Adaptive Estimation of Autoregression Models with Time-Varying Variances

31 Pages Posted: 17 Oct 2006

See all articles by Ke-Li Xu

Ke-Li Xu

Indiana University Bloomington

Peter C. B. Phillips

University of Auckland Business School; Yale University - Cowles Foundation; Singapore Management University - School of Economics

Date Written: October 2006


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.

Keywords: Adaptive estimation, Autoregression, Heterogeneity, Weighted regression

JEL Classification: C14. C22

Suggested Citation

Xu, Ke-Li and Phillips, Peter C. B., Adaptive Estimation of Autoregression Models with Time-Varying Variances (October 2006). Cowles Foundation Discussion Paper No. 1585, Available at SSRN:

Ke-Li Xu

Indiana University Bloomington ( email )

100 S. Woodlawn Ave.
Department of Economics, Wylie Hall
Bloomington, IN 47405-7104
United States


Peter C. B. Phillips (Contact Author)

University of Auckland Business School ( email )

12 Grafton Rd
Private Bag 92019
Auckland, 1010
New Zealand
+64 9 373 7599 x7596 (Phone)

Yale University - Cowles Foundation ( email )

Box 208281
New Haven, CT 06520-8281
United States
203-432-3695 (Phone)
203-432-5429 (Fax)

Singapore Management University - School of Economics

90 Stamford Road

Do you have a job opening that you would like to promote on SSRN?

Paper statistics

Abstract Views
PlumX Metrics