Monthly changes in the \\($US/\$AUS\) exchange rate \(S_{t}\) for the period 1985M7 to 2010M6 are stored in the file exrate5.

a. Plot the time series of the changes and their histogram. Are there periods of high volatility and periods of low volatility? Does the unconditional distribution of the changes appear to be normally distributed?

b. Estimate the GARCH \((1,1)\) model \(S_{t}=\beta_{0}+e_{t},\left(e_{t} \mid I_{t-1}\right) \sim N\left(0, h_{t}\right)\) and \(h_{t}=\delta+\alpha_{1} e_{t-1}^{2}+\beta_{1} h_{t-1}\). Comment on the results.

c. Estimate the conditional variance \(h_{t}\) for each observation and create the series \(v_{t}=\hat{e}_{t} / \sqrt{\hat{h}_{t}}\) where \(\hat{e}_{t}\) are the residuals \(\hat{e}_{t}=S_{t}-\hat{\beta}_{0}\). Create a histogram for the \(v_{t}\). Do they appear to be normally distributed?

d. Forecast the conditional mean and variance for 2010M7 and 2010M8.