We have used a standard GARCH(1,1) model with t distributed innovations for daily data of S&P index and JPM stock.
Question: is there any financial or statistical reason why the GARCH model would be more suitable for an index than for a stock? My initial guess was that returns "behave better" with indicies than stocks (for example JPM vary around +20% and -20% whereas S&P vary around +10% -10%) and since the return data is the input into the GARCH model this would make
To forecast the volatility we used a standard GARCH(1,1) model with t distributed innovations. Via the formula $VaR_t = \hat \mu_t + t \hat \sigma_t$ the volatility forecast was used in the calculation of the forecasted VaR.
We had two different datasets: returns on S&P index prices and returns on JPM stock prices. We used the same GARCH model for both return series, and labelled them $m.index$ and $m.stock$ respectively.
The VaR forecasts were backtested using standard backtesting procedures, the most important one is Christoffersen's test of independece (also called a cc test). The results from applying this test to our VaR forecasts showed that $m.index$ was better than $m.stock$ with respect to VaR backtesting. It was better because the p-value from Christoffersen's test of independece was $0.603$ for $m.stock$ but only $0.095$ for $m.stock$ suggesting that VaR violations were closer to being independent and correct number of violations when using index returns rather than stock returns.