Imagine an AR(1) has an autoregressive parameter which could change in time.

$y_t-\mu=\phi_t (y_{t-1}-\mu)+\varepsilon_t\,$, where $\phi_t$ is not always constant but still lies inside the usual bounds.

Is it always stationary? Sometimes stationary?


If you obtain the variance and covariances of the process $y_t$, you will see that they depend on $\phi_t$ (which is not constant). The covariance structure will therefore change over time and, hence, even if $\phi_t$ remains within the interval (-1,1) the process is not stationary.

In the particular case where $\phi_t$ takes on two values equal in absolute value (e.g., 0.4 and -0.4) the variance will remain constant (equal to $\sigma^2_\epsilon / (1 - \phi_t^2)$), but the autocovariances will still change over time with $\phi_t$ and hence the requirement for stationarity is not met in this case either.


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