The data file shanghai contains data on the daily returns to the Shanghai Stock Exchange Composite Index from July 7, 1995 to May 5, 2015.

a. Plot the time series of returns and their histogram. For what observations is volatility the greatest? Describe the shape of the distribution of returns. Does the Jarque-Bera test reject the null hypothesis that returns are normally distributed?

b. Estimate the GARCH model

\[y_{t}=\beta_{0}+e_{t} \quad\left(e_{t} \mid I_{t-1}\right) \sim N\left(0, h_{t}\right) \quad h_{t}=\delta+\alpha_{1} e_{t-1}^{2}+\beta_{1} h_{t-1}\]

Comment on the results. Plot the within-sample variance estimate \(\hat{h}_{t}\). Have the variance estimates captured the periods of high volatility noted in part (a)?

c. For the model estimated in part (b), compute the series \(z_{t}=\hat{e}_{t} / \sqrt{\hat{h}_{t}}\). Does a histogram for the \(z_{t}\) suggest the assumption \(z_{t} \sim N(0,1)\) is valid? Does the Jarque-Bera test support this assumption?

d. When the normality assumption is violated, the ordinary standard errors are not valid. However, valid robust standard errors can be used. \({ }^{1}\) Re-estimate the model in part (b) using the Bollerslev-Wooldridge robust standard errors. Does using these standard errors change any conclusions are about the precision of estimation?