How to test for invertibility in ARCH family models? I hope everyone is doing well.
Citing Enders (2014) in the book Applied Econometric Time Series: &quot;the Box–Jenkins approach also necessitates that the model be invertible&quot; while discussing necessary conditions for AR, MA and ARMA models. Both in algebra and in code (R), this is relatively simple to verify for smaller models.
How should one proceed when dealing with models of the ARCH family (ARCH, GARCH, GARCH-M)? Given one has two equations (that one describing the behavior of the mean and that one describing the behavior of the variance), must invertibility be verified for both, or must one only verify it for the mean equation?
Thank you!
It's not clear what you mean by &quot;invertibility of the mean equation&quot;. Invertibility, in the Box-Jenkins context, is a property of ARMA processes. A characterization of invertibility is that an ARMA process is invertible if and only if the MA polynomial has no roots inside the unit disk in the complex plane. Empirically speaking, invertibility means the current (unobservable) innovation is a weighted sum of what has been observed, although the weights in the sum are unknown ex ante. An ARCH type model usually do not assume any type of ARMA specification for the mean.
As for the variance $\sigma_t^2$ term in, say, GARCH, by assumption it follows an ARMA process, and its invertibility can be checked like any ARMA model.
For &quot;X&quot;-GARCH extensions that leaves the ARMA linear process framework, it's not clear how you would define invertibility.