这是一份 Imperial帝国理工大学 MATH97115作业代写的成功案例
To analyze the multivariate normal case, it is convenient to work in vector form. Let $\mathbf{r}=\left(r_{1}, r_{2}, \cdots, r_{T}\right)^{\prime}$ and $\eta=\left(\eta_{0}, \eta_{1}, \ldots, \eta_{T}\right)^{\prime}$ so that
$$
\tilde{\mathbf{r}}=\mathbf{r}+\mathbf{B} \eta
$$
Here B is a selection matrix with first row $[-1,-1,0, \cdots, 0]$. The covariance matrix of $r$ is $\Lambda=\operatorname{diag}\left(\sigma_{t}^{2}\right.$ ). The Kalman smoother equations (in vector form) are
$$
\hat{\mathbf{r}}=\Lambda\left(\Lambda+\sigma_{\eta}^{2} \mathbf{B B}^{\prime}\right)^{-1} \tilde{\mathbf{r}} \text { and } \Sigma=\sigma_{\eta}^{2} \mathbf{B}\left(\mathbf{I}+\sigma_{\eta}^{2} \mathbf{B}^{\prime} \Lambda^{-1} \mathbf{B}\right)^{-1} \mathbf{B}^{\prime}
$$
From Assumption 1, it follows that $r_{t} \mid \sigma_{t}^{2} \sim N\left(0, \sigma_{t}^{2}\right)$. If we extend the assumption to
$$
\left(\begin{array}{l}
\mathbf{r} \
\eta
\end{array}\right) \sim N_{T}\left(0,\left(\begin{array}{cc}
\Lambda & 0 \
0 & \sigma_{\eta}^{2} \mathbf{I}
\end{array}\right)\right)
$$
then it is simple to derive the conditional distribution of $\tilde{r} \mid r$. Specifically,
$$
\mathbf{r} \mid \tilde{\mathbf{r}} \sim N_{T}(\hat{\mathbf{r}}, \Sigma)
$$
MATH97115 COURSE NOTES ：
Specifically, we will focus on SV of order one $\left(L_{w}=1\right)$. Set
$$
\begin{gathered}
\theta=\left(a, r_{y}, r_{w}\right)^{\prime} \
v_{l}(\theta) \equiv \exp \left(\frac{a w_{l-1}+r_{w} v_{t}}{2}\right) r_{y} z_{l}, \quad \forall t
\end{gathered}
$$
may then be conveniently rewritten as the following identity:
$$
y_{t}-x_{t}^{\prime} \beta=v_{t}(\theta), \quad \forall t
$$