$$
\begin{aligned}
VAR(y_t|y_{t-1},\ldots,y_1)&= VAR(E(y_t|\theta_t,y_{t-1},\ldots,y_1)|y_{t-1},\ldots,y_1)\\
&+ E(VAR(y_t|\theta_t,y_{t-1},\ldots,y_1)|y_{t-1},\ldots,y_1)\\
&= VAR(E(y_t|\theta_t)| y_{t-1},\ldots,y_1) + E(VAR(y_t|\theta_t)|y_{t-1},\ldots,y_1) .
\end{aligned}
$$




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