Consider the three-dimensional return series of the previous exercise jointly. Build a multivariate time-varying volatility model for the data, using the Cholesky decomposition. Discuss the implications of the model and compute the 1-step-ahead volatility forecast at the forecast origin \(t=888\).

Previous Exercise:

Focus on the monthly log returns in percentages of GE stock and the S&P 500 index from January 1926 to December 1999. Build a time-varying correlation GARCH model for the bivariate series using the Cholesky decomposition. Check the adequacy of the fitted model, and obtain the 1 -step-ahead forecast of the covariance matrix at the forecast origin December 1999. Compare the model with the other model built in the previous exercise.

Previous Exercise:

The file \(\mathrm{m}\)-spibmge.txt contains the monthly log returns in percentages of the S&P composite index, IBM stock, and GE stock from January 1926 to December 1999. Focus on GE stock and the S&P 500 index. Build a time-varying correlation GARCH model for the bivariate series using a logistic function for the correlation coefficient. Check the adequacy of the fitted model, and obtain the 1-step-ahead forecast of the covariance matrix at the forecast origin December 1999.