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.
Han, Chirok; Phillips, Peter C.B.; and Sul, Donggyu, "Uniform Asymptotic Normality in Stationary and Unit Root Autoregression" (2010). Cowles Foundation Discussion Papers. 2074.