Consider the weekly world crude oil prices from January 3, 1997 to September 24, 2010. The data are available from the US Energy Information Administration and in the file w-petroprice.txt. Focus on the change series of the price.

- Build a pure ARMA model for the change series of the world oil prices.

- Is there any seasonality in the price change series? If yes, perform simple seasonal adjustment by removing seasonal impact.

- Build a pure ARMA model for the seasonally adjusted series of price change.

- Build an ARMA-GARCH model for the seasonally adjusted series of price change.

- Compare the models with and without GARCH components using backtesting on the last 56 data points.