Nonparametric Estimation of Large Auctions with Risk Averse Bidders

Abstract

This article studies the robustness of Guerre et al.'s (2000) two-step nonparametric estimation procedure in a first-price, sealed-bid auction with n (n ≫ 1) risk averse bidders. Based on an asymptotic approximation with precision of order O(n ⁻²) of the intractable equilibrium bidding function, we establish the uniform consistency with rates of convergence of Guerre et al.'s (2000) two-step nonparametric estimator in the presence of risk aversion. Monte Carlo experiments show that the two-step nonparametric estimator performs reasonably well with a moderate number of bidders such as six.

Keywords:

• Large auctions
• Nonparametric estimation
• Risk averse bidders

JEL Classification:

• C14
• D44

ACKNOWLEDGMENTS

I would like to thank Stephen Cosslett, Robert de Jong, Lung-Fei Lee, Carlos Martins-Filho, Yi Zhu, and two anonymous referees for their helpful comments at various stages of this work. All remaining errors are my own.

Notes

¹We assume in this article that the reservation price is nonbinding, and hence the number of potential bidders is equal to the number of actual bidders.

²We drop the subscript i from v _i in the equilibrium bidding function because of the symmetry nature of the game.

³For instance, if w = 0, the CRRA utility function given by (0 < θ _r  < 1) does not have a bounded derivative at the initial wealth level.

⁴Otherwise, a more complex bidding model with endogenous entry should be considered, which is outside the scope of this article.

⁶Using U.S. forest service timber auction data, the estimated relative risk aversion coefficient is around 0.3 in Lu and Perrigne (2008) and Campo et al. (2011), and the estimated absolute risk aversion coefficient is very small with the order of magnitude being 10⁻⁵.

⁷When a bidder's value for the auctioned item is very low, the bid is a good approximation of the value. For the extreme case that the value is at the lower bound of its support, i.e., , the bid will be . So, at the lower part of the support of the value distribution, the figures show that method 3 performs well. I thank an anonymous referee for pointing this out.

Method 1: the infeasible two-step estimator with the exact bidding function.

Method 2: the two-step estimator with the approximate bidding function.

Method 3: the one-step estimator assuming bid =value.

IAB [MISE].

Method 1: the infeasible two-step estimator with the exact bidding function.

Method 2: the two-step estimator with the approximate bidding function.

Method 3: the one-step estimator assuming bid =value.

IAB [MISE].

⁸I thank an anonymous referee for pointing this out.

Color versions of one or more of the figures in the article can be found online at www.tandfonline.com/lecr.