Why is AR(1) process with $|\phi| > 1$ not stationary? I saw the top answer on the following post: Stationarity of AR(1) model.
It says that an AR(1) process $X_{t} = \phi X_{t-1} + \epsilon_{t}$ where $\epsilon_{t} \sim WN(0, \sigma^{2})$ has a stationary solution if and only if $|\phi| < 1$. However, it is also commonly known that explosive processes with $|\phi| > 1$ can be represented as
$$
X_{t} = \sum_{j=1}^{\infty} -\phi^{-j} w_{t+j}
$$
which is a stationary time series. So does it not follow that all AR(1) processes other than when $|\phi| =1 $ have a stationary solution?

Take this $AR(1)$ process: $X_t-\phi X_{t-1}$=$Z_t$
If $\lvert {\phi} \rvert<1: X_t=\sum_{j=0}^{\infty}\phi^jZ_{t-j}$ is the unique stationary solution of the above $AR(1)$ process.
If $\lvert {\phi} \rvert>1: X_t=-\sum_{j=1}^{\infty}\phi^{-j}Z_{t+j}$ is the unique stationary solution of the above $AR(1)$ process.
So, yes, the $AR(1)$ process is stationary as long as $\lvert {\phi} \rvert\ne1$. Note however that when $\lvert {\phi} \rvert>1$, $X_t$ is expressed in terms of future values of $Z_{t+j}$. This is considered unnatural because $X_t$ is future dependent. In contrast when $\lvert {\phi} \rvert<1$, $X_t$ is future independent, also known as causal. And since every $AR(1)$ process with $\lvert {\phi} \rvert>1$ can be written as an $AR(1)$ process with $\lvert {\phi} \rvert<1$ and a different white noise sequence, not much is lost by focusing on only stationary causal $AR(1)$ models, which is what most textbooks do.
