The file m-gmsp5008.txt contains the dates and monthly simple returns of General Motors stock and the S\&P 500 index from 1950 to 2008.

(a) Build a GARCH model with Gaussian innovations for the log returns of GM stock. Check the model and write down the fitted model.

(b) Build a GARCH-M model with Gaussian innovations for the log returns of GM stock. What is the fitted model?

(c) Build a GARCH model with Student- \(t\) distribution for the \(\log\) returns of GM stock, including estimation of the degrees of freedom. Write down the fitted model. Let \(v\) be the degrees of freedom of the Student- \(t\) distribution. Test the hypothesis \(H_{0}: v=6\) versus \(H_{a}: v eq 6\), using the \(5 \%\) significance level.

(d) Build an EGARCH model for the log returns of GM stock. What is the fitted model?

(e) Obtain 1-step- to 6-step-ahead volatility forecasts for all the models obtained. Compare the forecasts.