I am reading Cochrane's (p.35) book on Time series and I'm a bit rusty on my linear algebra so I was hoping someone could help me derive the variance of a VAR process.

Consider a VAR(1):

$x_{t} = Ax_{t-1} + Cw_{t}$

where $C = [\sigma_{\epsilon} \quad0 \quad \sigma_{\epsilon}]'$ and $E(w_{t}w_{t}') = I$

Then he says

the forecast error variances are $x_{t+1} - E_{t}(x_{t+1}) = Cw_{t+1} \implies var_{t}(x_{t+1}) = CC'$

$x_{t+2} - E_{t}(x_{t+2}) = Cw_{t+2} +ACw_{t+1} \implies var_{t}(x_{t+2}) = CC' + ACC'A'$

I am able to follow up until the $\implies var_{t}(x_{})$ part. But I don't understand how/why he introduces the transpose of $C,A$.



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