When we test an autoregressive model for covariance nonstationarity, how is it possible that it could pass our tests and exhibit conditional heteroskedasticity? In other words, if we find conditional heteroskedacity to be present, isn’t our model already invalid, since variance is non-constant? My thought process is that conditional heteroskedasticity implies covariance nonstationarity which implies an invalid model, but not the other way around. I think I’m missing something very basic.
Short answer: conditional heteroskedasticity doesn’t imply a nonstationary series.
Longer answer: In general, think of conditional heteroskedasticity as applying to cross-sectional data and stationarity being applied to time-series data (they are more common in these contexts than in others, so this context might help).
Conditional heteroskedasticity means that the variance changes as the values of the independent variables change (for example, the variance is greater at lower values of the independent variables). Also note that the conditional heteroskedasticity doesn’t mean the variance changes with time (because we’re most likely dealing with cross-sectional data when we observe this–not a time-series).
One way to achieve covariance nonstationarity is to have the variance change over time (different than changing with the settings of the IVs). Think of 10 years of very low volatility in the stock market, followed by a period of higher volatility. Taken all together, this series would not be covariance stationary because the variance changed throughout time.