## Abstract

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.