Uniform Asymptotic Normality in Stationary and Unit Root Autoregression. Working Paper

While differencing transformations can eliminate nonstationarity, they typically reduce sig-nal 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 autoregres-sions with some very desirable properties. In rst order autoregression, a partially aggregated estimator based on moment conditions in differences is shown to have a limiting normal distri-bution which holds uniformly in the autoregressive coefcient including stationary and unit root cases. The rate of convergence is p nwhen jj < 1 and the limit distribution is the same as the Gaussian maximum likelihood estimator (MLE), but when = 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 articial shrinkage technique or otherwise accelerating convergence at unity at the cost of performance in the neighborhood of unity