# 宏观经济学原理 |ECON123 Principles of macroeconomics代写 warwick代写

$$q_{t}=\gamma_{0}+\gamma_{1} t+\xi_{t}$$
If $\gamma_{1} \neq 0$, the question is whether the deviation from the trend is stationary or if it is a driftless unit root process. If $\gamma_{1}=0$ and $\gamma_{0} \neq 0$, the question is whether the deviation of $q_{t}$ from a constant is stationary. Let’s ask the first question-whether the deviation from trend is stationary. Let
$$\xi_{t}=\rho \xi_{t-1}+\epsilon_{t},$$

where $0<\rho \leq 1$ and $\epsilon_{t} \stackrel{\text { iid }}{\sim} \mathrm{N}\left(0, \sigma_{\epsilon}^{2}\right)$. You want to test the null hypothesis $H_{o}: \rho=1$ against the alternative $H_{a}: \rho<1$. Under the null hypothesis
$$\Delta q_{t}=\gamma_{1}+\epsilon_{t}$$
and $q_{t}$ is a random walk with drift $\gamma_{1}$. Add the increments to get
$$q_{t}=\sum_{j=1}^{t} \Delta q_{j}=\gamma_{1} t+\left(\epsilon_{0}+\epsilon_{1}+\cdots+\epsilon_{t}\right)=\gamma_{0}+\gamma_{1} t+\xi_{t}$$
where $\gamma_{0}=\epsilon_{0}$ and $\xi_{t}=\left(\epsilon_{1}+\epsilon_{2}+\cdots+\epsilon_{t}\right)$. You can initialize by assuming

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