4. Which of the following MA(2) processes are invertible?

(a) X, = , - 0.9e, , + 0.2e, 2.

(b) X, = , - I.8e, , + 0.4e, 2-

(c) , = e, - 0.8e, , + 0.4e, 2.

5. Compute the acf for the following AR(2) processes and plot their correlograms.

(a) X, = 1.0Z, , - 0.5Z, 2 = e,. (The acf is given in the text. Just plot the correlogram.)

(b) X, = 0.9Z, , - 0.2Z, 2 + e,.

6. Table 13.2 at the end of the chapter gives the Beveridge index, and Figure 13.2 gives the correlogram for the trend-free index (with trend eliminated by the ratio to moving-average method). Plot the correlograms for the raw index and the first differences of the series.

7. Consider the ARMA model

X, = 1.0Z, , - 0.5Z,„2 + e, - 0.9e, , + 0.2e,„2

Express e, as a function of X, and lagged values of X, by expanding e, = (1 - 0.9L + 0.2U) \\ ~ l.OL + 0.5£2) :, in powers of L.

8. For a second-order AR process, show that the (theoretical) partial autocorrelation coefficient of order 2 is given by ( - p?)/(l - pf).

9. Suppose that the correlogram of a time series consisting of 100 observations has = 0.50, 2 = 0.63, = -0.10, 4 = 0.08, = -0.17, = 0.13, 7 = 0.09, = -0.05, - 0.12, r,o = -0.05. Suggest an ARMA model that would be appropriate. [Hint: The SE of each of the correlations ~ l/V = 0.10. Values greater than 2/\/7V are significant. Thus only the first 2 are significant. Hence an MA(2) process is appropriate.]

10. Table 13.7 gives data on monthly short-term interest rate (MRl) and the long-term interest rate (MR 240) for the period February 1950-December 1982. For each of the series estimate a regression on a time trend and seasonal dummies. Compute the R. Using the discussion of the different R measures suggested in Section 13.7, check whether these Rs are good or not.

11. In Exercise 10 compute the acf and plot the correlogram to determine whether the data are trend stationary or difference stationary. If they are difference stationary, determine the appropriate order of autoregression for the first differences of the two series using the methods in Table 13.1.

12. The seasonal effect in the two series in Table 13.7 can be taken into account by using seasonal dummies or by seasonal differencing as in the Box-Jenkins approach. Which of these two methods is appropriate for these data sets?

13. Repeat Exercise 10 with the quarterly data on consumption and income in Table 13.4.

14. Repeat Exercise 11 with the data in Table 13.4.

15. Repeat Exercise 12 with the data in Table 13.4.

16. Illustrate the use of the AIC and BIC criteria in the choice between different ARMA models using one of these data sets.

DATA SETS

Data Sets

In the following pages we present several time series. Table 13.2 presents data on the famous Beveridge price index. The data in Tables 13.3 to 13.7 were kindly provided by David N. DeJong. The data in Table 13.8 were provided by Harry Vroomen. In addition, time-series data are also provided in Table 4.10. All these data sets can be used to experiment with the techniques presented in the chapter.

Table 13.2 Beveridge Trend-Free Wheat Price Index

Table 13.2 (Cont.)