The behaviour of the maximum likelihood estimator of limited dependent variable models in the presence of fixed effects

Abstract

The nonlinear fixed‐effects model has two shortcomings, one practical and one methodological. The practical obstacle relates to the difficulty of computing the MLE of the coefficients of non‐linear models with possibly thousands of dummy variable coefficients. In fact, in many models of interest to practitioners, computing the MLE of the parameters of fixed effects model is feasible even in panels with very large numbers of groups. The result, though not new, appears not to be well known. The more difficult, methodological issue is the incidental parameters problem that raises questions about the statistical properties of the ML estimator. There is relatively little empirical evidence on the behaviour of the MLE in the presence of fixed effects, and that which has been obtained has focused almost exclusively on binary choice models. In this paper, we use Monte Carlo methods to examine the small sample bias of the MLE in the tobit, truncated regression and Weibull survival models as well as the binary probit and logit and ordered probit discrete choice models. We find that the estimator in the continuous response models behaves quite differently from the familiar and oft cited results. Among our findings are: first, a widely accepted result that suggests that the probit estimator is actually relatively well behaved appears to be incorrect; second, the estimators of the slopes in the tobit model, unlike the probit and logit models that have been studied previously, appear to be largely unaffected by the incidental parameters problem, but a surprising result related to the disturbance variance estimator arises instead; third, lest one jumps to a conclusion that the finite sample bias is restricted to discrete choice models, we submit evidence on the truncated regression, which is yet unlike the tobit in that regard—it appears to be biased towards zero; fourth, we find in the Weibull model that the biases in a vector of coefficients need not be in the same direction; fifth, as apparently unexamined previously, the estimated asymptotic standard errors for the ML estimators appear uniformly to be downward biased when the model contains fixed effects. In sum, the finite sample behaviour of the fixed effects estimator is much more varied than the received literature would suggest.