Again, consider the position and data of Exercise 1. Compute the VaR for the next trading day using the following methods:

(a) Empirical quantile with \(p=0.05\) and 0.01 .

(b) Quantile regression with \(1-p=0.95\) and 0.99 using lag-1 volatility of a Gaussian GARCH \((1,1)\) model and lag-1 absolute value of the log return as predictors. Write down the fitted quantile regression. Are the estimates statistically significant at the usual 5\% level?

Data From Exercise 1:

Consider a long position of \(\$ 1\) million on the Apple stock. To assess the risk of the position, we employ daily returns of the stock from January 2, 2001 to September 30, 2011 for 2704 observations. The daily simple returns are obtained from CRSP and in the file d-aapl-0111.txt. Let the tail probability be \(p=0.01\). Compute the VaR and ES of the position for the next trading day and the next 10 trading days using the following methods: