Matrix exponential GARCH

I propose a new multivariate GARCH specification that maintains positive definiteness of the conditional covariance matrix. The idea is to specify the dynamics in the matrix logarithm of the conditional covariance. Because the matrix exponential transformation ensures positive definiteness, the dynamics can be specified without the positive definiteness constraint. This affords a variety of specifications and, in particular, we can specify element-by-element the dynamics of the matrix logarithm. I discuss specifications with leverage effects, estimation with multivariate Gaussian and t-distributions, and diagnostics that evaluate the appropriateness of the matrix exponential specification.