= Covariance and cross-spectrum of a vector MA(1)
{title2=$\Gamma(0)=D+\Theta D\Theta^T,\quad\Gamma(1)=D\Theta^T$}
For $Z_t=W_t+\Theta W_{t-1}$ and vector <white noise> of <covariance matrix> $D$, the only nonzero covariance lags are $\Gamma(0)=D+\Theta D\Theta^T$, $\Gamma(1)=D\Theta^T$ and $\Gamma(-1)=\Theta D$. This follows by retaining only matching noise times in $\mathbb E[Z_tZ_{t+h}^T]$. Its spectral matrix, with this lag convention, is $(I+\Theta e^{i\lambda})D(I+\Theta^Te^{-i\lambda})/(2\pi)$. In two dimensions, with $D=\operatorname{diag}(a,b)$, its cross-spectrum is $[a\theta_{11}\theta_{21}+b\theta_{12}\theta_{22}+a\theta_{21}e^{-i\lambda}+b\theta_{12}e^{i\lambda}]/(2\pi)$.
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