Likelihood-based inference for dynamic panel data models

This paper considers maximum likelihood (ML)-based inferences for dynamic panel data models. We focus on the analysis of the panel data with a large number (N) of cross-sectional units and a small number (T) of repeated time series observations for each cross-sectional unit. We examine several different ML estimators and their asymptotic and finite-sample properties. Our major finding is that when data follow unit-root processes without or with drifts, the ML estimators have singular information matrices. This is a case of Sargan (Econometrica 51:1605–1634, 1983) in which the first-order condition for identification fails, but parameters are identified. The ML estimators are consistent, but they have non-standard asymptotic distributions, and their convergence rates are lower than N^1/2. In addition, the sizes of usual Wald statistics based on the estimators are distorted even asymptotically, and they reject the unit-root hypothesis too often. However, following Rotnitzky et al. (Bernoulli 6:243–284, 2000) we show that likelihood ratio (LR) tests for unit root follow mixtures of chi-square distributions. Our Monte Carlo experiments show that the LR tests with the p-values from the mixed distributions are much better sized than the Wald tests, although they tend to slightly over-reject the unit-root hypothesis in small samples. It is also shown that the LR tests for unit roots have good finite-sample power properties.