The purpose of the paper is to
(i) 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, that are not consistent with Full BEKK, and
(ii) provide the regularity conditions that arise from the underlying random coefficient autoregressive process, for which the (quasi-) maximum likelihood estimates (QMLE) have valid asymptotic properties under the appropriate parametric restrictions.
The paper provides a discussion of the stochastic processes that lead to the alternative specifications, regularity conditions, and asymptotic properties of the univariate and multivariate GARCH models. It is shown that the Full BEKK model, which in empirical practice is estimated almost exclusively compared with Diagonal BEKK (DBEKK), has no underlying stochastic process that leads to its specification, regularity conditions, or asymptotic properties, as compared with DBEKK. An empirical illustration shows the differences in the QMLE of the parameters of the conditional means and conditional variances for the univariate, DEBEKK and Full BEKK specifications.

Asymptotic properties, Conditional volatility, Diagonal BEKK, Fossil fuels and carbon emissions, Full BEKK, Off-diagonal parametric restrictions, Random coefficient stochastic process, Regularity conditions, Univariate and multivariate models
Time-Series Models; Dynamic Quantile Regressions (jel C22), Time-Series Models; Dynamic Quantile Regressions (jel C32), Model Evaluation and Testing (jel C52), Financial Econometrics (jel C58),
Finance Research Letters
Department of Econometrics

Chang, C-L, & McAleer, M.J. (2018). The fiction of full BEKK: Pricing fossil fuels and carbon emissions. Finance Research Letters. doi:10.1016/