Asymptotic Theory for ARCH Models_ Estimation and Testing

ANDREWA. WEISS
University of Southern California at Los Angeles
In the context of a linear dynamic model with moving average errors, we consider
a heteroscedastic model which represents an extension of the ARCH
model introduced by Engle [4]. We discuss the properties of maximum likelihood
and least squares estimates of the parameters of both the regression and
ARCH equations, and also the properties of various tests of the model that
are available. We do not assume that the errors are normally distributed

1. INTRODUCTION
We begin with the situation in which a researcher wishes to model the heteroscedasticity
in a time series regression. For this, Engle [4] has introduced
the concept of autoregressive conditional heteroscedasticity (ARCH). This
is seen as an extension of time series behavior in the mean, allowing the
variance of the errors to change if the process takes into account past experience
but assumes it constant if this experience is not known. In a process
with stochastic regressors, which is the case in most time series processes,
this corresponds to the usual properties of the mean of the output from the
regression model. Hence it is more appealing than the common assumption
of unconditional heteroscedasticity, i.e., where the unconditional variance
changes through time, which also implies, for example, that the stochastic
variables in the process cannot affect both the mean and variance of the
dependent variable.