COWLES FOUNDATION FOR RESEARCH IN
ECONOMICS Box 208281
COWLES FOUNDATION DISCUSSION PAPER NO. 1585R Adaptive Estimation of Autoregressive Models with Time-Varying Variances Ke-Li Xu and Peter C.B. Phillips 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
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. |
