Autocovariance of Explosive AR(1) model with $|\phi|>1$ expressed as a stationary process

Question

I am working through the book called Time Series Analysis and Its Applications by Shumway and Stoffer. I am stuck deriving an equation given in example 3.4 in the book (page 80 for the fourth edition), which I describe below. Now I do know a similar question was asked before (here: Autocovariance for an explosive AR(1) process), but I do not really get that response nor what is wrong with the derivation I show below.

Let us consider an AR(1) process that has $|\phi|>1$, also called an "Explosive AR(1)" process $$ x_t = \phi x_{t-1}+w_t,\;\;\;\;w_t \sim N(0,\sigma_w^2) $$

I know we can rewrite this as a "stationary process" as follows: $$ x_t = -\sum_{j=1}^{\infty}\phi^{-j}w_{t+j} $$

Now, I am trying to verify the equations that were given for the autocovariance of an Explosive AR(1) model. According to the book, the autocovariance for this explosive process is: $\gamma{(k)}=\sigma_w^2 \phi^{-2} \phi{-h} / (1-\phi^{-2})$.

However, when I try to derive this by myself, I get a slightly different equation. I will give my derivation below, and if anyone can show me what I did wrong, that would be great (or if I am right, and the equation in the book is wrong)!

\begin{align} \gamma_x(h)&=cov(x_{t+h},x_t)=cov\Big(-\sum_{j=1}^{\infty}\phi^{-j}w_{t+h+j},-\sum_{k=1}^{\infty}\phi^{-k}w_{t+k}\Big) \\ &= E\Bigg[\Big(-\sum_{j=1}^{\infty}\phi^{-j}w_{t+h+j}\Big)\Big(-\sum_{k=1}^{\infty}\phi^{-k}w_{t+k}\Big)\Bigg]\\ &= E\Big[\big(-\phi^{-1}w_{t+h+1}-\phi^{-2}w_{t+h+2}-\phi^{-3}w_{t+h+3}-...\big) \big(-\phi^{-1}w_{t+1}-...-\phi^{-h-1}w_{t+h+1}-\phi^{-h-2}w_{t+h+2}-\phi^{-h-3}w_{t+h+3}-...\big)\Big]\\ &=\phi^{-1}\phi^{-h-1}E(w_{t+h+1}^2)+\phi^{-2}\phi^{-h-2}E(w_{t+h+2}^2)+\phi^{-3}\phi^{-h-3}E(w_{t+h+3}^2)+...\\ &=\phi^{-h-2}\sigma_w^2+\phi^{-h-4}\sigma_w^2+\phi^{-h-6}\sigma_w^2+...\\ &=\sigma_w^2 \sum_{j=1}^{\infty} \phi^{-h-2j}\\ &=\sigma_w^2\phi^{-h} \sum_{j=1}^{\infty} \phi^{-2j}\\ &=\sigma_w^2\phi^{-h}\Bigg[ \frac{1}{1-\phi^{-2}}\Bigg]\\ \end{align}

I used the result for a geometric series in the last step because $|\phi|^{-2}<1$.

What I derived is close, but not the same to what the authors put in the book. Can someone verify what I did above, or I made a mistake?

Does this answer your question? Where is $|\theta|\lt1$ used in recursive method derivation of invertibility of MA(1)? — Ben, Mar 23 '22 at 05:48
I read through that, but I do not know if it helps me. I get that with $|\phi|>1$ the AR model explodes, which is why you have to do this trick to express it as a infinite sum of future values of $x$, and thus it is not causal. My issue was in the derivation of the autocovariance missing a $\phi^{-2}$ that the author of the textbook got when try to fill in the blanks in one of his steps. — insane_stats_student, Mar 24 '22 at 01:29
In that case I think this question is probably what you're after. It is asking about the exact same issue in the same book. — Ben, Mar 24 '22 at 06:11

Autocovariance of Explosive AR(1) model with $|\phi|>1$ expressed as a stationary process

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