The Bias of the Fixed Effects Estimator in Nonlinear Models

The nonlinear fixed effects model has two shortcomings, one practical, one methodological. The practical obstacle relates to the difficulty of estimating nonlinear models with possibly thousands of dummy variable coefficients. In fact, in many models of interest to practitioners, estimation of the 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 estimator. There is relatively little empirical evidence on the behavior of the fixed effects estimator 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 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 slopes in the tobit model, unlike the probit and logit models that have been studied previously, are largely unaffected by the incidental parameters problem, but a surprising result related to the disturbance variance estimator arises instead; lest one jump 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 is biased toward 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 fixed effects estimators appear uniformly to be downward biased. Finally, we consider directly the issue of “consistency” in the context of the tobit model and find that widely received perceptions to the contrary, at least in this model, the fixed effects estimator appears to be neither biased nor inconsistent. In sum, the finite sample behavior of the fixed effects estimator is much more varied than the received literature would suggest.