Robustness against incidental parameters and mixing distributions

Neyman and Scott (1948) define the incidental parameter problem. In panel data with T observations per individual and unobservable individual- specific effects, the inconsistency of the maximum likelihood estimator of the common parameters is in general of the order 1/T. This paper considers the integrated likelihood estimator and develops the integrated moment estimator. It shows that the inconsistency of the integrated likelihood estimator reduces from 1/T to 1/T2 if an information orthogonal parametrization is used. It derives information orthogonal moment functions for the general linear model and the index model with weakly exogenous regressors and thereby offers an approximate solution for the incidental parameter problem for a wide range of models. It argues that reparametrizations are easier in a Bayesian framework and shows how to use the 1/T2- result to increase the robustness against the choice of mixing distribution. The integrated likelihood estimator is consistent and adaptive for asympototics in which T proportional to N to the power alpha where alpha is larger than 1/3. The paper also shows that likelihood methods that use sufficient statistics for the individual-specific effects can be viewed as a special case of the integrated likelihood estimator.