1:n (Fn 1:n(v))) F n 1:n(v) = 1 ψ n 1:n(ψ 1 n 1:n(E θ n F n 1:n(v θ))) E θ n F n 1:n (v θ) = 1 ψ n 1:n(ψ 1 n 1:n(E θ n ψ n 1:n(F (v θ)))) E θ n ψ n 1:n (F (v θ)) = 1 ψ n 1:n(ψ 1 n 1:n(E θ n ψ n 1:n(F (v θ)))) E θ nψ n 1:n (F (v θ)) f(v θ) ψ 1 n 1:n F E θ n {ψ n 1:n (v ) = n 1:n (F (v θ)) f(v θ)} ψn 1:n ψ 1 n 1:n Eθ n {ψ n 1:n (F (v θ))} To calculate the denominator, we first fix θ, and suppress the dependence of F ( θ) and f( θ) on θ. The Taylor expansion of F (v θ) around v gives F (v ) =1 f(v) f (v) 2 + O( 3 ); 43

44 ψ n 1:n (s) can be written as ns n 1 (1 s)+s n,so ψ n 1:n (F (v )) = n 1 f(v) f (v) 2 + O( 3 ) n 1 f(v) 1 2 f (v) 2 + O( 3 ) + 1 f(v) f (v) 2 + O( 3 ) n = nf(v) n 2 f (v) 2 n(n 1)(f(v)) nf(v) + n 2 f (v) 2 + n(n 1) 2 (f(v)) O( 3 ) = 1 n(n 1) 2 (f(v)) O( 3 ) E θ n {ψ n 1:n (F (v ))} = 1 n(n 1) 2 2 E θ n {(f(v θ)) 2 } + O( 3 ) Next, we calculate y = ψn 1:n Eθ n {ψ n 1:n (F (v ))}. For small, the argument of ψn 1:n will be close to 1, so y will be close to 1; let x =1 y, and solve ψ n 1:n (1 x) = 1 n(n 1) 2 2 E θ n {(f(v θ)) 2 } + O( 3 ) n(1 x) n 1 x +(1 x) n = 1 n(n 1) 2 2 E θ n {(f(v θ)) 2 } + O( 3 ) nx n(n 1)x 2 +1 nx + n(n 1) 2 x 2 + O(x 3 ) = 1 n(n 1) 2 2 E θ n {(f(v θ)) 2 } + O( 3 ) 1 n(n 1) 2 x 2 + O(x 3 ) = 1 n(n 1) 2 2 E θ n {(f(v θ)) 2 } + O( 3 ) x 2 + O(x 3 ) = 2 E θ n {(f(v θ)) 2 } + O( 3 ) x = E θ n (f(v θ)) 2 + O( 2 ) (For the last equality, we re solving x 2 + O(x 3 )=z 2 + O(z 3 ). Suppose that x = z + βz α,withα<2. Then x 2 = z 2 +2βz 1+α + β 2 z 2α, giving a contradiction.) Finally, taking ψ n 1:n(1 x) to complete the denominator gives ψ n 1:n(s) = n(n 1)s n 2 (1 s) ψ n 1:n(1 x) = n(n 1)(1 x) n 2 x = n(n 1)(1 E θ n (f(v θ)) 2 + O( 2 )) n 2 E θ n (f(v θ)) 2 + O( 2 ) = n(n 1) E θ n (f(v θ)) 2 + O( 2 ) and so the denominator of ψ 1 n 1:n F n 1:n (v ) is ψn 1:n ψ 1 n 1:n Eθ n {ψ n 1:n (F (v θ))} = n(n 1) E θ n (f(v θ)) 2 + O( 2 ) To calculate the numerator, E θ n ψ n 1:n (F (v )) f(v ), we again fix θ and suppress the dependence of F and f on θ. F (v ) =1 f(v) + O( 2 ) and ψ n 1:n(s) =n(n 1)s n 2 (1 s), so ψ n 1:n(F (v )) = n(n 1) 1 f(v) + O( 2 ) n 2 f(v) + O( 2 ) = n(n 1)f(v) + O( 2 ) ψn 1:n(F (v ))f(v ) = n(n 1)f(v) + O( 2 ) f(v) f (v) + O( 2 ) = n(n 1)(f(v)) 2 + O( 2 ) E θ n ψ n 1:n (F (v θ))f(v θ) = n(n 1)E θ n {(f(v θ)) 2 } + O( 2 ) 44

45 Putting it together, then, ψ 1 n 1:n F E θ n ψ n 1:n (v ) = n 1:n (F (v θ)) f(v θ) ψn 1:n ψ 1 n 1:n Eθ n {ψ n 1:n (F (v θ))} = n(n 1)E θ n{(f(v θ)) 2 } + O( 2 ) n(n 1) E θ n (f(v θ)) 2 + O( 2 ) E θ n (f(v θ)) 2 = Eθ n (f(v θ)) + O() 2 = E θ n (f(v θ)) 2 + O() so ψn 1:n (F n 1:n(v)) = ψn 1:n (F n 1:n(v)) v v ψ 1 n 1:n F n 1:n(s) (s)ds Eθ n (f(v θ)) 2 + O (v v) ds = 1 v v = 1 (v v) E θ n (f(v θ)) 2 + O (v v) 2 completing the proof. Power Against Specific Entry Models Consider the well-known entry model of evin and Smith (1994). There are n potential bidders, with identical entry costs c. Bidders observe the realization of θ but not their valuations, decide simultaneously whether to enter, and those who do pay their entry costs, learn their valuations, and participate in an ascending auction. (Since entry decisions are made prior to learning valuations, evin and Smith interpret c as the cost of actually discovering one s valuation of the object.) The evin-smith model has a unique symmetric equilibrium: for each realized θ, all potential bidders mix with the same probability q θ of entering, where q θ is the solution to c = n 1 n 1 (q θ ) n (1 q θ ) n 1 n u(n +1,θ) n n=0 where u(n+1,θ) is the expected payoff to each bidder in an n+1-bidder auction with characteristics θ. (The term in square brackets is the probability that exactly n of a bidder s opponents enter, so this makes each potential bidder indifferent between entering and not entering, and therefore willing to mix.) Corollary 11 In addition to the continuity and differentiability assumptions of Theorem 5, suppose Θ and f(v θ) is increasing in θ. Ifq θ is increasing in θ, then the evin-smith entry model leads to a violation of (9) for v sufficiently close to v for every pair (n, n ). Proof. We need only show that if q θ is increasing in θ, thenp n FOSD p n for n>n. We show this for discrete θ; the proof is the same for continuous θ, with integrals replacing sums. etting 45

46 P (θ) denote the prior probability distribution of θ, we calculate the probability, given n, that θ is less than or equal to a particular value θ: Pr(θ θ n) = = θ θ P (θ) nc n qθ n(1 q θ) n n θ Θ P (θ) nc n qθ n(1 q θ) n n θ θ P (θ) nc n qθ n(1 q θ) n n θ θ P (θ) nc n qθ n(1 q θ) n n + θ> θ P (θ) nc n qθ n(1 q θ) n n = R θ> θ P (θ)qn θ (1 q θ) n n θ θ P (θ)qn θ (1 q θ) n n where R(x) = 1 1+x. As n increases, each term in the numerator gets multiplied by q θ 1 q θ > q θ, 1 q θ while each term in the denominator gets multiplied by q θ 1 q θ q θ, so the argument of R increases; 1 q θ since R is decreasing, this means Pr(θ θ n) is decreasing in n, meaning p n FOSD p n for n>n. (Note that q θ increasing in θ (positive selection) is not guaranteed under the evin-smith model. Since u(n, θ) is decreasing in n, a sufficient condition for positive selection is for u(n, θ) to be increasing in θ for each n. If q θ were decreasing in θ, (9) would not detect the endogeneity of N, but (8) could.) Another standard model of entry in auctions is that of Samuelson (1985). In this model, bidders again have homogeneous entry costs c, but learn both θ and their valuations before deciding whether to participate in the auction. In this case, for each realized θ, bidders play a cutoff strategy, entering if and only if their valuation is at least v (θ), where v (θ) solves c = v (F (v θ)) n 1. Suppose that for θ>θ, F ( θ) FOSD F ( θ ). It is easy to show that v (θ) is increasing in θ, but that F (v (θ) θ) is decreasing in θ for higher values of θ, bidders use a higher cutoff, but still enter with higher probability. Thus, auctions with higher θ have stochastically more bidders; using the same steps as above, we can show that p n FOSD p n for n>n. However, there is one added complication. Those bidders who do enter, no longer have values which are i.i.d. draws from the distribution F ( θ). Conditional on entering, they have values above v, which are therefore i.i.d. draws from the truncated distribution F ( θ) F(v (θ) θ) 1 F (v (θ) θ). It is therefore this distribution, not the original one, which must have tail density increasing in θ to satisfy the conditions of Theorem 5. Corollary 12 In addition to the continuity and differentiability assumptions of Theorem 5, suppose Θ and F ( θ) FOSD F ( θ ) if θ>θ. Consider the Samuelson entry model, with entry costs 46

47 c, and let v (θ) be the equilibrium cutoffs. If f(v θ) 1 F (v (θ) θ) is increasing in θ, then the Samuelson entry model leads to a violation of (9) for v sufficiently close to v for every pair (n, n ). So under fairly general conditions, both the evin-smith and the Samuelson entry games lead to data which is inconsistent with (9), and would therefore be rejected by our test given sufficient data. Numerical Examples Next, we give numerical examples of these two entry models. et θ take two values, H and, with equal probabilities, and F ( θ) be log-normal distributions: when θ = H, ln(v i ) N(2.5, 0.5), and when θ =, ln(v i ) N(2.0, 0.5). V i has mean 13.8 and median 12.2 whenθ = H, and mean 8.4 and median 7.4 whenθ =. The density functions are shown below. Figure 6: Marginal distributions of V i conditional on θ for numerical examples Consider the evin-smith entry game, with 12 potential bidders and entry costs of $1.50. This causes bidders to each enter with probability q H = when θ = H, and probability q = when θ =. Givenq θ, we can then calculate the distribution of N, conditional on θ: n Pr(N = n θ = H) 0.3% 2.1% 7.3% 15.4% 22.1% 22.4% 16.6% 9.0% 3.6% 1.0% 0.2% 0.0% 0.0% Pr(N = n θ = ) 2.7% 11.4% 22.0% 25.7% 20.3% 11.4% 4.7% 1.4% 0.3% 0.0% 0.0% 0.0% 0.0% And likewise, assuming the prior probability of θ = H is 1 2, the conditional probability distribution of θ given a value of N: n Pr(θ = H N = n) 9% 15% 25% 37% 52% 66% 78% 87% 92% 95% 97% 99% 99% Next, consider the Samuelson entry game, with 8 potential bidders and the same entry costs of $1.50. This leads to cutoff values v (H) = $16.12 and v () = $10.48; each bidder enters with 47

48 probability 1 F (v (H) H) =0.288 when θ = H, and probability 1 F (v () ) =0.242 when θ =. Again, we can calculate the distribution of N conditional on θ, and the distribution of θ conditional on N: n Pr(N = n θ = H) 6.6% 21.4% 30.3% 24.5% 12.3% 4.0% 0.8% 0.1% 0.0% Pr(N = n θ = ) 10.8% 27.8% 31.1% 19.9% 8.0% 2.0% 0.3% 0.0% 0.0% Pr(θ = H N = n) 38% 44% 49% 55% 61% 66% 71% 76% 80% Figure 7 shows ψ 1 n 1:n (F n 1:n(v)) for n =2, 3, 4, 5, 6, 7, 8. If ψ 1 n 1:n (F n 1:n(v)) is increasing in n everywhere, we would fail to reject exogenous N. If it is decreasing in n at any v, we would reject exogenous N given enough data. Both of these entry models would clearly lead to rejection given enough data. Figure 7: ψ 1 n 1:n (F n 1:n(v)) under two entry models. ψ 1 n 1:n (F n 1:n(v)) is decreasing in n for some v.) (Recall that we reject exogenous N if evin-smith Samuelson A.4 Monte Carlo Simulations To understand how much data is required to give our test power, we perform our test of (9) on simulated data, using the example above of the evin and Smith (1994) model as the data-generating process. We generated 1000 data sets of sizes = 200, = 400 and = 800 and calculated the test-statistic for each simulated data set. In every instance we constructed the statistic exactly as 48