A test for multivariate ARCH effects

Abstract

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 1% 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.

Notes

For a new approach to determine the optimal lag order in the VAR model see Hatemi-J (2003).

The adjustment is done by making use of an Edgeworth expansion suggested by Anderson (1958), which is generalized by Hatemi-J (2004) to multivariate cases. In our case, Δ = T – m × n + 0.5(n(m − 1) − 1).

Johansen (1996) suggested a test statistic similar to the test statistic in Equation 4 for testing multivariate autocorrelation, and indicates it is asymptotically distributed as χ² Johansen (1996) suggested a test statistic similar to the test statistic in Equation 4 for testing multivariate autocorrelation, and indicates it is asymptotically distributed as χ²with m × n ² degrees of freedom. Hatemi-J (2004) has suggested a modified version of this test for multivariate autocorrelation in stable and unstable VAR models. We use this test statistic for testing another constraint on the multivariate system, so it should have the same asymptotic distribution.

Notice that we have assumed that the covariance between the error terms across equations in the VAR model is equal to zero. However, in practical applications it is possible to allow for dependency between error terms. In such cases, the dimension of the system presented in Equation 6 will become 3 × 3 in this particular case.

For a proof of this result see Hatemi-J (2004).