The file \(m-3 m 4608\).txt contains two columns. They are date and the monthly simple return for \(3 \mathrm{M}\) stock. Transform the returns to \(\log\) returns.

(a) Is there any evidence of ARCH effects in the log returns? Use Ljung-Box statistics with 6 and 12 lags of autocorrelations and \(5 \%\) significance level to answer the question.

(b) Use the PACF of the squared returns to identify an ARCH model. What is the fitted model?

(c) There are 755 data points. Refit the model using the first 750 observations and use the fitted model to predict the volatilities for \(t\) from 751 to 755 (the forecast origin is 750).

(d) Build an ARCH-M model for the log return series of \(3 \mathrm{M}\) stock. Test the hypothesis that the risk premium is zero at the 5% significance level. Draw your conclusion.

(e) Build an EGARCH model for the log return series of \(3 \mathrm{M}\) stock using the first 750 observations. Use the fitted model to compute 1-step- to 5 -step-ahead volatility forecasts at the forecast origin \(h=750\).