Asymptotic Bias in Maximum Simulated Likelihood Estimation of Discrete Choice Models

Abstract

In this article, we have investigated a bias in an asymptotic expansion of the maximum simulated likelihood estimator introduced by Lerman and Manski for the estimation of discrete choice models. This bias occurs due to the nonlinearity of the derivatives of the log likelihood function and the statistically independent simulation errors of the choice probabilities across observations. This bias can be the dominated bias in an asymptotic expansion of the maximum simulated likelihood estimator when the number of simulated random variables per observationdoes not increase as fast as or faster than the sample size. The properly normalized maximum simulated likelihood estimator has even an asymptotic bias in its limiting distribution if the number of simulated random variables increase only as fast as the square root of the sample size. A bias-adjustment is introduced which can reduce the bias. Some Monte Carlo experiments have demonstrated the usefulness of the bias-adjustment procedure.