Asymptotic Bias in Simulated Maximum Likelihood Estimation of Discrete Choice Models
研究了离散选择模型模拟极大似然估计中因对数似然函数非线性及模拟误差导致的渐近偏误，发现当每观测模拟次数增长慢于样本量时偏误占主导，并提出了偏误调整方法。
In this article, we investigate a bias in an asymptotic expansion of the simulated maximum likelihood estimator introduced by Lerman and Manski (pp. 305–319 in C. Manski and D. McFadden (eds.), Structural Analysis of Discrete Data with Econometric Applications , Cambridge: MIT Press, 1981) 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 dominating bias in an asymptotic expansion of the simulated maximum likelihood estimator when the number of simulated random variables per observation does not increase at least as fast as the sample size. The properly normalized simulated maximum likelihood estimator even has an asymptotic bias in its limiting distribution if the number of simulated random variables increases only as fast as the square root of the sample size. A bias-adjustment is introduced that can reduce the bias. Some Monte Carlo experiments have demonstrated the usefulness of the bias-adjustment procedure.