ARIMA-GARCH instead of ARIMA for intervention analysis I'm looking to carry out an intervention time series analysis on the S&P500 to see how presidential elections affected the stock market.
I want to use an ARMA-GARCH process to model S&P500 return, and analyse different subsets of data between 1925 and present.
Below is a plot which shows the S&P500 returns against time, as well as the different subsets, each subset is a period of continuous republican/democratic administration.

I have searched online but cannot find any examples in academic papers or otherwise of people carrying out intervention time series analysis using ARIMA-GARCH rather than just ARIMA.
I'm using the "rugarch" package in R, and have fit models to the subsets, but I'm not sure how to quantify the intervention effect. For each subset, I was hoping to carry out a basic intervention analysis such as this one, and perhaps model the intervention effect.
Does anybody have any advice on doing this using an ARIMA-GARCH process; or should I stick to ARIMA despite it not being able to model financial data as well as ARIMA-GARCH?
 A: There is nothing inherently wrong with using a GARCH pattern to describe the time-varying conditional variance of a time series. Whether you are doing intervention analysis or some other stuff, accounting for conditional heteroskedasticity (if it is present) is sensible and advisable. If you fail to account for conditional heteroskedasticity, you are facing model misspecification. Nevertheless, the increase in model complexity due to the addition of a GARCH structure has the usual drawbacks such as increased chances of overfitting and high estimation variance. Therefore, you cannot be guaranteed to have, for example, better forecasts if you add a GARCH structure in your model.
If you are following the lecture notes as in the link you have provided, adding GARCH to ARIMA should be no problem. You would first check the residuals from the ARIMA model and see if they suffer from conditional heteroskedasticity; if they do, add a GARCH structure.
What you could also think about is intervention effects on the variance of the time series. Perhaps the unconditional variance increases or decreases following an intervention, or perhaps the autoregressive dynamics in the conditional variance is affected. 
In any case, the basic idea of intervention analysis is the same. You just may consider it in the conditional variance equation (GARCH) in addition to the conditional mean equation (ARIMA) or even on its own.
