Uniform Inference in Panel Autoregression

Abstract

This paper considers estimation and inference concerning the autoregressive coefficient (𝜌) in a panel autoregression for which the degree of persistence in the time dimension is unknown. Our main objective is to construct confidence intervals for 𝜌 that are asymptotically valid, having asymptotic coverage probability at least that of the nominal level uniformly over the parameter space. The starting point for our confidence procedure is the estimating equation of the Anderson–Hsiao (AH) IV procedure. It is well known that the AH IV estimation suffers from weak instrumentation when 𝜌 is near unity. But it is not so well known that AH IV estimation is still consistent when 𝜌=1. In fact, the AH estimating equation is very well-centered and is an unbiased estimating equation in the sense of Durbin (1960), a feature that is especially useful in confidence interval construction. We show that a properly normalized statistic based on the AH estimating equation, which we call the 𝕄 statistic, is uniformly convergent and can be inverted to obtain asymptotically valid interval estimates. To further improve the informativeness of our confidence procedure in the unit root and near unit root regions and to alleviate the problem that the AH procedure has greater variation in these regions, we use information from unit root pretesting to select among alternative confidence intervals. Two sequential tests are used to assess how close 𝜌 is to unity, and different intervals are applied depending on whether the test results indicate 𝜌 to be near or far away from unity. When 𝜌 is relatively close to unity, our procedure activates intervals whose width shrinks to zero at a faster rate than that of the confidence interval based on the 𝕄 statistic. Only when both of our unit root tests reject the null hypothesis does our procedure turn to the 𝕄 statistic interval, whose width has the optimal 𝑁−1/2𝑇−1/2 rate of shrinkage when the underlying process is stable. Our asymptotic analysis shows this pretest-based confidence procedure to have coverage probability that is at least the nominal level in large samples uniformly over the parameter space. Simulations confirm that the proposed interval estimation methods perform well in finite samples and are easy to implement in practice. A supplement to the paper provides an extensive set of new results on the asymptotic behavior of panel IV estimators in weak instrument settings.