Nonparametric identification in asymmetric second-price auctions: a new approach

Abstract

)>IJH=?J This paper proposes an approach to proving nonparametric identication for distributions of bidders' values in asymmetric second-price auctions. I consider the case when bidders have independent private values and the only available data pertain to the winner's identity and the transaction price. My proof of identication is constructive and is based on establishing the existence and uniqueness of a solution to the system of non-linear dierential equations that describes relationships between unknown distribution functions and observable functions. The proof is conducted in two logical steps. First, I prove the existence and uniqueness of a local solution. Then I describe a method that extends this local solution to the whole support. This paper delivers other interesting results. I show how this approach can be applied to obtain identication in more general auction settings, for instance, in auctions with stochastic number of bidders or weaker support conditions. Furthermore, I demonstrate that my results can be extended to generalized competing risks models. Moreover, contrary to results in classical competing risks (Roy model), I show that in this generalized class of models it is possible to obtain implications that can be used to check whether the risks in a model are dependent. Finally, I provide a sieve minimum distance estimator and show that it consistently estimates the underlying valuation distribution of interest.

Large sample properties for estimators based on the order statistics approach in auctions

“…2 Even though this paper focuses on estimation of auctions, the statistical problem is a prototype for other settings in which only the minimum, maximum, or other element of an ordered sample of realizations of a vector of random variables is observed. Athey and Haile (2002) and Komarova (2009) also pointed out the similarities between the problem of estimating valuation distributions from observations of the highest bid and competing risks models in duration analysis, so that estimators for failure time models with nonparametric baseline hazards under endogenous censoring should be expected to share some of the statistical properties of the procedures analyzed in this paper.…”

“…Much of the recent literature on nonparametric estimation of auctions has focused on identification (for a relatively recent survey, see Athey and Haile (2007)), where Athey and Haile (2002) and Komarova (2009) provided results on nonparametric identification from incomplete bidding data, and Haile and Tamer (2003) proposed a method of constructing nonparametric bounds on the distribution under weaker assumptions on bidding behavior by inverting the distribution of each bid separately. Guerre, Perrigne, and Vuong (2000) derived rate-optimal nonparametric estimators for first-price auctions when all bids are observed, a case for which the problem of inverting distributions of order statistics does not arise.…”

A new approach to identifying generalized competing risks models with application to second-price auctions

“…One possible approach, which is discussed in Komarova (2009), is to use a sieve method based on the minimization of a certain sample objective function over a chosen sieve space. One of them would be to develop procedures for the estimation of the distribution functions of private values.…”

“…Komarova (2009) shows that one can relax the support conditions and permit distributions to have different upper support points as well as holes in the support. Using the case of three bidders, I outline the specifics of proving identification in second-price auctions in which the set of actual bidders is unknown and varies exogenously.…”

Extremum sieve estimation in <i>k</i>-out-of-<i>n</i> systems

“…5 Consistency will hold if all conditions in Lemma A1 are satisfied. I divide these conditions into three groups, as in Newey and Powell (2003).…”