Consider, again, the SPY log return series of Problem 1.

(a) Fit an ARMA-APACRH model with Gaussian innovations to the data. Perform model checking and write down the model. You may ignore the lag-1 ARCH parameter as it is not statistically significant.]

(b) Fit an ARMA-APARCH model with Student \(t\) innovations to the data. Write down the fitted model and perform 1- to 5-step ahead predictions of the series and its volatility.

Data From Problem 1:

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.