ARIMA(p,d,q) + GARCH(p,d,q) model

Question

I found an article where they fit an ARIMA(p,d,q) model to a time series and then fit a GARCH(p,d,q) to the residual of the ARIMA (the parameters (p,d,q) are passed as the volatility model lags to the GARCH model and not as the mean model lags).

I know that the inconsistencies of doing them serially have been discussed here, here and in other posts. However, my question is if it is valid to reuse the lags of the ARIMA to fit the GARCH model. The traditional ARMA(p,q)-GARCH(r,s) does not do so and I could not find any reference supporting this.

Answer 1

Generally, there is no reason to expect the model order for ARIMA to be the same as for GARCH, so would really question the use of the same orders in practice. It could be noted that sometimes very simple models such as ARMA(1,1)-GARCH(1,1) are used as rough approximations to the underlying autoregressive structure in the conditional mean and variance of, say, financial returns time series (e.g. these orders are the default in the ugarchspec function in rugarch package in R). But I doubt there is any general justification for other orders not equal to (1,1) or perhaps (1,0) or (0,1) (though I have shown here that GARCH($p$,0) does not make much sense for any $p>0$).