# Stationary time series ar model

Given the AR(1) model $Y_t = ϕY_{t−1} + e_t$ . I want to show if |ϕ| = 1, the process cannot be stationary. I know to prove stationary it suffices to prove either mean function or autocovariance function is not independent of time. It is my first time approaching this type of problem so I really have no clue how to approach it~

As you said, to show that the process is not stationary it suffices to show that the autocovariance function is time dependent. In particular, let us look at the variance of the series.

$y_{t}=y_{t-1}+\varepsilon_{t}$

$Var\left(y_{t}\right)=Var\left(y_{t-1}+\varepsilon_{t}\right)$

$Var\left(y_{t}\right)=Var\left(y_{t-1}\right)+\sigma_{\varepsilon}^{2}$

$Var\left(y_{t}\right)=Var\left(y_{t-2}+\varepsilon_{t-1}\right)+\sigma_{\varepsilon}^{2}$

$Var\left(y_{t}\right)=Var\left(y_{t-2}\right)+2\sigma_{\varepsilon}^{2}$

...

$Var\left(y_{t}\right)=Var\left(y_{0}\right)+t\sigma_{\varepsilon}^{2}$

Assuming that the starting point of the series is constant and given:

$Var\left(y_{t}\right)=t\sigma_{\varepsilon}^{2}$

Since this last expression depends on $t$, it means that the variance of the series is time dependent and therefore not stationary.

Hope this is what you are looking for.