# ARMA-GARCH, invertibility, stationarity and insignificance

I am trying to forecast volatility out-of-sample using ARCH, GARCH, GJR and EGARCH. I used AIC to identify the ARMA and ARCH order and decided to stick with (1,1) for GARCH-type models. However, I have situation where my mean equation is either insignificant and/or non-invertible and non-stationary. The insignificance does not worry me too much, but I think the non-invertibility/non-stationarity will affect my forecast results out-of-sample. Should I remove the mean equation in the models where this happens? If no then how would this affect my forecast results out-of-sample?

• +1 because I find the idea of removing the mean equation to be thought provoking. – Richard Hardy Aug 5 '17 at 13:41

I agree that nonstationarity and noninvertibility are the two more serious concerns when it comes to forecasting. If you have a nonstationary process where nonstationarity is due to the level of the series (the unconditional expectation is undefined or changing over time), you will have trouble forecasting the level of the series.

Is that important when you are actually interested in forecasting volatility? Yes, it is. You either define volatility as the magnitude of deviations from the conditional mean (the level) or from zero. In both cases nonstationarity in mean will cause you trouble.

Should I remove the mean equation in the models where this happens?

Removing the mean equation will not solve the problem. In fact, you cannot remove the mean equation. A GARCH model specifies the conditional distribution of the time series, and that by definition specifies the conditional mean (unless it does not exist). In a GARCH(r,s) model for a time series $y_t$,

\begin{aligned} y_t &= \mu_t + u_t, \\ u_t &= \sigma_t \varepsilon_t, \\ \sigma_t^2 &= \omega + \alpha_1 u_{t-1}^2 + \dotsc + \alpha_s u_{t-s}^2 + \beta_1 \sigma_{t-1}^2 + \dotsc + \beta_r \sigma_{t-r}^2. \\ \varepsilon_t &\sim i.i.d(0,1), \\ \end{aligned}

you have the element $\mu_t$. It can be zero, it can be something more complicated, but it is there, it cannot just be removed. And it specifies the conditional mean of the distribution of $y_t$, the latter being $y_t \sim D(\mu_t,\sigma_t^2)$. (It has to do with the properties of a random variable. Ignoring some property of a distribution, such as the mean, does not make it disappear.)

What I suggest is trying to see why the series appears to be nonstationary and address that (by adjusting for a deterministic or a stochastic trend, a structural change or the like). Then you will have a solid ground for your volatility forecasting.

If no then how would this affect my forecast results out-of-sample?

If the conditional mean model is well specified and estimated well (e.g. the true process is explosive like $y_t = 1.1 y_{t-1} + u_t$ and it is estimated as such), your forecasts of the conditional mean should be fine. On the other hand, if there is, say, a neglected structural change, then the model will be misspecified and the forecasts will likely be poor.

• Thank you very much for your answer! I think I haven't expressed myself right. When I talk about the mean equation I talk about the ARIMA(3,3) which I estimated first. Basically I estimated an ARIMA(3,3) checked for ARCH effects and then estimated and ARIMA(3,3)+GARCH(1,1). Obviouslt I can drop the arima and estimate: $$r_t = \mu +\varepsilon_t$$ and the GARCH. If I drop the ARIMA however I need to drop it from all models, right? – Alex Aug 7 '17 at 11:09
• @Alex, You are welcome! Technically it does not matter whether you estimate the mean and variance equations simultaneously or stagewise -- the mean equation is implicitly there, even if it is just a constant. Now if you want to compare ARCH to GARCH, GJR-GARCH and EGARCH, it would make sense to have the same mean equation, thus also to "drop" it from all models rather than just one or two of them. – Richard Hardy Aug 7 '17 at 13:34