Stable autoregressive models of known ﬁnite order are considered with martingale diﬀerences errors scaled by an unknown nonparametric time-varying function generating heterogeneity. An important special case involves structural change in the error variance, but in most practical cases the pattern of variance change over time is unknown and may involve shifts at unknown discrete points in time, continuous evolution or combinations of the two. This paper develops kernel-based estimators of the residual variances and associated adaptive least squares (ALS) estimators of the autoregressive coeﬀicients. These are shown to be asymptotically eﬀicient, having the same limit distribution as the infeasible generalized least squares (GLS). Comparisons of the eﬀicient procedure and the ordinary least squares (OLS) reveal that least squares can be extremely ineﬀicient in some cases while nearly optimal in others. Simulations show that, when least squares work well, the adaptive estimators perform comparably well, whereas when least squares work poorly, major eﬀiciency gains are achieved by the new estimators.
Xu, Ke-Li and Phillips, Peter C.B., "Adaptive Estimation of Autoregressive Models with Time-Varying Variances" (2006). Cowles Foundation Discussion Papers. 1877.