COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1746 Uniform Asymptotic Normality in Stationary and Unit Root Autoregression Chirok Han, Peter C. B. Phillips and Donggyu Sul January 2010 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| < 1 and the limit distribution is the
same as the Gaussian maximum likelihood estimator (MLE), but when rho = 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. |
