Volatility forecast using ARIMA GARCH

Question

I am trying to forecast the volatility of the pair of currency USD/GBP. I am using python ans I used a GARCH model on the returns, but later on I found that I can fit an ARIMA-GARCH model to forecast the volatility too, except that I didn't find strong articles/references that explain if using an ARIMA-GARCH will give me the same results (a forecast of the volatility of the pair ). If I am using a GARCH(1.1) model then I will be applying it on the returns directly, however, if using an ARIMA-GARCH(1.1) model then I will be applying the GARCH model on the residuals of the ARIMA model. Which of the two methods will better help me achieve my purpose?

Answer 1

Regardless of the conditional mean model (constant, ARMA or something else), GARCH is a model for the conditional variance of the variable of interest (e.g. return on exchange rate).* Combined with a conditional mean model and a distributional assumption for the standardized innovations it becomes a model of the conditional distribution of the random variable of interest; see e.g. this.

Whether a constant mean or an ARMA will be a better fit for your data depends on... your data. Currency exchange rates are notoriously difficult to predict, so I doubt ARMA would do a better job than a constant mean when forecasting out of sample. (In sample ARMA will inevitably provide a better fit as it is a more flexible model, but it is usually the out-of-sample forecast ability that counts.) You may try both models and compare the them in terms of AIC and/or using time-series cross validation via rolling windows.

*Equivalently, GARCH is a model of the variance of the zero-mean residual that is obtained by subtracting the conditional mean from the variable of interest.