# Regarding explosive AR processes and stationarity

I often see this:

If we have an $$\text{AR}(1)$$ process,$$x_t=\phi x_{t-1}+w_t,$$

$$x_t$$ is:

stationary if $$|\phi|<1$$

an unit root (nonstationary) if $$|\phi|=1$$

explosive (and nonstationary) if $$|\phi|>1$$

But then I have seen the following idea in other sources (a course that uses the book Time Series Analysis and its Applications with R examples, the book itself, and a few other lecture notes online)

$$\text{AR}(1)$$ is causal and stationary if $$|\phi|<1$$

is an unit root, nonstationary, if $$|\phi|=1$$

is explosive, but future dependent stationary if $$|\phi|>1$$

The explanation given is this:

If we unravel the $$\text{AR}(1)$$ equation , we get:

$$x_t=\phi^k x_{t-k}+\sum_{j=0}^k \phi^j w_{t-j}$$

If $$|\phi|<1$$, then we can send $$k \rightarrow \infty$$ in the last display to get $$x_t=\sum_{j=0}^{\infty} \phi^j w_{t-j}$$ This is called the stationary representation of the $$\mathrm{AR}(1)$$ process (4) (from Tibshirani)

They later shows that autocovariance depends on lag difference, so it is (weakly) stationary.

Causal is defined as " does not depend on the future" (from here) and more formally

a series $$x_t, t=0, \pm 1, \pm 2, \pm 3, \ldots$$ is causal provided that it can be written in the form $$x_t=\sum_{j=0}^{\infty} \psi_j w_{t-j}$$ for a white noise sequence $$w_t, t=0, \pm 1, \pm 2, \pm 3, \ldots$$, and coefficients such that $$\sum_{j=0}^{\infty}\left|\psi_j\right|<\infty$$" (from Tibshirani)

causality actually implies stationarity... causality actually tells us more than stationary: it is stationary “plus” a representation a linear filter of past white noise variates, with summable coefficients

The specific section that states an explosive $$\text{AR}(1)$$ process is stationary is:

it was discovered that the random walk $$x_t=x_{t-1}+w_t$$ is not stationary. We might wonder whether there is a stationary $$\operatorname{AR}(1)$$ process with $$|\phi|>1$$. Such processes are called explosive because the values of the time series quickly become large in magnitude. Clearly, because $$|\phi|^j$$ increases without bound as $$j \rightarrow > \infty, \sum_{j=0}^{k-1} \phi^j w_{t-j}$$ will not converge (in mean square) as $$k \rightarrow \infty$$, so the intuition used to get (3.6) will not work directly. We can, however, modify that argument to obtain a stationary model as follows. Write $$x_{t+1}=\phi x_t+w_{t+1}$$, in which case, \begin{aligned} x_t & =\phi^{-1} x_{t+1}-\phi^{-1} w_{t+1}=\phi^{-1}\left(\phi^{-1} x_{t+2}-\phi^{-1} w_{t+2}\right)-\phi^{-1} w_{t+1} \\ & \vdots \\ & =\phi^{-k}x_{t+k}-\sum_{j=1}^{k-1} \phi^{-j} w_{t+j} \end{aligned} by iterating forward $$k$$ steps. Because $$|\phi|^{-1}<1$$, this result suggests the stationary future dependent $$\mathrm{AR}(1)$$ model $$x_t=-\sum_{j=1}^{\infty} \phi^{-j} w_{t+j} .$$

The reader can verify that this is stationary and of the $$\mathrm{AR}(1)$$ form $$x_t=\phi x_{t-1}+w_t$$. Unfortunately, this model is useless because it requires us to know the future to be able to predict the future. When a process does not depend on the future, such as the $$\operatorname{AR}(1)$$ when $$|\phi|<1$$, we will say the process is causal. In the explosive case of this example, the process is stationary, but it is also future dependent, and not causal. (from the Shumway book)

And furthermore my professor (from a few years back) said going forward in time, $$|\phi|<1$$ is stationary, and $$|\phi|>1$$ is stationary if we are going backwards in time.

I am thinking the way to reconcile the sources that say it is nonstationary vs sources that say it is "future dependent stationary". Perhaps we are typically looking forward in time, so changing an explosive AR process to be in a future dependent form is pointless, real life does not operate that way. And so the sources that say an explosive $$\boldsymbol{\text{AR}(1)}$$ process is nonstationary are only really considering the scenario of going forward in time, and excluding the idea of going backwards? Another question is, for like DF/ADF tests, we test unit root vs not unit root. Is there a reason we don't consider the case of $$\boldsymbol{|\phi|>1}$$? Is it also because "future dependent" stationary processes aren't typically what we are observing?

Is there a reason we don't consider the case of $$|\phi|>1$$? Is it also because "future dependent" stationary processes aren't typically what we are observing?
This is exactly what I had in mind. The point of forecasting is to investigate whether past realisations of a process $$x_t$$ have some form of stability (to be modelled) that enables us to form predictions about the future. We don't have access to future realisations of a process, otherwise we wouldn't need to forecast.