This set of exercises uses data from the file entitled QUARTELY.XLS in order to estimate the dynamic interrelationships among the level of industrial production, the unemployment rate, and interest rates. you created the interest rate spread (st) as the difference between the 10-year rate and the T-bill rate. Now create the logarithmic change in the index of industrial production (ip) as ?lipt = ln(ipt) - ln(ipt-1) and the seasonal difference of the unemployment rate as ?4urt = urt - urt-4.

Estimate the three-variable VAR using eight lags of each variable and a constant and save the residuals. Explain why the estimation cannot be begin earlier than 1963Q1. What are the potential advantages of using the variables ?lipt and ?4urt instead of ipt and urt?

Verify that ln(|S8|) = -13.968 and (assuming normality) that the log of the likelihood function is 493.647. Calculate the multivariate AIC and SBC using the formulas AIC = T ln(|S|) + 2N and SBC = Tln(|S|) + Nln(T). Calculate the multivariate AIC and SBC using the formulas AIC* = -2ln(L)/T + 2n/T and SBC* = -2ln(L)/T + n ln(T)/T.

Estimate the model using three lags of each variable and save the residuals. Show that the AIC selects the eight-lag model and that the SBC selects the three-lag model. Show that the same ambiguity applies to the AIC* and SBC*. Why is it important to estimate the three-variable VAR beginning with 1963Q1?

Construct the likelihood ratio test for the null hypothesis of eight lags against the alternative of three lags. How many restrictions are there in the system? How many regressors are there in each of the unrestricted equations? If you answer correctly, you should find that the calculated value ?2 with 45 degrees of freedom is 95.20, with a significance level smaller than 0.0001. Hence, the restriction of three lags is binding.

Now estimate the model with six lags. You should find that the likelihood ratio test selects the eight-lag model, the AIC selects the six-lag model, and the SBC selects the three-lag model.