Say I have the following model:
$$
y_t = c+\phi y_{t-1} +\epsilon_t \,, \epsilon_t|\Omega_{t-1} \tilde{} WN(0,\sigma_t^2 )
$$
$$
\sigma_t^2=\alpha_0+\alpha_1\epsilon_{t-1}^2
$$
$$
|\phi|<1 \,, \alpha_0 \geqslant 0, \ 0<\alpha_1<1 \,.
$$
I know that an AR(1) is covariance stationary if $|\phi|<1$.
I also know that an ARCH(1) is covariance stationary if $\alpha_0 > 0$ and $0 < \alpha_1 < 1$.
If those conditions hold, does that imply that an AR(1)-ARCH(1) is also covariance stationary?
Or do I instead need to work out the unconditional mean, variance and autocovariance to check that they are constant? Although I am not sure how to do that without first assuming stationarity.
