# 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: "the Box–Jenkins approach also necessitates that the model be invertible" 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!

• 1. "the Box–Jenkins approach also necessitates that the model be invertible..."--- Restricting to invertible models is not a requirement for inference, but a choice made for empirical interpretation. E.g. An MA unit root model can be estimated exactly as an invertible MA but if fitting an MA(1) model gives you a non-invertible MA, you may rule out MA(1) and try a difference specification. 2. Could you provide definition of invertibility for, e.g. ARCH, and reference? Oct 15 '20 at 22:58
• Hi: Here is a link to some nice answers as far as the meaning of invertibility. stats.stackexchange.com/questions/50682/…. The link doesn't answer your specific question but I would just google for "invertibility of ARCH models" and a lot of things should come up. Oct 16 '20 at 1:56
• This link looks more relevant to your question. web.mit.edu/14.384/www/ps1%20solutions.pdf Oct 16 '20 at 2:00

## 1 Answer

It's not clear what you mean by "invertibility of the mean equation". 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 "X"-GARCH extensions that leaves the ARMA linear process framework, it's not clear how you would define invertibility.

• The $\sigma_t^2$ follows what looks like an ARMA process without the contemporaneous error term. It is very similar to how the conditional mean follows an ARMA-looking process without the contemporaneous error term. A detailed comparison of ARMA and GARCH is available in this thread on Cross Validated. Oct 30 '20 at 14:37