Hostname: page-component-77c89778f8-9q27g Total loading time: 0 Render date: 2024-07-24T22:03:52.956Z Has data issue: false hasContentIssue false

TESTING FOR A UNIT ROOT IN THE PRESENCE OF A POSSIBLE BREAK IN TREND

Published online by Cambridge University Press:  01 December 2009

David Harris
Affiliation:
University of Melbourne
David I. Harvey
Affiliation:
University of Nottingham
Stephen J. Leybourne
Affiliation:
University of Nottingham
A.M. Robert Taylor*
Affiliation:
University of Nottingham
*
*Address correspondence to Robert Taylor, School of Economics, University of Nottingham, Nottingham NG7 2RD, UK; e-mail: robert.taylor@nottingham.ac.uk.

Abstract

We consider the issue of testing a time series for a unit root in the possible presence of a break in a linear deterministic trend at an unknown point in the series. We propose a new break fraction estimator which, where a break in trend occurs, is consistent for the true break fraction at rate Op(T−1). Unlike other available estimators, however, when there is no trend break, our estimator converges to zero at rate Op(T−1/2). Used in conjunction with a quasi difference (QD) detrended unit root test that incorporates a trend break regressor, we show that these rates of convergence ensure that known break fraction null critical values are asymptotically valid. Unlike available procedures in the literature, this holds even if there is no break in trend (the break fraction is zero). Here the trend break regressor is dropped from the deterministic component, and standard QD detrended unit root test critical values then apply. We also propose a second procedure that makes use of a formal pretest for a trend break in the series, including a trend break regressor only where the pretest rejects the null of no break. Both procedures ensure that the correctly sized (near-) efficient unit root test that allows (does not allow) for a break in trend is applied in the limit when a trend break does (does not) occur.

Type
ARTICLES
Copyright
Copyright © Cambridge University Press 2009

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

Andrews, D.W.K. (1993) Tests for parameter instability and structural change with unknown change point. Econometrica 61, 821–856.Google Scholar
Bai, J. (1994) Least squares estimation of a shift in a linear process. Journal of Time Series Analysis 15, 453–472.CrossRefGoogle Scholar
Bai, J. (1997) Estimation of a change point in multiple regression models. Review of Economics and Statistics 79, 551–563.CrossRefGoogle Scholar
Banerjee, A., Lumsdaine, R., & Stock, J. (1992) Recursive and sequential tests of the unit root and trend break hypotheses: Theory and international evidence. Journal of Business and Economics Statistics 10, 271–288.Google Scholar
Berk, K.N. (1974) Consistent autoregressive spectral estimates. Annals of Statistics 2, 489–502.Google Scholar
Beveridge, S. & Nelson, C.R. (1981) A new approach to the decomposition of economic time series into permanent and transitory components with particular attention to measurement of the business cycle. Journal of Monetary Economics 7, 151–174.CrossRefGoogle Scholar
Carrion-i-Silvestre, J.L., Kim, D., & Perron, P. (2009) GLS-based unit root tests with multiple structural breaks under both the null and the alternative hypotheses. Econometric Theory 25, 1754–1792 (this issue).CrossRefGoogle Scholar
Cavaliere, G. & Georgiev, I. (2007) Testing for unit roots in autoregressions with multiple level shifts. Econometric Theory 23, 1162–1215.CrossRefGoogle Scholar
Cavaliere, G. & Taylor, A.M.R. (2007) Testing for unit roots in time series models with non-stationary volatility. Journal of Econometrics 140, 919–947.Google Scholar
Chang, Y. & Park, Y.J. (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.Google Scholar
Cheng, X. & Phillips, P.C.B. (2008) Cointegration Rank Selection in Models with Time-Varying Variance. Manuscript, Department of Economics, Yale University.Google Scholar
Christiano, L.J. (1992) Searching for a break in GNP. Journal of Business and Economic Statistics 10, 237–50.Google Scholar
Elliott, G., Rothenberg, T.J. & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.Google Scholar
Hajek, J. & Renyi, A. (1955) A generalization of an inequality of Kolomogorov. Acta Mathematica Academic Scientarixm Hungaricae 6, 281–284.Google Scholar
Hansen, B.E. (2009) Averaging estimators for regressions with a possible structural break. Econometric Theory 25, 1498–1514 (this issue).CrossRefGoogle Scholar
Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2009a) Simple, robust, and powerful tests of the breaking trend hypothesis. Econometric Theory 25, 995–1029.Google Scholar
Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2009b) Unit root testing in practice: Dealing with uncertainty over the trend and initial condition. Econometric Theory 25, 587–667.Google Scholar
Kim, D. & Perron, P. (2009) Unit root tests allowing for a break in the trend function at an unknown time under both the null and alternative hypotheses. Journal of Econometrics 148, 1–13.CrossRefGoogle Scholar
Kim, J.-Y. (1998) Large sample properties of posterior densities, Bayesian information criterion and the likelihood principle in nonstationary time series models. Econometrica 66, 359–380.Google Scholar
Kwiatkowski, D., Phillips, P.C.B., Schmidt, P., & Shin, Y. (1992) 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, 159–178.CrossRefGoogle Scholar
Marsh, P. (2009) The properties of Kullback–Labler divergence for the unit root hypothesis. Econometric Theory 25, 1662–1681 (this issue).Google Scholar
Ng, S. & Perron, P. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.CrossRefGoogle Scholar
Nunes, L.C., Kuan, C.-M., & Newbold, P. (1995) Spurious break. Econometric Theory 11, 736–749.Google Scholar
Perron, P. (1989) The great crash, the oil price shock, and the unit root hypothesis. Econometrica 57, 1361–1401.Google Scholar
Perron, P. (1997) Further evidence of breaking trend functions in macroeconomic variables. Journal of Econometrics 80, 355–385.CrossRefGoogle Scholar
Perron, P. & Rodríguez, G. (2003) GLS detrending, efficient unit root tests and structural change. Journal of Econometrics 115, 1–27.Google Scholar
Perron, P. & Yabu, T. (2009) Testing for shifts in trend with an integrated or stationary noise component. Journal of Business and Economic Statistics, forthcoming.CrossRefGoogle Scholar
Perron, P. & Zhu, X. (2005) Structural breaks with deterministic and stochastic trends. Journal of Econometrics 129, 65–119.Google Scholar
Phillips, P.C.B. (1991a) Bayesian routes and unit roots: De rebus prioribus semper est disputandum. Journal of Applied Econometrics 6, 435–473.CrossRefGoogle Scholar
Phillips, P.C.B. (1991b) To criticize the critics: An objective Bayesian analysis of stochastic trends. Journal of Applied Econometrics 6, 333–364.Google Scholar
Phillips, P.C.B. (2008) Unit root model selection. Journal of the Japan Statistical Society 38, 65–74.CrossRefGoogle Scholar
Phillips, P.C.B. & Ploberger, W. (1994) Posterior odds testing for a unit root with data-based model selection. Econometric Theory 10, 774–808.Google Scholar
Rodríguez, G. (2007) Finite sample behavior of the level shift model using quasi-differenced data. Journal of Statistical Computation and Simulation 77, 889–905.Google Scholar
Sayginsoy, O. & Vogelsang, T.J. (2004) Powerful Tests of Structural Change that are Robust to Strong Serial Correlation. working paper 04/08, SUNY at Albany.Google Scholar
Stock, J.H. (1999) A class of tests for integration and cointegration. In Engle, R.F. & White, H. (eds.), Cointegration, Causality and Forecasting. A Festschrift in Honour of Clive W.J. Granger, pp. 137–167. Oxford University Press.Google Scholar
Stock, J.H. & Watson, M.W. (1996) Evidence on structural instability in macroeconomic time series relations. Journal of Business and Economic Statistics 14, 11–30.Google Scholar
Stock, J.H. & Watson, M.W. (1999) A comparison of linear and nonlinear univariate models for forecasting macroeconomic time series. In Engle, R.F. & White, H. (eds.), Cointegration, Causality and Forecasting: A Festschrift in Honour of Clive W.J. Granger, pp. 1–44. Oxford University Press.Google Scholar
Stock, J. & Watson, M.W. (2005) Implications of Dynamic Factor Analysis for VAR Models, NBER Working paper 11467.Google Scholar
Vogelsang, T.J. (1998) Trend function hypothesis testing in the presence of serial correlation. Econometrica 66, 123–148.Google Scholar
Vogelsang, T.J. & Perron, P. (1998) Additional tests for a unit root allowing the possibility of breaks in the trend function. International Economic Review 39, 1073–1100.CrossRefGoogle Scholar
Zivot, E. & Andrews, D.W.K. (1992) Further evidence on the great crash, the oil-price shock, and the unit-root hypothesis. Journal of Business and Economic Statistics 10, 251–270.Google Scholar