I recently have a task related to non-stationary model in time series but I'm running out of ideas how to solve it, may be someone can help?

I am tasked to find the auto-covariance for the model $$ W_t=Z_t - Z_{t-1} $$ where $Z_t$ is an AR(1) model with parameters $-1<\phi_1<1$ and $\sigma^2$. And in particular, I must verify that the variance of $W_t$ is $2\sigma^2/(1 + \phi_1)$.

  • $\begingroup$ Let me know if there's anything that I edited incorrectly. $\endgroup$
    – Taylor
    Oct 14 '17 at 16:43
  • $\begingroup$ @Taylor Thank you for your help, It looks better now. What did you use to convert it into mathematical form? Is there any way I can do the same via phone (android)? $\endgroup$
    – user24024
    Oct 14 '17 at 17:21
  • $\begingroup$ @user24024 it's called $\LaTeX$ (pronounced "lay-tek"). There are expressions you can type that usually start with a backslash that can get rendered as mathematical symbols. Click "edit" on your post and you can see what I typed. $\endgroup$
    – Taylor
    Oct 14 '17 at 17:52
  • $\begingroup$ @The Laconic I found out the model above has the similar pattern with The random walk model, so I tried to solve it by modifying it with nonzero constant term but I couldn't get any further. I also tried with standard AR 1 form, then calculate the expectation value in order to find the auto-covariance but useless. $\endgroup$
    – user24024
    Oct 14 '17 at 17:53
  • $\begingroup$ @Taylor Okay, I'll try it later in my laptop. Anyway, do you have any idea how to solve the problem above? $\endgroup$
    – user24024
    Oct 14 '17 at 18:10

A hint: bilinearity of $\operatorname{Cov}(\cdot,\cdot)$.

For the $Z_t$ series, \begin{align*} \gamma_Z(h) &= \operatorname{Cov}(Z_{t+h},Z_t) \\ &= \operatorname{Cov}(\phi Z_{t+h-1} + \epsilon_{t+h},Z_t) \\ &= \operatorname{Cov}(\phi Z_{t+h-1} ,Z_t) \\ &= \phi \gamma_Z(h-1). \end{align*} and it is usually assumed that $\gamma_Z(0) = \operatorname{Var}(Z_1) = \frac{\sigma^2}{1- \phi^2}$.

For the $W_t$ series: \begin{align*} \gamma_W(h) &= \operatorname{Cov}(W_{t+h},W_t) \\ &= \operatorname{Cov}(Z_{t+h} - Z_{t+h-1},Z_t-Z_{t-1}) \\ &= \operatorname{Cov}(Z_{t+h},Z_t) - \operatorname{Cov}(Z_{t+h},Z_{t-1}) - \operatorname{Cov}(Z_{t+h-1},Z_{t}) + \operatorname{Cov}(Z_{t+h-1},Z_{t-1})\\ &= \gamma_Z(h) - \gamma_Z(h+1) - \gamma_Z(h-1) + \gamma_Z(h) \\ &= 2\gamma_Z(h) - \gamma_Z(h+1) - \gamma_Z(h-1). \end{align*}


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