Hostname: page-component-77c89778f8-sh8wx Total loading time: 0 Render date: 2024-07-22T07:01:24.405Z Has data issue: false hasContentIssue false
  • English
  • Français

ON TESTS FOR DOUBLE DIFFERENCING: METHODS OF DEMEANING AND DETRENDING AND THE ROLE OF INITIAL VALUES

Published online by Cambridge University Press:  05 March 2004

Paulo M.M. Rodrigues  and
A.M. Robert Taylor
Paulo M.M. Rodrigues
Affiliation:
University of Algarve
A.M. Robert Taylor
Affiliation:
University of Birmingham
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we investigate the impact that starting values have upon the double differencing tests of Hasza and Fuller (1979, Annals of Statistics 7, 1106–1120) and Sen and Dickey (1987, Journal of Business and Economic Statistics 5, 463–473), based on ordinary least squares (OLS) and simple symmetric least squares (SSLS) estimation, respectively. We demonstrate that, contrary to what is observed for conventional unit root tests, when based on data that have been demeaned, either directly or by the inclusion of a constant in the test regression, these tests are not exact similar to the starting values of the process, except where they are fixed and equal. We show that such test statistics can also fail to be asymptotically pivotal, contrary to claims made previously in the literature. We demonstrate that OLS tests based on data that have been detrended, either directly or by the inclusion of a constant and linear time trend in the test regression, do provide exact similar inference. In the context of the SSLS-based approach, we demonstrate that one obtains exact similar tests only where direct detrending is used. We highlight another error that appears in the literature, demonstrating that the two demeaned OLS-based test statistics do not coincide, even asymptotically, and that the same holds for the OLS- and SSLS-based test statistics for detrended data. We use Monte Carlo methods to quantify the finite-sample dependence of these tests on the starting values. These results suggest that the SSLS-based tests are considerably more sensitive to the nature of the starting values than are the OLS-based tests.We are grateful to Pentti Saikkonen and two anonymous referees for their helpful and constructive comments on an earlier version of this paper. We are especially grateful to one of the referees for the suggestion implemented in Remark 3.1 of the paper.

Type
Research Article
Information
Econometric Theory , Volume 20 , Issue 1 , February 2004 , pp. 95 - 115
Copyright
© 2004 Cambridge University Press

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

REFERENCES

Billingsley, P. (1968) Convergence of Probability Measures. Wiley.
Canjels, E. & M. Watson (1997) Estimating deterministic trends in presence of serially correlated errors. Review of Economics and Statistics 79, 184–200.Google Scholar
Dickey, D.A. & W.A. Fuller (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Dickey, D.A. & W.A. Fuller (1981) Likelihood ratio statistics for autoregressive time series with a unit root. Econometrica 49, 1057–1072.Google Scholar
Fuller, W.A. (1996) Introduction to Statistical Time Series, 2nd ed. Wiley.
Haldrup, N. (1994) Semi-parametric tests for double unit roots. Journal of Business and Economic Statistics 12, 109–122.Google Scholar
Haldrup, N. (1998) A review of the econometric analysis of I(2) variables. Journal of Economic Surveys 12, 595–650.Google Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.
Hasza, D.P. & W.A. Fuller (1979) Estimation of autoregressive processes with unit roots. The Annals of Statistics 7, 1106–1120.Google Scholar
Phillips, P.C.B. (1987) Time series regression with a unit root. Econometrica 55, 277–301.Google Scholar
Phillips, P.C.B. & C.C. Lee (1996) Efficiency gains from quasi-differencing under nonstationarity. In P.M. Robinson and M. Rosenblatt (eds.), Athens Conference, Applied Probability and Time Series: Essays in Memory of E.J. Hannan. Springer-Verlag.
Phillips, P.C.B. & P. Perron (1988) Testing for a unit root in time series regression. Biometrika 75, 335–346.Google Scholar
Rodrigues, P.M.M. & A.M.R. Taylor (2002) On Tests for Double Differencing: Some Extensions and the Role of Initial Values, Discussion paper 02-05, Department of Economics, University of Birmingham.
Sen, D.L. & D.A. Dickey (1987) Symmetric test for second differencing in univariate time series. Journal of Business and Economic Statistics 5, 463–473.Google Scholar
Shin, D.W. & H.J. Kim (1999) Semiparametric tests for double unit roots based on symmetric estimators. Journal of Business and Economic Statistics 17, 67–73.Google Scholar