I'm becoming confused by this. Say I have the following model:
\(\displaystyle y_t = c + \phi y_{t-1} + \epsilon_t, \epsilon_{t}|\Omega_{t-1} ~ \tilde{} ~ WN(0,\sigma^2_t),\)
\(\displaystyle \sigma^2_t = \alpha_0 + \alpha_1 + \epsilon^2_{t-1},\)
\(\displaystyle |\phi|<1, \alpha_1<1, \alpha_0 \ge 0, \alpha_1>0.\)
I know that an AR(1) is covariance stationary if \(\displaystyle |\phi|<1\).
I also know that an ARCH(2) is covariance stationary if \(\displaystyle \alpha_0>0, \alpha_1>0\) and \(\displaystyle \alpha_1<1 \).
If those conditions hold does that imply that an AR(1)-ARCH(1) is also covariance stationary?
\(\displaystyle y_t = c + \phi y_{t-1} + \epsilon_t, \epsilon_{t}|\Omega_{t-1} ~ \tilde{} ~ WN(0,\sigma^2_t),\)
\(\displaystyle \sigma^2_t = \alpha_0 + \alpha_1 + \epsilon^2_{t-1},\)
\(\displaystyle |\phi|<1, \alpha_1<1, \alpha_0 \ge 0, \alpha_1>0.\)
I know that an AR(1) is covariance stationary if \(\displaystyle |\phi|<1\).
I also know that an ARCH(2) is covariance stationary if \(\displaystyle \alpha_0>0, \alpha_1>0\) and \(\displaystyle \alpha_1<1 \).
If those conditions hold does that imply that an AR(1)-ARCH(1) is also covariance stationary?