COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1780 First Difference MLE and Dynamic Panel Estimation Chirok Han and Peter C. B. Phillips January 2011 First difference maximum likelihood (FDML) seems an attractive
estimation methodology in dynamic panel data modeling because differencing eliminates
fixed effects and, in the case of a unit root, differencing transforms the data to
stationarity, thereby addressing both incidental parameter problems and the possible
effects of nonstationarity. This paper draws attention to certain pathologies that arise
in the use of FDML that have gone unnoticed in the literature and that affect both finite
sample peformance and asymptotics. FDML uses the Gaussian likelihood function for first
differenced data and parameter estimation is based on the whole domain over which the
log-likelihood is defined. However, extending the domain of the likelihood beyond the
stationary region has certain consequences that have a major effect on finite sample and
asymptotic performance. First, the extended likelihood is not the true likelihood even in
the Gaussian case and it has a finite upper bound of definition. Second, it is often
bimodal, and one of its peaks can be so peculiar that numerical maximization of the
extended likelihood frequently fails to locate the global maximum. As a result of these
pathologies, the FDML estimator is a restricted estimator, numerical implementation is not
straightforward and asymptotics are hard to derive in cases where the peculiarity occurs
with non-negligible probabilities. We investigate these problems, provide a convenient new
expression for the likelihood and a new algorithm to maximize it. The peculiarities in the
likelihood are found to be particularly marked in time series with a unit root. In this
case, the asymptotic distribution of the FDMLE has bounded support and its density is
infinite at the upper bound when the time series sample size T approaching infinity. As
the panel width n approaching infinity the pathology is removed and the limit theory is
normal. This result applies even for T fixed and we present an expression for the
asymptotic distribution which does not depend on the time dimension. When n,T approaching
infinity, the FDMLE has smaller asymptotic variance than that of the bias corrected MLE,
an outcome that is explained by the restricted nature of the FDMLE. |
