这是一份nottingham诺丁汉大学ECON1045作业代写的成功案例

Finally, the maximized likelihood function is found from
$$
L_{\max }^{-2 T}=|\hat{Q}|=\left|S_{o o}\right| \prod_{i=1}^{r}\left(1-\hat{\lambda}{i}\right) $$ and the likelihood ratio test of the hypothesis $H{1}(r)$ is given by the trace test statistic
$$
-2 \log Q\left[H_{1}(r) \mid H_{o}\right]=-T \sum_{i=r+1}^{p} \log \left(1-\hat{\lambda}{i}\right) $$ An alternative test (called the maximum eigenvalue test, $\lambda{\max }$ ) is based on the comparison of $H_{1}(r-1)$ against $H_{1}(r)$ :
$$
-2 \log Q\left[H_{1}(r-1) \mid H_{1}(r)\right]=-T \log \left(1-\hat{\lambda}_{r+1}\right)
$$

ECON1045 COURSE NOTES ：

If the individual series have a stochastic trend, we can explore for shared stochastic trends between the series. In particular, if the stochastic trend of $x_{t}$ is shared with the $y_{t}$ series (i.e., $\tau_{x t}$ is linearly related to $\tau_{y t}$ ), then we have the following structure
$$
\begin{aligned}
&y_{t}=\tau_{y t}+c_{y t}+\epsilon_{y t} \
&x_{t}=\alpha \tau_{y t}+c_{x t}+\epsilon_{x t}
\end{aligned}
$$
where $\alpha$ is the factor of proportionality between the two trends. In this case there is a unique coefficient $\lambda$, such that the following linear combination of $y_{t}$ and $x_{t}$
$$
z_{t}=y_{t}-\lambda x_{t}
$$
is a stationary series – see Engle and Granger (1987). In fact, if there is a shared stochastic trend, the linear combination $z_{t}$ can be written as
$$
\begin{aligned}
z_{t} &=\tau_{y t}+c_{y t}+\epsilon_{y t}-\lambda\left(\alpha \tau_{y t}+c_{x t}+\epsilon_{x t}\right) \
&=\tau_{y t}-\lambda \alpha \tau_{y t}+c_{y t}-\lambda c_{x t}+\epsilon_{y t}-\lambda \epsilon_{x t}
\end{aligned}
$$