# ARIMA + Rolling Window

I'm currently working on building an ARIMA+GARCH model using R. My dataset consists of the logarithmic returns of the Dow Jones index for a period of 11 years 2005-2016, however, it's worth noting that the time period selected is arbitrary and I do plan on using different time periods to compare forecasting results.

With regards to the GARCH variance model I plan on using a GARCH(1,1) model and for the GARCH mean model I plan on using the optimised p and q values from my ARIMA model. However, I have few questions regarding the optimisation of the ARIMA model.

As far as I understand the two key criteria in ensuring a good ARIMA model are to:

1) Optimise p and q i.e. the lag periods used for both the autoregressive and moving average components of the ARIMA model

2) Optimise the respective AR and MA coefficients

Again, as far as I know, the easiest way to achieve this in R is through the use of the auto.arima function which chooses the best model according to AIC criterion.

However, I've recently come across an article where for each day, n, the previous k days of the differenced logarithmic returns of a stock market index are used as a window for fitting an optimal ARIMA and GARCH model with their chosen k equal to 500 i.e for each day the previous 500 days of the differenced logarithmic returns are used to optimise the GARCH and ARIMA model.

Firstly, I'm having a hard time understanding how the rolling window would work, I can't seem to grasp it conceptually and secondly, I'm having a hard time understanding how exactly the rolling window would enhance the performance the ARIMA model, specifically how it would optimise the selected lag periods and the AR and MA coefficients.

If someone could help clarify these two points, I'd be very appreciative.