# 17-015/III (2017-01-23)

Author(s)
Chia-Lin Chang, National Chung Hsing University, Taiwan; Michael McAleer, National Tsing Hua University, Taiwan; Erasmus University Rotterdam, The Netherlands; Complutense University of Madrid, Spain; Yokohama National University, Japan
Keywords:
Random coefficient stochastic process, Off-diagonal parametric restrictions, Diagonal and Full BEKK, Regularity conditions, Asymptotic properties, Conditional volatility, Univariate and multivariate models.
JEL codes:
C22, C32, C52, C58

The purpose of the paper is to show that univariate GARCH is not a special case of multivariate GARCH, specifically the Full BEKK model, except under parametric restrictions on the off-diagonal elements of the random coefficient autoregressive coefficient matrix, provides the regularity conditions that arise from the underlying random coefficient autoregressive process, and for which the (quasi-) maximum likelihood estimates have valid asymptotic properties under the appropriate parametric restrictions. The paper provides a discussion of the stochastic processes, regularity conditions, and asymptotic properties of univariate and multivariate GARCH models. It is shown that the Full BEKK model, which in practice is estimated almost exclusively, has no underlying stochastic process, regularity conditions, or asymptotic properties.