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.
This paper introduces a new estimation method for dynamic panel models with ﬁxed eﬀects and AR(p) idiosyncratic errors. The proposed estimator uses a novel form of systematic diﬀerencing, called X-diﬀerencing, that eliminates ﬁxed eﬀects and retains information and signal strength in cases where there is a root at or near unity. The resulting “panel fully aggregated” estimator (PFAE) is obtained by pooled least squares on the system of X-diﬀerenced equations. The method is simple to implement, free from bias for all parameter values, including unit root cases, and has strong asymptotic and ﬁnite sample performance characteristics that dominate other procedures, such as bias corrected least squares, GMM and system GMM methods. The asymptotic theory holds as long as the cross section (n) or time series (T) sample size is large, regardless of the n/T ratio, which makes the approach appealing for practical work. In the time series AR(1) case (n = 1), the FAE estimator has a limit distribution with smaller bias and variance than the maximum likelihood estimator (MLE) when the autoregressive coeﬀicient is at or near unity and the same limit distribution as the MLE in the stationary case, so the advantages of the approach continue to hold for ﬁxed and even small n. For panel data modeling purposes, a general-to-speciﬁc selection rule is suggested for choosing the lag parameter p and the procedure works in a standard manner, aiding practical implementation. The PFAE estimation method is also applicable to dynamic panel models with exogenous regressors. Some simulation results are reported giving comparisons with other dynamic panel estimation methods.
While diﬀerencing transformations can eliminate nonstationarity, they typically reduce signal strength and correspondingly reduce rates of convergence in unit root autoregressions. The present paper shows that aggregating moment conditions that are formulated in diﬀerences provides an orderly mechanism for preserving information and signal strength in autoregressions with some very desirable properties. In ﬁrst order autoregression, a partially aggregated estimator based on moment conditions in diﬀerences is shown to have a limiting normal distribution which holds uniformly in the autoregressive coeﬀicient rho including stationary and unit root cases. The rate of convergence is root of n when |τ| < 1 and the limit distribution is the same as the Gaussian maximum likelihood estimator (MLE), but when τ = 1 the rate of convergence to the normal distribution is within a slowly varying factor of n. A fully aggregated estimator is shown to have the same limit behavior in the stationary case and to have nonstandard limit distributions in unit root and near integrated cases which reduce both the bias and the variance of the MLE. This result shows that it is possible to improve on the asymptotic behavior of the MLE without using an artiﬁcial shrinkage technique or otherwise accelerating convergence at unity at the cost of performance in the neighborhood of unity.
Statistics are developed to test for the presence of an asymptotic discontinuity (or inﬁnite density or peakedness) in a probability density at the median. The approach makes use of work by Knight (1998) on L1 estimation asymptotics in conjunction with non-parametric kernel density estimation methods. The size and power of the tests are assessed, and conditions under which the tests have good performance are explored in simulations. The new methods are applied to stock returns of leading companies across major U.S. industry groups. The results conﬁrm the presence of inﬁnite density at the median as a new signiﬁcant empirical evidence for stock return distributions.
This paper provides a ﬁrst order asymptotic theory for generalized method of moments (GMM) estimators when the number of moment conditions is allowed to increase with the sample size and the moment conditions may be weak. Examples in which these asymptotics are relevant include instrumental variable (IV) estimation with many (possibly weak or uninformed) instruments and some panel data models covering moderate time spans and with correspondingly large numbers of instruments. Under certain regularity conditions, the GMM estimators are shown to converge in probability but not necessarily to the true parameter, and conditions for consistent GMM estimation are given. A general framework for the GMM limit distribution theory is developed based on epiconvergence methods. Some illustrations are provided, including consistent GMM estimation of a panel model with time varying individual eﬀects, consistent LIML estimation as a continuously updated GMM estimator, and consistent IV structural estimation using large numbers of weak or irrelevant instruments. Some simulations are reported.
Keywords: Epiconvergence, GMM, Irrelevant instruments, IV, Large numbers of instruments, LIML estimation, Panel models, Pseudo true value, Signal, Signal Variability, Weak instrumentation