Consider the quarterly U.S. federal government debt from the first quarter of 1970 to the third quarter of 2012. The first series is the federal debt held by foreign and international investors and the second series is federal debt held by Federal Reserve Banks. The data are from Federal Reserve Bank of St Louis, in billions of dollars, and are not seasonally adjusted. See the file q-fdebt . txt.

- Construct the bivariate time series \(z_{t}\) of the first difference of \(\log\) federal debt series. Plot the data.

- Fit a VAR(6) model to the \(z_{t}\) series. Perform model checking. Is the model adequate? Why?

- Perform a chi-square test to verify that all coefficient estimates of the VAR(6) model with \(t\)-ratios less than 1.96 can indeed be removed based on an approximate \(5 \%\) type I error.

- Based on the simplified model, is there any Granger causality between the two time series? Why?