Consider the monthly simple excess returns of 10 U.S. stocks from January 1990 to December 2003 for 168 observations. The 3-month Treasury bill rate on the secondary market is used to compute the excess returns. The data are in the file m-excess-10c-9003.txt. The 10 tick symbols of the stocks are

(1) pharmaceutical: ABT, LLY, MRK, and PFE;

(2) automobile: F and GM;

(3) oil companies: BP, CVX, RD, and XOM. Let \(z_{t}\) be the 10 -dimensional excess return series.

(a) Perform PCA on \(z_{t}\). Obtain the scree plot and select the number of command factors.

(b) Fit an orthogonal factor model to \(z_{t}\) using the PCA approach.

(c) Fit an orthogonal factor model to \(z_{t}\) using the maximum likelihood method.

(d) Fit a constrained factor model, based on the industrial sectors, to \(z_{t}\).

(e) Compare the three fitted factor models.