First Difference MLE and Dynamic Panel Estimation

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.Asymptote, Bounded support, Dynamic panel, Efficiency, First difference MLE, Likelihood, Quartic equation, Restricted extremum estimator

GMM with Many Moment Conditions

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.Epiconvergence, GMM, Irrelevant instruments, IV, Large numbers of instruments, LIML estimation, Panel models, Pseudo true value, Signal, Signal Variability, Weak instrumentation

GMM Estimation for Dynamic Panels with Fixed Effects and Strong Instruments at Unity

This paper develops new estimation and inference procedures for dynamic panel data models with fixed effects and incidental trends. A simple consistent GMM estimation method is proposed that avoids the weak moment condition problem that is known to affect conventional GMM estimation when the autoregressive coefficient (rho) is near unity. In both panel and time series cases, the estimator has standard Gaussian asymptotics for all values of rho in (-1, 1] irrespective of how the composite cross section and time series sample sizes pass to infinity. Simulations reveal that the estimator has little bias even in very small samples. The approach is applied to panel unit root testing.Asymptotic normality, Asymptotic power envelope, Moment conditions, Panel unit roots, Point optimal test, Unit root tests, Weak instruments

Uniform Asymptotic Normality in Stationary and Unit Root Autoregression

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 |rho|Aggregating information, Asymptotic normality, Bias Reduction, Differencing, Efficiency, Full aggregation, Maximum likelihood estimation

LAD Asymptotics under Conditional Heteroskedasticity with Possibly Infinite Error Densities

Least absolute deviations (LAD) estimation of linear time-series models is considered under conditional heteroskedasticity and serial correlation. The limit theory of the LAD estimator is obtained without assuming the finite density condition for the errors that is required in standard LAD asymptotics. The results are particularly useful in application of LAD estimation to financial time series data.Asymptotic leptokurtosis, Convex function, Infinite density, Least absolute deviations, Median, Weak convergence

Gaussian Inference in AR(1) Time Series with or without a Unit Root

This note introduces a simple first-difference-based approach to estimation and inference for the AR(1) model. The estimates have virtually no finite sample bias, are not sensitive to initial conditions, and the approach has the unusual advantage that a Gaussian central limit theory applies and is continuous as the autoregressive coefficient passes through unity with a uniform / n rate of convergence. En route, a useful CLT for sample covariances of linear processes is given, following Phillips and Solo (1992). The approach also has useful extensions to dynamic panels

GMM Estimation for Dynamic Panels with Fixed Effects and Strong Instruments at Unity

This paper develops new estimation and inference procedures for dynamic panel data models with fixed effects and incidental trends. A simple consistent GMM estimation method is proposed that avoids the weak moment condition problem that is known to affect conventional GMM estimation when the autoregressive coefficient (rho) is near unity. In both panel and time series cases, the estimator has standard Gaussian asymptotics for all values of rho in (-1, 1] irrespective of how the composite cross section and time series sample sizes pass to infinity. Simulations reveal that the estimator has little bias even in very small samples. The approach is applied to panel unit root testing

True Limit Distributions of the Anderson-Hsiao IV Estimators in Panel Autoregression

This note derives the correct limit distributions of the Anderson Hsiao (1981) levels and differences instrumental variable estimators, provides comparisons showing that the levels IV estimator has uniformly smaller variance asymptotically as the cross section ( n ) and time series ( T ) sample sizes tend to infinity, and compares these results with those of the first difference least squares (FDLS) estimator

Infinite Density at the Median and the Typical Shape of Stock Return Distributions

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 L_1 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.Asymptotic leptokurtosis, Infinite density at the median, Least absolute deviations, Kernel density estimation, Stock returns, Stylized facts