Chang, Chia-LinMcAleer, Michael2023-06-172023-06-172018-032341-2356https://hdl.handle.net/20.500.14352/17428Revised: March 2018. The authors are most grateful to the Editor and a reviewer for very helpful comments and suggestions. For financial support, the first author wishes to thank the National Science Council, Ministry of Science and Technology (MOST), Taiwan, and the second author acknowledges the Australian Research Council and the National Science Council, Ministry of Science and Technology (MOST), TaiwanThe 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.engtechnical reporthttps://www.ucm.es/icaeopen accessC22C32C52C58Random coefficient stochastic processOff-diagonal parametric restrictionsDiagonal BEKKFull BEKKRegularity conditionsAsymptotic propertiesConditional volatilityUnivariate and multivariate modelsFossil fuels and carbon emissions.Econometría (Economía)Indicadores económicos5302 Econometría5302.01 Indicadores Económicos