Consider the daily returns of the exchange trade fund (ETF) SPDR S\&P 500 of State Street Global Advisors from September 4, 2001, to September 30, 2011. The tick symbol is SPY and there are 2535 observations. The simple returns are available from CRSP and in the file d-spy-0111.txt. Transform the simple returns to \(\log\) returns.

(a) Is the expected log return zero? Are there any serial correlations in the log returns? Is there ARCH effect in the log returns?

(b) Fit a Gaussian ARMA-GARCH model for the log return series. Perform model checking, obtain the QQ plot of the standardized residuals, and write down the fitted model.

(c) Build an ARMA-GARCH model with Student \(t\) innovations for the \(\log\) return series. Perform model checking and write down the fitted model.