Internally Consistent Estimation of Nonlinear Panel Data Models with Correlated Random Effects

This paper investigates identification and estimation of semi-parametric nonlinear panel data models with correlated random effects (CRE). It is shown that under the Mundlak-type CRE specification, the average (or integrated) likelihood is the convolution of the proposed models and the conditional distribution of the unobserved heterogeneity. Then the conditional distribution of the unobserved heterogeneity can be recovered by means of Fourier transformation without imposing any distributional assumptions on it. Combining the proposed the conditional distributions of the outcome variables with the recovered distribution of the unobserved heterogeneity, we can construct a parametric family of average likelihood functions of observables and then show that the parameter vector is identifiable. Based on the identification condition, we propose a semi-parametric two-step maximum likelihood estimator which is root-n consistent and asymptotically normal. Compared with the conventional parametric CRE approaches, the advantage of our method is that it is not subject to the function form misspecification. We investigate the finite sample properties of the proposed estimator through a Monte Carlo study and apply our method to determine the persistence effects of union membership.