Multivariate Simultaneous Generalized Arch

This paper presents theoretical results on the formulation and estimation of
multivariate generalized ARCH models within simultaneous equations systems.
A new parameterization of the multivariate ARCH process is proposed, and
equivalence relations are discussed for the various ARCH parameterizations.
Constraints sufficient to guarantee the positive definiteness of the conditional
covariance matrices are developed, and necessary and sufficient conditions
for covariance stationaritp are presented. Identification and maximum likelihood
estimation of the parameters in the simultaneous equations context are
also covered.
1. INTRODUCTION
Although economists have long been interested in the analysis of behavior
under uncertainty, econometricians have only recently begun developing an
analytical framework to deal with uncertainty. A central feature of this
framework is the modeling of second and possiblj~h igher moments, as well.
One of the most prominent tools used to model the second moments is due
to Engle (1982). Engle (1982) suggested that these unobservable second
moments could be modeled by specifying a functional form for the conditional
variance and modeling the first and second moments jointly, giving
what is called in the literature the Autoregressive Conditional Heteroskedasticity
(ARCH) model. Of course, many different functional forms are possible,
but Engle's (1982) suggestion that the conditional variances depend on
elements in the information set in an autoregressive manner has become perhaps
the most common. This linear ARCH model was generalized by Bollerslev (1986) in a manner analogous to the extension from AR to ARMA
models in traditional times series by allowing past conditional variances to
appear in the current conditional variance equation. The resulting model is
called Generalized ARCH, or GARCH. These models have been applied
extensively in the literature (see, e.g., the survey by Bollerslev, Chou, and
Kroner, 1992).