E16.3 In the data file USMacro_Quarterly, you will find the data on U.S. real GDP

(GDPC1) that was analyzed in Chapter 15. In this exercise, you will construct a 95% confidence interval for the mean growth rate of real GDP in the United States; in addition, you will construct a 95% forecast interval for the average growth rate of real GDP for 2018:Q1–2067:Q4. Before attempting this empirical exercise, you should answer Exercise 16.12.

a. Compute the growth rate of real GDP: Yt = 400 * [ln(GDPC1t) -

ln(GDPC1t-1)]. Plot the series from 1960 through 2017, and verify that the data are the same as plotted in Figure 15.1b.

b. Using the data from 1960:Q1 through 2017:Q4:

i. Estimate an AR(1) model for Yt . In the notation of Exercise 16.12, denote the estimated AR(1) coefficient by f n 1 and the standard error of the regression as sn u.
ii. Compute the sample mean of Yt.

c. Assuming that Yt follows an AR(1), use the results you derived in Exercise 16.12, the estimated values of f1 and s2 u from (b.i), and the sample mean from (b.ii) to i. Construct a 95% confidence interval for μY, the mean growth rate of real GDP.
ii. Construct a 95% forecast interval for the average growth rate of real GDP over the period 2018:Q1–2067:Q4—that is, for Y2018Q1:2067Q4.

d. Using the data from 1960:Q1 through 2017:Q4:
i. Regress Yt on a constant (with no other regressors). Construct the standard error for the estimated constant using the Newey–West HAC estimator with four lags.
ii. Use the results from this regression to construct a 95% confidence interval for μY, the mean growth rate of real GDP.
iii. Use the results from this regression to construct a 95% forecast interval for the average growth rate of real GDP over the period 2018:Q1–2067:Q4—that is, for Y2018Q1:2067Q4.

e. Are the intervals constructed in (d.ii) and (d.iii) similar to the intervals constructed in (c.i) and (c.ii)? Should they be? Explain.