A test for multivariate ARCH effects

This paper extends Engle's LM test for ARCH affects to multivariate cases. The size and power properties of this multivariate test for ARCH effects in VAR models are investigated based on asymptotic and bootstrap distributions. Using the asymptotic distribution, deviations of actual size from nominal size do not appear to be very excessive. Nevertheless, there is a tendency for the actual size to overreject the null hypothesis when the nominal size is I % and underreject the null when the nominal size is 5% or 10%. It is found that using a bootstrap distribution for the multivariate LM test is generally superior in achieving the appropriate size to using the asymptotic distribution when (1) the nominal size is 5%; (2) the sample size is small (40 observations) and/or the VAR system is stable. With a small sample, the power of the test using the bootstrap distribution also appears better at the 5% nominal size.