Sample Autocorrelation Functions of Dally Hog Perce Data

...

3:

tionary. Only first-differencing will yield stationary series. Second. the answer has implications for our understanding of the economy and for forecasting. If a variable like CNP follows a random walk. the effects of a temporary shock (such as an increase in oil prices or a drop in government spending) will not dissipate after Several years, but instead will be permanent.[pic 23]

In a provocative study. Charles Nelson and Charles Plosser found evidence that GNP and other macroeconomic time series behave like random walks.' The work spawned a series of studies that investigate whether economic and financial variables are random walks or trend-reverting. Several of these studies show that many economic time series do appear to be random walks. or at least have random walk components.' Most of these studies use unit root tests introduced by David Dickey and Wayne Fullcr.9

Suppose we believe that a variable Yr. which has been growing over time. can be described by the following equation:

Yt = a + ßt + p Yr—I + e, (15.34)

[pic 24]One possibility is that Y, has been growing because it has a positive trend (P > O). but would be stationa:y after detrending (i.e.. p [pic 25]it follows a random walk with a positivc drift (i.e.. > O p = O. and p = l). la this case. one would want to work with a Y, Detrending would not make the [pic 26]series stationary. and inclusion of Y, in a regression (even if detrended) could lead to spurious results.[pic 27]

One might think that Eq. (15.34) could be estimated by OLS. and the t statistic on $5 could then be used to test whetherþ is significantly different from l. However, as we saw in Chapter 9, if the true value of p is indeed I, then the OLS estimator is biased toward zero. Thus the use of OLS in this manner can-lead onc to incorrectly reject the random walk hypothesis.

Dickey and Fuller derived the distribution for the estilnator þ that holds when p = I. and generated statistics for a simple F test of the random walk hypothesis. i.e.. of the hypothesis that p = O and p = l . The Dickey-Fuller test is easy to

[pic 28]

C. R. Nelson and C. Plosscc. • "Trcaus and Random "Valks in Macroeconomic Time Scrics: Some Evidcnce and Implications.•• Journal of Monetary Ecønomia. vol. 10. pp. 139—162, 1982.

• Examples of these studies includc J. Y. Campbell and N. G. Mankiw, "Arc Output Fluctuations Transitory?.•• Quarterly Journal of Economics. vol. 102. pp. 857—880. 1987: J. Y. Camptxll and N. G. Mankiw. "Permanent and Transitory Components In Macroeconomic Fluctuations." American Eco• nomx Review papers and Ptxeedinys. vol. 77. pp. I I t—117. 1987; and G. W. Gardner and K. P. Kimbrough. •The Behavior of U.S. Tariff Rates." American Economic Review. vol. 79. pp. 211—218. 1989.

[pic 29]D. A. Dickey and W. A. Fuller. "Distribution or the Estimators for Autoregressive nme-Series with a Unit Root:•_ Journal Of the Amertcart Statistical R.gxiation. vol. 74. pp. 427—01. 1979; D. A. Dickey and W. A. Fuller, • 'Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root." Econometrica. vol- 49. pp. 1057—1072. 198 1; and W. A. Fuller. Introduction to Statistüal Time Series (New York: Wiley. 1976).

226

a-qAPTER Of [pic 30]

TABLE 15.1

OCSTRtB1-mON OF F FOR (a. ß.-p) - (a. O. 1) r, - a + + + [pic 31]

Sample stz•

Probablltty of a smatter value

.05

.10

.95

.975

.99

25

50

250

Standard

.74

.76

.76

.76

.76

.77

.004

.90

.93

.94

.94

.94

.94

.004

1.08

1.11

1.13

1.13

1.13

1.33

1.37

1.38

1.39

1.39

1.39

5.91

5.61

5.47

6.39

5.36

5.34

.015

7.24

6.73

6.49

6.34

6.30

6.25

.020

8.65

7.81

7.44

7.25

7.16

032

10.61

9.38

833

8.43

8.34

8 27

058