Housing markets in the United States have been under pressure since the 2007 subprime financial crisis. In this problem, we consider the housing starts for the West and South regions of U.S. Census. The data are from FRED of the Federal Reserve Bank of St. Louis, in thousands of units and not seasonally adjusted. The sample period is from January 1959 to December 2012 for 648 observations. See the file m-houst-nsa.txt, which contains data for all four regions. Let \(z_{t}\) be the log series of the housing starts for West and South regions.

(a) Build a seasonal VARMA model for \(\boldsymbol{z}_{t}\). Perform model checking and discuss the implications of the model.

(b) Use the fitted model to obtain 1-step to 3-step forecasts of \(\boldsymbol{z}_{t}\) at the forecast origin December 2012.