Likelihood Expansion for Panel Regression Models with Factors

In this paper we provide a new methodology to analyze the (Gaussian) profile quasi likelihood function for panel regression models with interactive fixed effects, also called factor models. The number of factors is assumed to be known. Employing the perturbation theory of linear operators, we derive a power series expansion of the likelihood function in the regression parameters. Using this expansion we work out the first order asymptotic theory of the quasi maximum likelihood estimator (QMLE) in the limit where both the cross sectional dimension and the number of time periods become large. We find that there are two sources of asymptotic bias of the QMLE: bias due to correlation or heteroscedasticity of the idiosyncratic error term, and bias due to weak (as opposed to strict) exogeneity of the regressors. For idiosyncratic errors that are independent across time and cross section we provide an estimator for the bias and a bias corrected QMLE. We also discuss estimation in cases where the true parameter is on the boundary of the parameter set, and we provide bias corrected versions of the three classical test statistics (Wald, LR and LM test) and show that their asymptotic distribution is a χ-distribution. Monte Carlo simulations show that the bias correction of the QMLE and of the test statistics also work well for finite sample sizes.