Incidental parameters, initial conditions and sample size in statistical inference for dynamic panel data models

We use a quasi-likelihood function approach to clarify the role of initial values and the relative size of the cross-section dimension N and the time series dimension T in the asymptotic distribution of dynamic panel data models with the presence of individual- specific effects. We show that the quasi-maximum likelihood estimator (QMLE) treating initial values as fixed constants is asymptotically biased of order square root of N divided by T squared as T goes to infinity for a time series models and asymptotically biased of order square root of N divided by T for a model that also contains other covariates that are correlated with the individual-specific effects. Using Mundlak- Chamberlain approach to condition the effects on the covariates can reduce the asymptotic bias to the order of square root of N divided by T cubed, provided the data generating processes for the covariates are homogeneous across cross-sectional units. On the other hand, the QMLE combining the Mundlak-Chamberlain approach with the proper treatment of initial value distribution is asymptotically unbiased if N goes to infinity whether T is fixed or goes to infinity. Monte Carlo studies are conducted to demonstrate the importance of properly treating initial values in getting valid statistical inference. The results also suggest that when using the conditional approach to get around the issue of incidental parameters, in finite sample it is perhaps better to follow Mundlak's (1978) suggestion to simply condition the individual effects or initial values on the time series average of individual's observed regressors under the assumption that our model is correctly specified.