一个MATLAB 的程序学习

$$
\lambda_{SP} = (T_1+T_2) \left\{ \log \det( \widehat  \Sigma_{1,2}) - \log \det\left[ \left( \frac{1}{T_1+T_2} (T_1 \widehat \Sigma_{1} + T_2  \widehat \Sigma_{2})\right) \right] \right\}
$$

$$
\lambda_{BP} = (T_1+T_2) \log \det( \widehat \Sigma_{(1,2)}) - T_1 \log \det( \widehat \Sigma_{1}) - T_2 \log \det( \widehat \Sigma_{2}),
$$

$$
\widehat \Sigma_{(1,2)} = \frac{1}{T_1} \sum_{t=1}^{T_1} \widehat u_t \widehat u_t^\top + \frac{1}{T_2} \sum_{t=T-T_2+1}^{T_2} \widehat u_t \widehat
        u_t^\top,
$$

$$
\widehat \Sigma_{1} =
        \frac{1}{T_1} \sum_{t=1}^{T_1} \widehat{u}^{(1)}_t \widehat{u}^{(1)^\top}_t, \mbox{ and }
        \widehat \Sigma_{2}
        \, = \, \frac{1}{T_2} \sum_{t=T_1+1}^T \widehat{u}_t^{(2)} \widehat{u}_t^{(2)^\top}.
$$

$$
\lambda_{W, ij} = \frac{(\lambda_{ii} - \lambda_{jj})^2}{\mbox{Var}(\lambda_{ii}) + \mbox{Var}(\lambda_{jj}) - 2\mbox{Cov}(\lambda_{ii}, \lambda_{jj})} \sim \chi^2_{(2)},
$$

$$
\lambda_{LR} = 2\left[\log \mathcal{L}\left(\mbox{vec}(\widetilde{B})\right) - \log \mathcal{L}\left(\mbox{vec}(\widetilde{B}_r)\right)\right] \sim \chi^2_{(N)},
$$

$$
\begin{align*}
A(L)y_t&= \mu + B\varepsilon_t\\
y_t&=A(L)^{-1}\mu+A(L)^{-1}B\varepsilon_t\\
&=\nu+\Phi(L)B\varepsilon_t=\nu+\sum_{i=0}^{\infty}\Phi_{i}B\varepsilon_{t-i}=\nu+\sum_{i=0}^{\infty}\Theta_{i}\varepsilon_{t-i},
\end{align*}
$$

Forecast error variance decompositions (FEVD) highlight the relative contribution of each shock to the variation a variable under scrutiny. For the multivariate series $y_t$, the corresponding $h$-step ahead forecast error is  $y_{t+h}-y_{t|t}(h)=\Theta_0\varepsilon_{t+h}+\ldots+\Theta_h\varepsilon_{t+1}$, and the forecast error variance of the $k$-th variable is $\sigma_k^2(h)=\sum_{j=0}^{h-1}(\Theta^2_{k1,j}+\ldots+\Theta^2_{kK,j})$ \citep{IntroductionMultipleTS}. Since $\Sigma_{\varepsilon}=I_K$ holds by assumption, the relative contribution of shock $\varepsilon_{it}$ to the $h$-step forecast error variance of variable $y_{kt}$ is

$$
FEVD_{ki}(h)=(\Theta^2_{ki,0}+\ldots+\Theta^2_{ki,h-1})/\sigma_k^2(h).
$$

$$
\tag{1.1}
\beta^2=\sigma_t * \omega \sim \mathbb{n}
$$

$$
y_t = A y_{t-1} + \varepsilon_t \,\,\, \text{for} \,\,\, t = 1, 2, ..., T
$$