# Why does a "stationary" ARCH(1) process make sense?

For the simplest ARCH model $x_t = \sigma_t \epsilon_t$ where $\sigma_t^2 = a_0 + a_1x_{t-1}^2$, $x_t$ is stationary when $a_1 <1$. However, isn't the whole point of using an ARCH model is to model unstationary process (the variance of $x_t$ changes)? How could we end up with modeling a stationary process with an ARCH model?

• When they say stationary they are referring to the unconditional distribution of the process. These models are conditional models (autoregressive "conditional" heteroscedasticity), and it is permitted to have short term divergences as long as the long run behavior is preserved. Jul 13, 2017 at 19:22
• @CagdasOzgenc Thanks, and that makes sense. If you move this comment to an answer, I'm happy to accept it, or let me know if you think this question is trivial, and I'll just delete it Jul 13, 2017 at 19:29
• This may be useful stats.stackexchange.com/a/72628/28746 Aug 5, 2017 at 15:41
• @AlecosPapadopoulos Indeed! Upvoted your answer since I had the same doubt as the OP of that question did Aug 9, 2017 at 13:07

No, the point of ARCH is to model time-varying conditional variance which does not have to be nonstationary. Stationarity is not the same as constancy. A time series can be stationary without being constant. E.g. the variance $\sigma_t$ of a variable $x_t$ can be stationary without being constant, i.e. without $\sigma_t\equiv \sigma$ for some fixed value $\sigma$.