The file mstateunrate.

txt contains the monthly unemployment rates of the states of California, Illinois, Michigan, New York, Pennsylvania, and Texas from January 1976 to August 2017 for 500 observations. The data are seasonally adjusted and available from FRED. For simplicity, focus on the first differenced series and denote the differenced series by yt.

(a) Perform a principal component analysis on yt using a sample covariance matrix. Obtain the scree plot. What percentage of total variations is explained by the first two principal components?

(b) Let zt be the first two principal component series. Obtain a suitableVAR model to zt.

(c) Put the fitted VAR model for zt into a SSM.

(d) Fit a dynamic factor model to yt using two common factors similar to that of Example 6.4.

(e) Obtain the time plots of smoothed factors based on the fitted SSM.