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# Nonparametric identification and semi-nonparametric estimation of first-price auctions

Abstract (Summary)
iii In this thesis, we study nonparametric identification of first-price auction models and propose a semi-nonparametric simulated integrated moment estimation method to recover the underlying value distribution. 1 In the first essay, we investigate the nonparametric identification of the first-price auction model. In most cases in the nonparametric auction literature, the support of the biddersâ€™ values is assumed to be bounded. We show via an alternative nonparametric identification proof that the boundedness assumption can be relaxed to the condition that the value distribution has a finite expectation. In the first instance, we show this for the case of independent and identical first-price auctions, and then we extend the proof to the case of first-price auctions with observed auction-specific heterogeneity. Also, we consider the case where the log of the values is modeled as a median regression model, and the case where the bidders know ex-ante the actual number of bidders rather than the number of potential bidders. In the second essay, we propose a semi-nonparametric simulated integrated moment (SNP-SIM) to estimate the value distribution of independently repeated identical first-price auctions. First, we construct an increasing sequence of compact metric spaces of distribution functions (the sieve), based on the approach in Bierens (2007). Given a candidate value distribution function in the sieve, we simulate bids according to the equilibrium bid function involved. We take the difference of the empirical characteristic 1The three essays in this thesis are co-authored with Herman Bierens. iv functions of the actual and simulated bids as the moment function. The objective function is then the integral of the squared moment function over an interval. Minimizing this integral to the distribution functions in the sieve then yields a uniformly consistent semi-nonparametric estimator of the actual value distribution. Also, we propose an integrated moment test for the validity of the first-price auction model, and a data-driven method for the choice of the sieve order. Finally, we conduct a few numerical experiments to check the performance of our approach. In the third essay, we propose to estimate first-price auction models with observed auction-specific heterogeneity via a semi-nonparametric simulated integrated conditional moment (SNP-SICM) method. The auction-specific heterogeneity will be incorporated via a median regression model for the log values with unknown error distribution. The latter distribution will be modeled semi-nonparametrically using orthonormal Legendre polynomials, similar to the approach in Bierens (2007). Given a parametric specification of the median function, the semi-nonparametric conditional value distribution involved can be estimated consistently by minimizing the integrated square distance between the empirical characteristic functions of the actual bids and the simulated bids, together with the covariates, via an integrated conditional moment criterion. This approach yields as a by-product an integrated conditional moment test for the validity of the model. Moreover, we apply the SNP-SICM estimation method to the US timber auction data and test the validity of the first-price auction model for this data.
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