covariance stationarity

[fadimohamed]

Can anyone help me with the following covariance stationarity contion question? The BEKK model is covariance stationary when the eigen values of A ⨂ A
+ B ⨂ B are less than 1, how can i calculate the covariance
stationarity in the presence of asymmetry (the Matrix D) or in the presence of exogenous variables?

[TomDoan]

If you have exogenous variables, you can't really talk about "stationarity" since the variances will change with the values of those.

I don't believe there's a simple condition for the asymmetric effect. You get the relatively simple condition for the standard BEKK because the expected value of A'u(t-1)u(t-1)'A is linear in the (presumed) stationary covariance matrix. By contrast, the expected value of the asymmetry term is a very unpleasant non-linear function of it involving integrals with correlated multivariate Normals.