 Multivariate moving-average model

A multivariate moving-average model filters vector white noise through fixed matrix coefficients. If the noise has zero mean and lag-zero covariance matrix $D$, with all nonzero-lag cross-covariances zero, then the filtered process is a weakly stationary process. With the convention $\Gamma (h)=\mathbb{E}[Z_t{Z}_{t+h}^T]$, its covariance matrix is $\Gamma (h)={\sum }_j{\Theta }_{j}D{\Theta }_{j+h}^T$, omitting terms outside the filter range. Positive and negative lags satisfy $\Gamma (-h)=\Gamma (h{)}^T$.