# Spurious Regression Dividends And Profit

Questions

21.1. What is meant by weak stationarity?

21.2. What is meant by an integrated time series?

Models

Econometrics: Some Basic Concepts

CHAPTER TWENTY-ONE: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS 831

21.3. What is the meaning of a unit root?

21.4. If a time series is 7(3), how many times would you have to difference it to make it stationary?

21.5. What are Dickey-Fuller (DF) and augmented DF tests?

21.6. What are Engle-Granger (EG) and augmented EG tests?

21.7. What is the meaning of cointegration?

21.8. What is the difference, if any, between tests of unit roots and tests of cointegration?

21.9. What is spurious regression?

21.10. What is the connection between cointegration and spurious regression?

21.11. What is the difference between a deterministic trend and a stochastic trend?

21.12. What is meant by a trend-stationary process (TSP) and a difference-stationary process (DSP)?

21.13. What is a random walk (model)?

21.14. "For a random walk stochastic process, the variance is infinite." Do you agree? Why?

21.15. What is the error correction mechanism (ECM)? What is its relation with cointegration?

21.16. Using the data given in Table 21.1, obtain sample correlograms up to 25 lags for the time series PCE, PDI, Profits, and Dividends. What general pattern do you see? Intuitively, which one(s) of these time series seem to be stationary?

21.17. For each of the time series of exercise 21.16, use the DF test to find out if these series contain a unit root. If a unit root exists, how would you characterize such a time series?

21.18. Continue with exercise 21.17. How would you decide if the ADF test is more appropriate than the DF test?

21.19. Consider the dividends and profits time series given in Table 21.1. Since dividends depend on profits, consider the following simple model:

a. Would you expect this regression to suffer from the spurious regression phenomenon? Why?

b. Are Dividends and Profits time series cointegrated? How do you test for this explicitly? If, after testing, you find that they are cointegrated, would your answer in a change?

c. Employ the error correction mechanism (ECM) to study the short-and long-run behavior of dividends in relation to profits.

d. If you examine the Dividends and Profits series individually, do they exhibit stochastic or deterministic trends? What tests do you use?

*e. Assume Dividends and Profits are cointegrated. Then, instead of regressing dividends on profits, you regress profits on dividends. Is such a regression valid?

Problems

*Optional.

832 PART FOUR: SIMULTANEOUS-EQUATION MODELS

21.20. Take the first differences of the time series given in Table 21.1 and plot them. Also obtain a correlogram of each time series up to 25 lags. What strikes you about these correlograms?

21.21. Instead of regressing dividends on profits in level form, suppose you regress the first difference of dividends on the first difference of profits. Would you include the intercept in this regression? Why or why not? Show the calculations.

21.22. Continue with the previous exercise. How would you test the first-difference regression for stationarity? In the present example, what would you expect a priori and why? Show all the calculations.

21.23. From the U.K. private sector housing starts (X) for the period 1948 to 1984, Terence Mills obtained the following regression results*:

AXt = 31.03 - 0.188Xt-1 se = (12.50) (0.080) (t = )t (-2.35)

Note: The 5 percent critical t value is -2.95 and the 10 percent critical t value is -2.60.

a. On the basis of these results, is the housing starts time series stationary or nonstationary? Alternatively, is there a unit root in this time series? How do you know?

b. If you were to use the usual t test, is the observed t value statistically significant? On this basis, would you have concluded that this time series is stationary?

c. Now consider the following regression results:

A2Xt = 4.76 - 1.39AXt-1 + 0.313A2Xt-1 se = (5.06) (0.236) (0.163) (t = )t (-5.89)

where A2 is the second difference operator, that is, the first difference of the first difference. The estimated t value is now statistically significant. What can you say now about the stationarity of the time series in question?

Note: The purpose of the preceding regression is to find out if there is a second unit root in the time series.

21.24. Generate two random walk series as indicated in (21.7.1) and (21.7.2) and regress one on the other. Repeat this exercise but now use their first differences and verify that in this regression the R2 value is about zero and the Durbin-Watson d is close to 2.

*Terence C. Mills, op. cit., p. 127. Notation slightly altered.

CHAPTER TWENTY-ONE: TIME SERIES ECONOMETRICS: SOME BASIC CONCEPTS 833

21.25. To show that two variables, each with deterministic trend, can lead to spurious regression, Charemza et al. obtained the following regression based on 30 observations*:

where Y1 = 1, Y2 = 2, . . . , Yn = n and X1 = 1, X2 = 4, . . . , Xn = n2.

a. What kind of trend does Y exhibit? and X?

b. Plot the two variables and plot the regression line. What general conclusion do you draw from this plot?

21.26. From the data for the period 1971-I to 1988-IV for Canada, the following regression results were obtained:

1. lnM1t = —10.2571 + 1.5975 lnGDPt t = (—12.9422) (25.8865)

2. A^M! = 0.0095 + 0.5833A lnGDPt t = (2.4957) (1.8958)

where M1 = M1 money supply, GDP = gross domestic product, both measured in billions of Canadian dollars, ln is natural log, and ut represent the estimated residuals from regression 1.

a. Interpret regressions 1 and 2.

b. Do you suspect that regression 1 is spurious? Why?

c. Is regression 2 spurious? How do you know?

d. From the results of regression 3, would you change your conclusion in b? And why?

e. Now consider the following regression:

What does this regression tell you? Does this help you decide if regression 1 is spurious or not?

834 PART FOUR: SIMULTANEOUS-EQUATION MODELS

21.27. The following regressions are based on the CPI data for the United States for the period 1960-1999, for a total of 40 annual observations:

R2 0.4483 d 0.7969 RSS 115.8579

where RSS = residual sum of squares.

a. Examining the preceding regressions, what can you say about sta-tionarity of the CPI time series?

b. How would you choose among the three models?

c. Equation (1) is Eq. (3) minus the intercept and trend. Which test would you use to decide if the implied restrictions of model 1 are valid? (Hint: Use the Dickey-Fuller t and F tests. Use the approximate values given in Appendix D, Table D.7.)