integration versus trend stationary in time series.pdf

1、Integration Versus Trend Stationary in Time SeriesDavid N. DeJong; John C. Nankervis; N. E. Savin; Charles H. WhitemanEconometrica, Vol. 60, No. 2. (Mar., 1992), pp. 423-433.Stable URL:http:/links.jstor.org/sici?sici=0012-9682%28199203%2960%3A2%3C423%3AIVTSIT%3E2.0.CO%3B2-UEconometrica is currently

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6、 JSTOR, please contact supportjstor.org.http:/www.jstor.orgTue Oct 30 17:00:14 2007Econornetrica, Vol. 60, No. 2, (March, 1992), 423-433 NOTES AND COMMENTS INTEGRATION VERSUS TREND STATIONARITY IN TIME SERIES 1. INTRODUCTION A WELL-KNOWN APPROACH to modeling macroeconomic time series is to assume th

7、at the natural logarithm of the series can be represented by the sum of a deterministic time trend and a stochastic term. The trend need not literally be part of the data generation process, but may be viewed as a substitute for a complicated and unknown function of population, capital accumulation,

8、 technical progress, etc. Within this approach there are two competing models; in the trendatationary specification the stochastic term follows a stationary process, while in the integrated specification the stochastic term follows a random walk. The essential difference between the models is the na

9、ture of the process driving the stochastic component, not whether the series is trended. The conclusion of this study is that it is difficult to discriminate between the two models using classical testing methods. This is the consequence of low power: the powers of integration tests against plausibl

10、e trend-stationary alternatives can be quite low, as can the powers of trend-stationarity tests against integrated alternatives. Our analysis thus suggests that it is premature to accept the integration hypothesis as a stylized fact of macroeconomic time series. The leading case we examine is a mode

11、l with linear trend and iid normal innovations, which we use to study the power of integration and trend-stationarity tests. This strategy is motivated by the idea that the study should begin with the case which is most favorable to high power; the presumption is that the finite sample powers of tes

12、ts designed for this case are superior to the powers of tests designed for models with more general innovation sequences. 2. LEADING CASE Let the time series y, be the stochastic process generated by the linear model and the first-order autoregressive (AR) process It is assumed that the innovation s

13、equence u, is iid N(0, a), and xo is an unknown constant. Thus model (2.1)-(2.2) can be interpreted as a random walk about a linear trend when /3 = 1 and an asymptotically stationary AR(1) process about a linear trend when I 0.7953 and to unity when j3 0.95, and Ix,*1 2, it is useful to recognize th

14、at such values are quite likely. When j3 1, in treating x;i as fixed we are conditioning on a drawing from a Gaussian distribution with mean zero and variance (1 -p2)-l. Thus with j3 = 0.85, such a drawing is one standard deviation away from its mean at 1.9, two standard deviations at 3.8, three sta

15、ndard deviations at 5.7. ?Schmidt and Phillips (1989) show that the distributions of related LM test statistics depend only upon p and xg, and note that the empirical powers of the LM tests depend only upon the absolute value of x,*. 7hisfeature of the limiting distribution of K(1) is confirmed by t

16、he small-a asymptotic theory in Evans and Savin (1984). D. N. DEJONG, J. C. NANKERVIS, N. E. SAVIN, AND C. H. WHITEMAN TABLE I1 POWERSOF UNITROOT TESTS FOR T = 100 Exact powers of the K(1)test, critical value = -20.47 P I X I 10.75 0.80 0.85 0.90 0.95 0.99 1.00 Powers of the t(1) test, critical valu

17、e = -3.45 P 1x81 10.75 0.80 0.85 0.90 0.95 0.99 1.00 0.00 0.86 0.65 0.39 0.19 0.08 0.05 0.05 1.00 0.87 0.66 0.40 0.19 0.08 0.05 0.05 2.00 0.88 0.68 0.41 0.19 0.08 0.05 0.05 3.00 0.90 0.71 0.43 0.20 0.08 0.05 0.05 4.00 0.93 0.75 0.46 0.21 0.08 0.05 0.05 5.00 0.95 0.79 0.50 0.22 0.08 0.05 0.05 6.00 0.

18、97 0.84 0.54 0.24 0.09 0.05 0.05 7.00 0.98 0.89 0.60 0.26 0.09 0.05 0.05 8.00 0.99 0.92 0.66 0.29 0.09 0.05 0.05 9.00 1.00 0.95 0.72 0.32 0.09 0.05 0.05 10.00 1.00 0.97 0.78 0.35 0.10 0.05 0.05 Powers of the F(O,1) test, critical value = 6.49 P lx$1 0.75 0.80 0.85 0.90 0.95 0.99 1.00 0.00 0.80 0.56

19、0.31 0.14 0.06 0.05 0.05 1.00 0.80 0.57 0.32 0.14 0.06 0.05 0.05 2.00 0.83 0.60 0.34 0.15 0.07 0.05 0.05 3.00 0.86 0.64 0.37 0.16 0.07 0.05 0.05 4.00 0.90 0.70 0.41 0.18 0.07 0.05 0.05 5.00 0.94 0.77 0.46 0.20 0.07 0.05 0.05 6.00 0.97 0.83 0.53 0.23 0.08 0.05 0.05 7.00 0.99 0.90 0.61 0.27 0.08 0.05

20、0.05 8.00 1.00 0.94 0.70 0.32 0.09 0.05 0.05 9.00 1.00 0.97 0.79 0.37 0.10 0.05 0.05 10.00 1.00 0.99 0.86 0.45 0.11 0.05 0.05 Estimates based on 20,000 replications From the power tables, as xg ranges from 0 to 5.7, the power of K(1) against P = 0.85 falls from 0.49 to 0.41, the power of t(1) rises

21、from 0.39 to about 0.54, and the power of F(0,l) rises from 0.31 to over 0.5. Thus likely variation in xg appears to be important for power of the unit root tests. Finally, for all tests the powers when T = 50 (not reported) are substantially lower than those at T = 100. For example, when xg = 0 and

22、 /3 = 0.8 the power of the a = 0.05 size F(0,l) test is 0.16 and the power of the LY = 0.05 size lower-tail t(1) test is 0.20. In INTEGRATION VS. TREND STATIONARITY TABLE 111 POWERSOF TESTSOF H,: P =0.85 FOR T = 100 Powers of the tA(0.85) test, critical value = 0.56 lx; l 1 0.85 0.90 P 0.95 0.99 1.0

23、0 Estimates based on 20,000 replications Exact Powers of the SA(0.85)test, critical value = 0.506 P 1x8 l 0.85 0.90 0.95 0.99 1.00 0.00 0.05 0.20 0.45 0.60 0.61 1.00 0.05 0.20 0.45 0.60 0.61 2.00 0.05 0.20 0.45 0.60 0.61 3.00 0.05 0.20 0.45 0.60 0.61 4.00 0.05 0.20 0.46 0.60 0.61 5.00 0.05 0.21 0.46

24、 0.60 0.61 6.00 0.05 0.21 0.46 0.60 0.61 7.00 0.05 0.21 0.47 0.60 0.61 8.00 0.05 0.21 0.47 0.60 0.61 9.00 0.05 0.21 0.48 0.60 0.61 10.00 0.05 0.21 0.48 0.60 0.61 general, the powers of the integration tests are so low when T = 50 that they are not worth reporting in detail. In light of the low power

25、s we do not recommend performing the integration tests for T 100. B. Trend-Stationarity Tests The powers of 0.05 one-sided similar tests of H: P = 0.85 for T = 100 are given in Table 111. Several features of the tables stand out. For example, the powers of the two tests are nearly the same, and incr

26、ease with p. Also, for the tabled values, the powers of the tests are nearly invariant with respect to x,*.For the tabled values of x,*,the exact power of detecting a unit root with the SA(.85) test is 0.61; the empirical power of the tA(.85) test at P = 1 is 0.61. These results indicate that the tr

27、end-stationarity tests have somewhat better than a 50% chance of detecting a unit root for plausible values of xg. C. Comparison Our analysis has revealed that both types of tests are beset by lower power at the alternatives of interest, with the performance of the trend-stationarity tests somewhat

28、better than that of the integration tests. Figure 1 illustrates this situation. The tests examined are the LY = 0.05 lower-tail t(1) test and the LY = 0.05 upper-tail tA(.85) test. The powers of the tests are illustrated for T = 100 using four distributions: the distribu- D. N. DEJONG, J. C. NANKERV

29、IS, N. E. SAVIN, AND C. H. WHITEMAN t(l i.85) + t(lj1) 0 t4.85 / .85) A ta(.85 / 1) FIGURE 1.-Empirical distributions of t(. I .) statistics. tions of tA(.85) when (Ix: 1, P) = (2,0235) and (0,l); and the distributions of t(1) when (Ix: 1, P) = (0,l) and (2,0.85). Denote the first two distributions

30、by tA(.85 1.85) and tA(.85 11) and the second two by t(ll1) and t(11.85): tA(.851.85) and t(l(1)are the distributions of tA(.85)and t(1)under the respective nulls, and tA(.85 11) and t(11.85) are the distributions under the alternatives. The low power-about 0.4-of the t(1) test is shown by the overl

31、ap of the t(l I I) and t(11.85) distributions. The power-about 0.6-of the tA(.85)test is due to the difference in the shapes of the tA(.85 1.85) and tA(.8511) distributions. These power comparisons emphasize the importance of considering the nature of the alternative in hypothesis testing. In partic

32、ular, the powers of unit root and stationarity tests may not be high enough to settle the integration vs. trend-stationarity issue in many practically relevant situations. 6. CONCLUSIONS There are three major findings in this paper. First, unit root tests have low power against plausible trend-stati

33、onary alternatives. Second, tests of an empirically plausible trend-stationarity hypothesis have moderate power against the unit root alternative. Third, there are many cases in which neither test will reject. This suggests that inferences based exclusively on tests for integration may be fragile. D

34、epartment of Economics, University of Pittsburgh, Pittsburgh, PA 15260, U.S.A., Department of Economics, City of London Polytechnic, 84 Moorgate, London EC2M 6SQ, U.K., and Department of Economics, University of Iowa, Iowa City, IA 52246, U.S.A. Manuscript received February, 1989;final revision rece

35、ived June, 1991. INTEGRATION VS. TREND STATIONARITY REFERENCES BHARGAVA,A. (1986): “On the Theory of Testing for Unit Roots in Observed Time Series,“ Review of Economic Studies, 52, 369-384. DAVIES,R. B. (1980): “The Distribution of a Linear Combination of Chi-Square Random Variables,“ (Algorithm AS

36、155), Applied Statistics, 29, 323-333. DEJONG, D. N., J. C. NANKERVIS, N. E. SAVIN, AND C. H. WHITEMAN (1988): “Integration Versus Trend-Stationarity in Macroeconomic Time Series,“ Department of Economics Working Paper 88-27, University of Iowa. Revised as Working Paper 88-27a (1988) and 89-31 (1989

37、). DICKEY,D. A. (1976): “Estimation and Hypothesis Testing for Nonstationary Time Series,“ Ph.D. Thesis, Iowa State University. -(1984): “Powers of Unit Root Tests,“ Proceedings of the American Statistical Association, Business and Economics Section, 489-492. DICKEY,D. A, AND W. A. FULLER (1979): “D

38、istribution of the Estimators for Autoregressive Time Series With a Unit Root,“ Journal of the American Statistical Association, 74, 427-431. -(1981): “Likelihood Ratio Statistics for Autoregressive Time Series with a Unit Root,“ Econometrica, 49, 1057-1072. DHRYMES,P. J. (1981): Distributed Lags, 2

39、nd ed. Amsterdam: North Holland. EVANS,G. B. A, AND N. E. SAVIN(1981): “Testing for Unit Roots I,“ Econometrica, 49, 753-779. -(1984): “Testing for Unit Roots 11,“ Econometrica, 52, 1241-1269. FULLER,W. A. (1976): Introduction to Statistical Time Series. New York: John Wiley and Sons. NANKERVIS,J. C

40、., AND N. E. SAVIN (1985): “Testing the Autoregressive Parameter With the t Statistic,“ Journal of Econometrics, 27, 143-161. -(1987): “Finite Sample Distributions of t and F Statistics in an AR(1) Model with an Exogenous Variable,“ Economic Theory, 3, 387-408. PARK, J. Y., AND B. CHOI (1988): “A Ne

41、w Approach to Testing for a Unit Root,“ Cornell University, manuscript. PHILLIPS,P. C. B., AND P. PERRON (1988): “Testing for a Unit Root in Time Series Regression,“ Biometrika, 75, 335-346. SCHMIDT, P., AND P. C. B. PHILLIPS(1989): “Testing for a Unit Root in the Presence of Deterministic Trends,“

42、Michigan State University, manuscript. You have printed the following article:Integration Versus Trend Stationary in Time SeriesDavid N. DeJong; John C. Nankervis; N. E. Savin; Charles H. WhitemanEconometrica, Vol. 60, No. 2. (Mar., 1992), pp. 423-433.Stable URL:http:/links.jstor.org/sici?sici=0012-

43、9682%28199203%2960%3A2%3C423%3AIVTSIT%3E2.0.CO%3B2-UThis article references the following linked citations. If you are trying to access articles from anoff-campus location, you may be required to first logon via your library web site to access JSTOR. Pleasevisit your librarys website or contact a li

44、brarian to learn about options for remote access to JSTOR.Footnotes3 Likelihood Ratio Statistics for Autoregressive Time Series with a Unit RootDavid A. Dickey; Wayne A. FullerEconometrica, Vol. 49, No. 4. (Jul., 1981), pp. 1057-1072.Stable URL:http:/links.jstor.org/sici?sici=0012-9682%28198107%2949

45、%3A4%3C1057%3ALRSFAT%3E2.0.CO%3B2-47 Testing for Unit Roots: 2G. B. A. Evans; N. E. SavinEconometrica, Vol. 52, No. 5. (Sep., 1984), pp. 1241-1269.Stable URL:http:/links.jstor.org/sici?sici=0012-9682%28198409%2952%3A5%3C1241%3ATFUR2%3E2.0.CO%3B2-WReferencesOn the Theory of Testing for Unit Roots in

46、Observed Time SeriesAlok BhargavaThe Review of Economic Studies, Vol. 53, No. 3. (Jul., 1986), pp. 369-384.Stable URL:http:/links.jstor.org/sici?sici=0034-6527%28198607%2953%3A3%3C369%3AOTTOTF%3E2.0.CO%3B2-Ahttp:/www.jstor.orgLINKED CITATIONS- Page 1 of 2 -NOTE: The reference numbering from the orig

47、inal has been maintained in this citation list.Algorithm AS 155: The Distribution of a Linear Combination of #2 Random VariablesRobert B. DaviesApplied Statistics, Vol. 29, No. 3. (1980), pp. 323-333.Stable URL:http:/links.jstor.org/sici?sici=0035-9254%281980%2929%3A3%3C323%3AAA1TDO%3E2.0.CO%3B2-HDi

48、stribution of the Estimators for Autoregressive Time Series With a Unit RootDavid A. Dickey; Wayne A. FullerJournal of the American Statistical Association, Vol. 74, No. 366. (Jun., 1979), pp. 427-431.Stable URL:http:/links.jstor.org/sici?sici=0162-1459%28197906%2974%3A366%3C427%3ADOTEFA%3E2.0.CO%3B

49、2-3Likelihood Ratio Statistics for Autoregressive Time Series with a Unit RootDavid A. Dickey; Wayne A. FullerEconometrica, Vol. 49, No. 4. (Jul., 1981), pp. 1057-1072.Stable URL:http:/links.jstor.org/sici?sici=0012-9682%28198107%2949%3A4%3C1057%3ALRSFAT%3E2.0.CO%3B2-4Testing For Unit Roots: 1G. B. A. Evans; N. E. SavinEconometrica, Vol. 49, No. 3. (May, 1981), pp. 753-779.Stable URL:http:/links.jstor.org/sici?sici=0012-9682%28198105%2949%3A3%3C753%3ATFUR1%3E2.0.CO%3B2-ATesting for Unit Roots: 2G. B. A. Evans; N. E. SavinEconometrica, Vol. 52, No. 5. (Sep., 1984)