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On the Asymptotic Power of Unit Root Tests

Published online by Cambridge University Press:  11 February 2009

Karim M. Abadir
Karim M. Abadir
Affiliation:
The American University in Cairo
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Abstract

Closed forms for the distribution of some conventional statistics are given as a prelude to deriving their asymptotic power functions as unit root tests. In the process, an important distinction is drawn between two classes of statistics: one which relies on deterministic normalizations and the other which uses stochastic normalizations. When the data follow a driftless autoregression, a t test (which belongs to the second class) for a unit root is found to perform better than the other tests in small to moderate effective samples.

Type
Articles
Information
Econometric Theory , Volume 9 , Issue 2 , April 1993 , pp. 189 - 221
Copyright
Copyright © Cambridge University Press 1993

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References

1.Abadir, K.M.The limiting distribution of the t ratio under a unit root. Mimeo, American University in Cairo, 1991.Google Scholar
2.Abadir, K.M.A distribution generating equation for unit-root statistics. Oxford Bulletin of Economics and Statistics 54 (1992): 305–323.CrossRefGoogle Scholar
3.Abadir, K.M.The limiting distribution of the autocorrelation coefficient under a unit root. Annals of Statistics 21 (1993), No. 2.Google Scholar
4.Abadir, K.M.OLS bias in a nonstationary autoregression. Econometric Theory 9 (1993): 81–93.CrossRefGoogle Scholar
5.Abramowitz, M. & Stegun, I.A.. Handbook of Mathematical Functions. New York: Dover Publications, 1965.Google Scholar
6.Ahtola, J. & Tiao, G.C.. Parameter inference for nearly nonstationary first-order autoregressive model. Biometrika 71 (1984): 263–272.Google Scholar
7.Breiman, L.Probability. Reading, MA: Addison-Wesley, 1968.Google Scholar
8.Chan, N.H.The parameter inference for nearly nonstationary time series. Journal of the American Statistical Association 83 (1988): 857–862.CrossRefGoogle Scholar
9.Craig, C.C.On the frequency function of xy. Annals of Mathematical Statistics 7 (1936): 1–15.Google Scholar
10.De Bruijn, N.G.Asymptotic Methods in Analysis. New York: Dover Publications, 1981.Google Scholar
11.Deaton, A. & Muellbauer, J.. Economics and Consumer Behavior. Cambridge: Cambridge University Press, 1980.Google Scholar
12.Dickey, D.A. & Fuller, W.A.. Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74 (1979): 427–431.Google Scholar
13.Dickey, D.A. & Fuller, W.A.. Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49 (1981): 1057–1072.CrossRefGoogle Scholar
14.Donsker, M.D. & Varadhan, S.R.S.. On laws of the iterated logarithm for local times. Communications in Pure and Applied Mathematics 30 (1977): 707–753.Google Scholar
15.Engle, R.F. Wald, likelihood ratio, and Lagrange multiplier tests in econometrics. In Griliches, Z. and Intriligator, M.D. (eds.), Handbook of Econometrics, Vol. 2. Amsterdam: North-Holland, 1984.Google Scholar
16.Erdélyi, A. (ed.). Higher Transcendental Functions, Vols. 1–2. New York: McGraw-Hill, 1953.Google Scholar
17.Erdélyi, A. (ed.). Higher Transcendental Functions, Vol. 3. New York: McGraw-Hill, 1955.Google Scholar
18.Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 1. Econometrica 49 (1981): 753–779.CrossRefGoogle Scholar
19.Evans, G.B.A. & Savin, N.E.. Testing for unit roots: 2. Econometrica 52 (1984): 1241–1269.Google Scholar
20.Fountis, N.G. & Dickey, D.A.. Testing for a unit root nonstationarity in multivariate autoregressive time series. Annals of Statistics 17 (1989): 419–428.Google Scholar
21.Fuller, W.A.Introduction to Statistical Time Series. New York: Wiley, 1976.Google Scholar
22.Gradshteyn, I.S. & Ryzhik, I.M.. Table of Integrals, Series, and Products. New York: Academic Press, 1980.Google Scholar
23.Hendry, D.F. Monte Carlo experimentation in econometrics. In Griliches, Z. and Intriligator, M.D. (eds.), Handbook of Econometrics, Vol. 2. Amsterdam: North-Holland, 1984.Google Scholar
24.Jahnke, E. & Emde, F.. Tables of Functions. New York: Dover Publications, 1945.Google Scholar
25.Jain, N.C. & Pruitt, W.E.. The other law of the iterated logarithm. Annals of Probability 3 (1975): 1046–1049.Google Scholar
26.Johnson, N. & Kotz, S.. Continuous Univariate Distributions−2. Boston: Houghton Mifflin Co., 1970.Google Scholar
27.Karlin, S. & Taylor, H.M.. A First Course in Stochastic Processes, 2nd ed.New York: Academic Press, 1975.Google Scholar
28.Kendall, M. & Stuart, A.. The Advanced Theory of Statistics, Vol. 1, 4th ed.London: Charles Griffin & Co., 1977.Google Scholar
29.Lai, T.L. & Wei, C.Z.. Least squares estimates in stochastic regression models with applications to identification and control of dynamic systems. Annals of Statistics 10 (1982): 154–166.Google Scholar
30.Mood, A., Graybill, F. & Boes, D.. Introduction to the Theory of Statistics, 3rd ed.Singapore: McGraw-Hill, 1974.Google Scholar
31.Nabeya, S. & Tanaka, K.. Limiting power of unit root tests in time-series regression. Journal of Econometrics 46 (1990): 247–271.Google Scholar
32.Oberhettinger, F.Tables of Mellin Transforms. Berlin: Springer-Verlag, 1974.CrossRefGoogle Scholar
33.Oberhettinger, F. & Badii, L.. Tables of Laplace Transforms. Berlin: Springer-Verlag, 1973.CrossRefGoogle Scholar
34.Pantula, S.G.Testing for unit roots in time series data. Econometric Theory 5 (1989): 256–271.CrossRefGoogle Scholar
35.Perron, P.The calculation of the limiting distribution of the least-squares estimator in a near-integrated model. Econometric Theory 5 (1989): 241–255.Google Scholar
36.Phillips, P.C.B.Approximations to some finite sample distributions associated with a first-order stochastic difference equation. Econometrica 45 (1977): 463–485.Google Scholar
37.Phillips, P.C.B.Edgeworth and saddlepoint approximations in the first-order noncircular autoregression. Biometrika 65 (1978): 91–98.Google Scholar
38.Phillips, P.C.B. A note on the saddlepoint approximation in the first order non-circular autoregression. Cowles Foundation Discussion Paper No. 487, Yale University, 1978.Google Scholar
39.Phillips, P.C.B. Exact small sample theory in the simultaneous equations model. In Griliches, Z. and Intriligator, M.D. (eds.), Handbook of Econometrics, Vol. 1. Amsterdam: North-Holland, 1983.Google Scholar
40.Phillips, P.C.B.Towards a unified asymptotic theory for autoregression. Biometrika 74 (1987): 535–547.Google Scholar
41.Phillips, P.C.B.Time series regression with a unit root. Econometrica 55 (1987): 277–301.Google Scholar
42.Phillips, P.C.B.Regression theory for near-integrated time series. Econometrica 56 (1988): 1021–1043.Google Scholar
43.Phillips, P.C.B. & Perron, P.. Testing for a unit root in time series regression. Biometrika 75 (1988): 335–346.Google Scholar
44.Rao, M.M.Asymptotic distribution of an estimator of the boundary parameter of an unstable process. Annals of Statistics 6 (1978): 185–190.Google Scholar
45.Solo, V.The order of differencing in ARIMA models. Journal of the American Statistical Association 79 (1984): 916–921.Google Scholar
46.Stock, J.H.Asymptotic properties of least squares estimators of cointegrating vectors. Econometrica 55 (1987): 1035–1056.Google Scholar
47.White, J.S.The limiting distribution of the serial correlation coefficient in the explosive case. Annals of Mathematical Statistics 29 (1958): 1188–1197.Google Scholar
48.White, J.S.The limiting distribution of the serial correlation coefficient in the explosive case II. Annals of Mathematical Statistics 30 (1959): 831–834.Google Scholar
49.Whittaker, E.T. & Watson, G.N.. A Course of Modern Analysis, 4th ed.Cambridge: Cambridge University Press, 1927.Google Scholar