Again, consider the percentage daily log returns of GM stock and the S&P 500 index from 1999 to 2008 as before, but we shall investigate whether the volatility of GM stock has any contribution in modeling the S\&P index volatility. Follow the steps below to perform the analysis.

(a) Fit a \(\operatorname{GARCH}(1,1)\) model with generalized error distribution to the percentage \(\log\) returns of GM stock. Denote the fitted volatility by gmvol. Build an adequate GARCH model plus gmvol as the exogenous variable for the log S\&P return series. Write down the fitted model.

(b) Is the volatility of GM stock returns helpful in modeling the volatility of the S\&P index returns? Why?