Consider the U.S. populations of men and women from January 1948 to December 2012. The data are available from FRED of the Federal Reserve Bank of St. Louis and are in thousands. See also the files m-popmen.txt and m-popwomen.txt. Let \(z_{t}\) be the \(\log\) series of the U.S. monthly populations of men and women.

(a) Are the two population series cointegrated? Why?

(b) Build VARMA model for \(\boldsymbol{z}_{t}\), including model checking. Write down the fitted model.

(c) Use the fitted model to produce one-step to three-step ahead predictions of the \(z_{t}\) series at the forecast origin December 2012.