Adaptive Estimation in Arch Models

OLIVERLINTON
Nuffield College
We construct efficient estimators of the identifiable parameters in a regression
model when the errors follow a stationary parametric ARCH(P)process. We
do not assume a functional form for the conditional density of the errors, but
do require that it be symmetric about zero. The estimators of the mean parameters
are adaptive in the sense of Bickel [2]. The ARCH parameters are not
jointly identifiable with the error density. We consider a reparameterization of
the variance process and show that the identifiable parameters of this process
are adaptively estimable.

1. INTRODUCTION
We consider the problem of obtaining efficient estimators of the identifiable
parameters in the following linear regression model where the errors are conditionally
heteroskedastic according to an ARCH(P) process:
This specification of the error process was originally suggested in Engle
[lo], and was employed there to model United Kingdom inflation rates. It
has been used in countless empirical studies-see the survey papers of Engle
and Bollerslev [12] and Bollerslev, Chou, and Kroner [7] for references.
The ARCH specification rationalizes two well-established empirical regularities
about financial and macroeconomic time series. When el, j =
1,2,. . . ,P are all positive, the process a: is positively serially dependent.
This is an important feature: Many financial and macroeconomic time series
are characterized by episodic bursts of volatility followed by more tranquil
periods. Uncertainty about future events-and the consequent risk to
investors -varies over time yet typically is closely related to previous assessments
of uncertainty.