Hostname: page-component-745bb68f8f-mzp66 Total loading time: 0 Render date: 2025-01-12T14:05:30.811Z Has data issue: false hasContentIssue false
  • English
  • Français

Residual-Based Tests for the Null of Stationarity with Applications to U.S. Macroeconomic Time Series

Published online by Cambridge University Press:  11 February 2009

In Choi
In Choi
Affiliation:
Kookmin University
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper proposes residual-based tests for the null of level- and trend-stationarity, which are analogs of the LM test for an MA unit root. Asymptotic distributions of the tests are nonstandard, but they are expressed in a unified manner by expressing stochastic integrals. In addition, the tests are shown to be consistent. By expressing the distributions expressed as a function of a chi-square variable with one degree of freedom, the exact limiting probability density and cumulative distribution functions are obtained, and the exact limiting cumulative distribution functions are tabulated. Finite sample performance of the proposed tests is studied by simulation. The tests display stable size when the lag truncation number for the long-run variance estimation is chosen appropriately. But the power of the tests is generally not high at selected sample sizes. The test for the null of trend-stationarity is applied to the U.S. macroeconomic time series along with the Phillips-Perron Z(⋯) test. For some monthly and annual series, the two tests provide consistent inferential results. But for most series, the two contradictory nulls of trend-stationarity and a unit root cannot be rejected at the conventional significance levels.

Type
Articles
Information
Econometric Theory , Volume 10 , Issue 3-4 , August 1994 , pp. 720 - 746
Copyright
Copyright © Cambridge University Press 1994

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCE

1.Anderson, T.W. & Darling, D.A.. Asymptotic theory of certain goodness of fit criteria based on stochastic processes. Annals of Mathematical Statistics 23 (1952): 193–212.CrossRefGoogle Scholar
2.Andrews, D.W.K.Heteroskedasticity and autocorrelation consistent covariance matrix estimation. Econometrica 58 (1991): 817–858.Google Scholar
3.Andrews, D.W.K. & Monahan, C.. An improved heteroskedasticity and autocorrelation consistent covariance matrix estimator. Econometrica 60 (1992): 953–966.Google Scholar
4.Berger, J.O.Statistical Decision Theory and Bayesian Analysis. New York: Springer-Verlag, 1985.CrossRefGoogle Scholar
5.Bernard, A.B. & Durlauf, S.. Interpreting Tests of Convergence Hypothesis. Working paper, Stanford University, 1992.Google Scholar
6.Bhargava, A.On the theory of testing for unit roots in observed time series. Review of Economic Studies 52 (1986): 369–384.CrossRefGoogle Scholar
7.Bierens, H.Testing Stationarity Against the Unit Root Hypothesis. Working paper, Vrije Universiteit and Texas A&M University, 1991.Google Scholar
8.Choi, I.Asymptotic Theory for Noninvertible MA(1) Processes. Working paper, The Ohio State University, 1991.Google Scholar
9.DeJong, D., Nankervis, J., Savin, N. & Whiteman, C.. The power problems of unit root tests in time series with autoregressive errors. Journal of Econometrics 53 (1992): 323–343.CrossRefGoogle Scholar
10.DeJong, D., Nankervis, J., Savin, N. & Whiteman, C.. Integration versus trend-stationarity in time series. Econometrica 60 (1992): 423–433.Google Scholar
11.Engle, R. & Granger, C.W.J.. Co-integration and error correction: Representation, estimation and testing. Econometrica 55 (1987): 251–276.CrossRefGoogle Scholar
12.Hannan, E.J.Multiple Time Series. New York: John Wiley, 1970.CrossRefGoogle Scholar
13.Hannan, E.J. & Heyde, C.. On limit theorem for quadratic functions of discrete time series. Annals of Mathematical Statistics 43 (1972): 2058–2066.Google Scholar
14.Herce, M.Stationarity Tests for Time Series. Working paper, University of North Carolina at Chapel Hill, 1991.Google Scholar
15.Kwiatkowski, D., Phillips, P.C.B., Schmidt, P. & Shin, Y.. Testing the null hypothesis of stationarity against the alternative of a unit root: How sure are we that economic time series have a unit root? Journal of Econometrics 54 (1992): 159–178.Google Scholar
16.Nabeya, S. Application of the Theory of Integral Equations to the Derivation of Asymptotic Distributions on Time Series Analysis. Ph.D. Dissertation, Hitotsubashi University, 1987.Google Scholar
17.Nabeya, S. & Tanaka, K.. A general approach to the limiting distribution of estimators in time series regression with nonstable autoregressive errors. Econometrica 58 (1990): 145–164.Google Scholar
18.Nelson, C. & Plosser, C.. Trends and random walks in macro in macroeconomic time series: Some evidence and implications. Journal of Monetary Economics 10 (1982): 139–162.Google Scholar
19.Park, J.Y. & Choi, B.. A New Approach to Testing for a Unit Root. Working paper, Cornell University, 1988.Google Scholar
20.Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: Part 1. Econometric Theory 4 (1988): 468–497.Google Scholar
21.Park, J.Y. & Phillips, P.C.B.. Statistical inference in regressions with integrated processes: Part 2. Econometric Theory 5 (1989): 95–131.Google Scholar
22.Perron, P.Test consistency with varying sampling frequency. Econometric Theory 7 (1991): 341–368.Google Scholar
23.Perron, P. & Phillips, P.C.B.. Does GNP have a unit root? Economic Letters 23 (1987): 139–145.Google Scholar
24.Phillips, P.C.B.Time series regression with a unit root. Econometrica 55 (1987): 277–301.Google Scholar
25.Phillips, P.C.B. Spectral regression for cointegrated time series. In Barnett, W. (ed.), Nonparametric and Semiparametric Methods in Economics and Statistics. New York: Cambridge University Press, 1992.Google Scholar
26.Phillips, P.C.B. & Perron, P.. Testing for a unit root in time series regression. Biometrika 74 (1988): 535–547.Google Scholar
27.Phillips, P.C.B. & Solo, V.. Asymptotics for linear processes. Annals of Statistics 20 (1992): 971–1001.Google Scholar
28.Saikkonen, P. & Luukkonen, R.. Testing for a moving average unit root. Journal of the American Statistical Association 88 (1989): 596–601.Google Scholar
29.Schmidt, P. & Phillips, P.C.B.. Testing for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54 (1992): 257–287.CrossRefGoogle Scholar
30.Shiller, R.Do stock prices move too much to be justified by subsequent changes in dividends? American Economic Review 71 (1981): 421–436.Google Scholar
31.Shiller, R. & Perron, P.. Testing the random walk hypothesis: Power versus frequency of observations. Economics Letters 18 (1985): 381–386.Google Scholar
32.Stock, J.Deciding Between I(1) and I(0). Working paper, Harvard University, 1992.CrossRefGoogle Scholar
33.Tanaka, K.Testing for a moving average unit root. Econometric Theory 6 (1990).CrossRefGoogle Scholar
34.Tanaka, K.The Fredholm determinant approach to asymptotic inference on nonstationary and noninvertible time series models. Econometric Theory 6 (1990).Google Scholar
35.Varberg, D.E.Convergence of quadratic forms in independent random variables. Annals of Mathematical Statistics 37 (1966): 567–576.Google Scholar
36.West, K.A note on the power of least squares tests for a unit root. Economics Letters 24 (1987): 249–252.CrossRefGoogle Scholar