Strong Consistency of Estimators for Multivariate Arch Models

This paper deals with the asymptotic properties of quasi-maximum likelihood estimators
for multivariate heteroskedastic models. For a general model, we give
conditions under which strong consistency can be obtained; unlike in the current
literature, the assumptions on the existence of moments of the error term are weak,
and no study of the various derivatives of the likelihood is required. Then, for a
particular model, the multivariate GARCH model with constant correlation, we
describe the set of parameters where these conditions hold.
1. INTRODUCTION
As a result of the paper by Mandelbrot (1963), we know that for certain time series,
and especially economic and financial time series, the conditional variance is
not constant over time. Therefore, several models trying to take into account this
particular behavior have been introduced. The most successful ones are undoubtedly
the autoregressive conditional heteroskedastic (ARCH) model, introduced by
Engle (1 982), and some of its derivative models (GARCH, GARCH-M, EGARCH,
etc.). The implementation of these parametric models is relatively simple. And,
from a practical point of view, it is well known now how to identify, estimate, and
test this kind of model (for a description of these methods and some empirical evidence,
see the survey of Bollerslev, Chou, and Kroner, 1992).

From a theoretical point of view, however, the problem of statistical inference
for these models remains partially open. Indeed, Weiss (1986) gave the first proof
of consistency and asymptotic normality of the maximum likelihood estimator
for univariate ARCH model but under strong conditions on the existence of the
moments of the error term. On the other hand, a paper of Nelson (1990) showed
that for a particular model, the GARCH(1,l) model, the process can be strictly
stationary with infinite second moment, and the empirical studies (see Bollerslev
et al., 1992) suggest that some financial time series seem to be characterized by
such a process. Therefore, statistical inference under weak conditions on the existence
of moments is needed.