Semiparametric Estimation of First-Price Auction Models

We propose a semiparametric method to estimate the density of private values in first-price auctions. Specifically, we model private values through a set of conditional moment restrictions and use a two-step procedure. In the first step we recover a sample of pseudo private values using Local Polynomial Estimator. In the second step we use a GMM procedure to estimate the parameter(s) of interest. We show that the proposed semiparametric estimator is consistent, has an asymptotic normal distribution, and attains the parametric ("root-n") rate of convergence.Comment: 66 pages, 2 figure

Misspecification and Conditional Maximum Likelihood Estimation

Recently White (1982) studied the properties of Maximum Likelihood estimation of possibly misspecified models. The present paper extends Andersen (1970) results on Conditional Maximum Likelihood estimators (CMLE) to such a situation. In particular, the asymptotic properties of CMLE's are derived under correct and incorrect specification of the conditional model. Robustness of conditional inferences and estimation with respect to misspecification of the model for the conditioning variables is emphasized. Conditions for asymptotic efficiency of CMLE's are obtained, and specification tests a la Hausman (1978) and White (1982) are derived. Examples are also given to illustrate the use of CMLE's properties. These examples include the simple linear model, the multinomial logit model, the simple Tobit model, and the multivariate logit model

Nonidentification of Insurance Models with Probability of Accidents

In contrast to Aryal, Perrigne and Vuong (2009), this note shows that in an insurance model with multidimensional screening when only information on whether the insuree has been involved in some accident is available, the joint distribution of risk and risk aversion is not identified.

Cramer-Rao Bounds for Misspecified Models

In this paper, we derive some lower bounds of the Cramer-Rao type for the covariance matrix of any unbiased estimator of the pseudo-true parameters in a parametric model that may be misspecified. We obtain some lower bounds when the true distribution belongs either to a parametric model that may differ from the specified parametric model or to the class of all distributions with respect to which the model is regular. As an illustration, we apply our results to the normal linear regression model. In particular, we extend the Gauss-Markov Theorem by showing that the OLS estimator has minimum variance in the entire class of unbiased estimators of the pseudo-true parameters when the mean and the distribution of the errors are both misspecified

Generalized Inverses and Asymptotic Properties of Wald Tests

We consider Wald tests based on consistent estimators of g-inverses of the asymptotic covariance matrix ∑ of a statistic that is n^1/2-asymptotically normal distributed under the null hypothesis. Under the null hypothesis and under any sequence of local alternatives in the column space of ∑, these tests are asymptotically equivalent for any choice of g-inverses. For sequences of local alternatives not in the column space of ∑ and for a suitable choice of g- inverse, the asymptotic power of the corresponding Wald test can be made equal to zero or arbitrarily large. In particular, the test based on a consistent estimator of the Moore-Penrose inverse of ∑ has zero asymptotic power against sequences of local alternatives in the orthogonal complement to the column space of ∑

Identification of Insurance Models with Multidimensional Screening

We study the identification of an insurance model with multidimensional screening, where insurees are characterized by risk and risk aversion. The model is solved using the concept of certainty equivalence under constant absolute risk aversion and an unspecified joint distribution of risk and risk aversion. The paper then analyzes how data availability constraints identification under four data scenarios from the ideal situation to a more realistic one. The observed number of accidents for each insuree plays a key role to identify the model. In a first part, we consider the case of a continuum of coverages offered to each insuree whether the damage distribution is fully observed or truncated. Truncation arises from that an insuree files a claim only when the accident involves a damage above the deductible. Despite bunching due to multidimensional screening, we show that the joint distribution of risk and risk aversion is identified. In a second part, we consider the case of a finite number of coverages offered to each insuree. When the full damage distribution is observed, we show that despite additional pooling due to the finite number of contracts, the joint distribution of risk and risk aversion is identified under a full support assumption and a conditional independence assumption involving the car characteristics. When the damage distribution is truncated, the joint distribution is identified up to the probability that the damage is above the deductible. In a third part, we derive the restrictions imposed by the model on observables for the fourth scenario. We also propose several identification strategies for the damage probability at the deductible. These identification results are further exploited in a companion paper developing an estimation method with an application to insurance data

Probability Feedback in a Recursive System of Probability Models

This paper presents a general model for qualitative endogenous variables that is defined by a recursive system of conditional probability models in which the probabilities of some outcomes may depend on the probabilities of posterior outcomes. The model is related to, but conceptually different from C. D. Mallar's (1977) simultaneous probability model. It has as special cases the multivariate logit model (M. Nerlove and S. J. Press (1973, 1976)) and the constrained nested logit model (D. McFadden (1981)). The model can also be used to analyze outcomes of some game situations. Two examples are in particular considered: a game against Nature and a Stackelberg game under uncertainty. Identification of the structural parameters in the first example is seen to be related to the classical problem of stochastic revealed preference as studied by M. K. Richter and L. Shapiro (1978)