Integrated-Quantile-Based Estimation for First-Price Auction Models

Abstract

This article considers nonparametric estimation of first-price auction models under the monotonicity restriction on the bidding strategy. Based on an integrated-quantile representation of the first-order condition, we propose a tuning-parameter-free estimator for the valuation quantile function. We establish its cube-root-n consistency and asymptotic distribution under weaker smoothness assumptions than those typically assumed in the empirical literature. If the latter are true, we also provide a trimming-free smoothed estimator and show that it is asymptotically normal and achieves the optimal rate of Guerre, Perrigne, and Vuong (2000). We illustrate our method using Monte Carlo simulations and an empirical study of the California highway procurement auctions. Supplementary materials for this article are available online.

KEY WORDS:

• First price auctions
• Monotone bidding strategy
• Nonparametric estimation
• Tuning-Parameter-free

ACKNOWLEDGMENTS

We thank the editor, the associate editor, and two anonymous referees for their useful suggestions that improved this article. We thank the valuable comments from Victor Aguirregabiria, Tim Armstrong, Yanqin Fan, Christian Gourieroux, Emmanuel Guerre, Sung Jae Jun, Shakeeb Khan, Nianqing Liu, Ruixuan Liu, Vadim Marmer, Ismael Mourifie, Joris Pinkse, and Quang Vuong. We benefited from discussions with participants at CMES 2014, the 2nd CEC (Nanjing), the 5th Shanghai Econometrics Workshop at SUFE, ESWC 2015, and the CEME 2015 Conference at Cornell. Luo gratefully acknowledges the financial support from the National Natural Science Foundation of China (NSFC-71373283 and NSFC-71463019). All errors are ours.

Notes

See Hickman and Hubbard (2014) for a modified version of the GPV estimator which replaces trimming with boundary correction.

In general, the first price auction model is not identified if there is unobserved heterogeneity across auctions, see Armstrong (2013b).

Please see the online supplementary materials (Luo and Wan 2016) for the proof of all corollaries in this article.

Marmer, Shneyerov, and Xu (2013) considered entry. In our model, all the potential bidders enter with probability one. Let $h_{\text{MSX}}$ be the bandwidth of Marmer, Shneyerov, and Xu (2013) and h_LW be ours. Then, the equality of variances holds by letting $h_{\text{MSX}}=h_{LW}{Q}^{\text{'}}\left({\alpha }_0\right)$ and observing that Lh^r → 1⇔nh^r → I since n = LI and I is fixed. As a matter of fact, taking derivative of a smoothed quantile estimator and taking reciprocal of a kernel density estimator are two general approaches to estimate quantile derivatives. For general comparison of these two approaches, see Jones (1992).

Under a similar set of smoothness assumptions to ours, Armstrong (2013a) proposed to estimate the bidding strategy by maximizing the sample analog of the bidder’s objective function and subsequently estimates the valuation distribution function at cube-root-n rate. Our approach is based on the integrated-quantile representation of the first-order condition and imposes monotonicity restriction. Both estimators are tuning-parameter-free and robust to the degree of smoothness in the model.