Consider the monthly unemployment rate and the industrial production index of United States from January 1967 to December 2012 for 552 observations. The data are seasonally adjusted and obtained from FRED of the Federal Reserve Bank of St. Louis. We focus on modeling \(z_{t}\), which consists of the change series of unemployment rate and industrial production index. We also consider two input variables. They are the PMI composite index of the Institute of Supply Management and total capacity utilization (TCU). These series are also seasonally adjusted. Let \(x_{t}\) be the change series of PMI index and TCU. The data are in the file m-unippmitcu-6712.txt.

(a) Build a VAR model for \(z_{t}\), including model checking.

(b) Build a VARX model for \(\boldsymbol{z}_{t}\) using the input variable \(\boldsymbol{x}_{t}\). In model selection, you may use maximum \(p=11\) and maximum \(m=6\). Check the fitted model.

(c) Build a regression model with time series errors for \(\boldsymbol{z}_{t}\) using \(\boldsymbol{w}_{t}=\) \(\left(\boldsymbol{x}_{t}^{\prime}, \boldsymbol{x}_{t-1}^{\prime}\right)^{\prime}\) as the input variable. Check the fitted model.

(d) Compare the three fitted models for \(\boldsymbol{z}_{t}\).