Hostname: page-component-cb9f654ff-mnl9s Total loading time: 0 Render date: 2025-08-29T09:04:33.434Z Has data issue: false hasContentIssue false
  • English
  • Français

UNIT ROOT TESTING IN PRACTICE: DEALING WITHUNCERTAINTY OVER THE TREND AND INITIALCONDITION

Published online by Cambridge University Press:  01 June 2009

David I. Harvey ,
Stephen J. Leybourne  and
A.M. Robert Taylor
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we focus on two major issues thatsurround testing for a unit root in practice,namely, (i) uncertainty as to whether or not alinear deterministic trend is present in the dataand (ii) uncertainty as to whether the initialcondition of the process is (asymptotically)negligible or not. In each case simple testingprocedures are proposed with the aim of maintaininggood power properties across such uncertainties. Forthe first issue, if the initial condition isnegligible, quasi-differenced (QD) detrended(demeaned) Dickey–Fuller-type unit root tests arenear asymptotically efficient when a deterministictrend is (is not) present in the data generatingprocess. Consequently, we compare a variety ofstrategies that aim to select the detrended variantwhen a trend is present, and the demeaned variantotherwise. Based on asymptotic and finite-sampleevidence, we recommend a simple union ofrejections-based decision rule whereby the unit rootnull hypothesis is rejected whenever either of thedetrended or demeaned unit root tests yields arejection. Our results show that this approachgenerally outperforms more sophisticated strategiesbased on auxiliary methods of trend detection. Forthe second issue, we again recommend a union ofrejections decision rule, rejecting the unit rootnull if either of the QD or ordinary least squares(OLS) detrended/demeaned Dickey–Fuller-type testsrejects. This procedure is also shown to performwell in practice, simultaneously exploiting thesuperior power of the QD (OLS) detrended/demeanedtest for small (large) initial conditions.

Information

Type
Article Commentary
Information
Econometric Theory , Volume 25 , Issue 3 , June 2009 , pp. 587 - 636
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.)

Article purchase

Temporarily unavailable

Footnotes

We are extremely grateful to Peter Phillips andfive anonymous referees for their helpful andencouraging comments on the scope and content ofearlier drafts of this paper. These have enabledus to make significant improvements to thepaper.

References

REFERENCES

Ayat, L. & Burridge, P. (2000) Unit root tests in the presence of uncertainty about the non-stochastic trend. Journal of Econometrics 95, 71–96.10.1016/S0304-4076(99)00030-5CrossRefGoogle Scholar
Bhargave, A. (1986) On the theory of testing for unit roots in observed time series. Review of Economic Studies 53, 369–384.10.2307/2297634CrossRefGoogle Scholar
Bunzel, H. & Vogelsang, T.J. (2005) Powerful trend function tests that are robust to strong serial correlation with an application to the Prebisch-Singer hypothesis. Journal of Business & Economic Statistics 23, 381–394.10.1198/073500104000000631CrossRefGoogle Scholar
Chang, Y. & Park, Y.J. (2002) On the asymptotics of ADF tests for unit roots. Econometric Reviews 21, 431–447.10.1081/ETC-120015385CrossRefGoogle Scholar
Dickey, D.A. & Fuller, W.A. (1979) Distribution of the estimators for autoregressive time series with a unit root. Journal of the American Statistical Association 74, 427–431.Google Scholar
Elliott, G. (1999) Efficient tests for a unit root when the initial observation is drawn from its unconditional distribution. International Economic Review 40, 767–783.CrossRefGoogle Scholar
Elliott, G. & Müller, U.K. (2006) Minimizing the impact of the initial condition on testing for unit roots. Journal of Econometrics 135, 285–310.10.1016/j.jeconom.2005.07.024CrossRefGoogle Scholar
Elliott, G., Rothenberg, T.J., & Stock, J.H. (1996) Efficient tests for an autoregressive unit root. Econometrica 64, 813–836.10.2307/2171846CrossRefGoogle Scholar
Hamilton, J.D. (1994) Time Series Analysis. Princeton University Press.CrossRefGoogle Scholar
Harris, D., Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2007) Testing for a Unit Root in the Presence of a Possible Break in Trend. Granger Centre Discussion paper 07/04, Granger Centre for Time Series Econometrics, University of Nottingham. Downloadable from http://www.nottingham.ac.uk/economics/grangercentre/publications.htm.Google Scholar
Harvey, D.I. & Leybourne, S.J. (2005) On testing for unit roots and the initial observation. Econometrics Journal 8, 97–111.10.1111/j.1368-423X.2005.00154.xCrossRefGoogle Scholar
Harvey, D.I. & Leybourne, S.J. (2006) Unit root test power and the initial condition. Journal of Time Series Analysis 27, 739–752.10.1111/j.1467-9892.2006.00486.xCrossRefGoogle Scholar
Harvey, D.I., Leybourne, S.J., & Taylor, A.M.R. (2007) A simple, robust and powerful test of the trend hypothesis. Journal of Econometrics 141, 1302–1330.10.1016/j.jeconom.2007.02.005CrossRefGoogle Scholar
Jansson, M. (2007) Semiparametric Power Envelopes for Tests of the Unit Root Hypothesis. School of Economics, University of California, Berkeley. Downloadable from http://www.econ.berkeley.edu/~mjansson/Papers/SemiparametricUnitRoot.pdf.10.2139/ssrn.1149960CrossRefGoogle 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, 159–178.10.1016/0304-4076(92)90104-YCrossRefGoogle Scholar
Marsh, P. (2007) The available information for invariant tests of a unit root. Econometric Theory 23, 686–710.10.1017/S0266466607070296CrossRefGoogle Scholar
Müller, U.K. & Elliott, G. (2003) Tests for unit roots and the initial condition. Econometrica 71, 1269–1286.10.1111/1468-0262.00447CrossRefGoogle Scholar
Newey, W.K. & West, K.D. (1994) Automatic lag selection in covariance matrix estimation. Review of Economic Studies 61, 631–653.CrossRefGoogle Scholar
Ng, S. & Perron, P. (2001) Lag length selection and the construction of unit root tests with good size and power. Econometrica 69, 1519–1554.10.1111/1468-0262.00256CrossRefGoogle Scholar
Park, J.Y. (1990) Testing for unit roots and cointegration by variable addition. In Fomby, T. & Rhodes, F. (eds.), Advances in Econometrics: Cointegration, Spurious Regression and Unit Roots, pp. 107–133. Jai Press.Google Scholar
Park, J.Y. & Choi, B. (1988) A New Approach to Testing for a Unit Root. CAE Working paper 88–23, Cornell University.Google Scholar
Perron, P. & Qu, Z. (2007) A simple modification to improve the finite sample properties of Ng and Perron's unit root tests. Economics Letters 94, 12–19.CrossRefGoogle Scholar
Phillips, P.C.B. (1987a) Time series regression with a unit root. Econometrica 55, 277–301.10.2307/1913237CrossRefGoogle Scholar
Phillips, P.C.B. (1987b) Towards a unified asymptotic theory for autoregression. Biometrika 74, 535–547.10.1093/biomet/74.3.535CrossRefGoogle Scholar
Phillips, P.C.B. (1991a) To criticize the critics: An objective Bayesian analysis of stochastic trends. Journal of Applied Econometrics 6, 333–364.10.1002/jae.3950060402CrossRefGoogle Scholar
Phillips, P.C.B. (1991b) Bayesian routes and unit roots: De rebus prioribus semper est disputandum. Journal of Applied Econometrics 6, 435–473.CrossRefGoogle Scholar
Phillips, P.C.B. (1998) New tools for understanding spurious regressions. Econometrica 66, 1299–1326.CrossRefGoogle Scholar
Phillips, P.C.B. & Perron, P. (1988) Testing for a unit root in time series regression. Biometrika 75, 335–346.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.CrossRefGoogle Scholar
Phillips, P.C.B. & Solo, V. (1992) Asymptotics for linear processes. Annals of Statistics 20, 971–1001.10.1214/aos/1176348666CrossRefGoogle Scholar
Phillips, P.C.B. & Xiao, Z. (1998) A primer on unit root testing. Journal of Economic Surveys 12, 423–470.10.1111/1467-6419.00064CrossRefGoogle Scholar
Said, S.E. & Dickey, D.A. (1984) Testing for unit roots in autoregressive-moving average models of unknown order. Biometrika 71, 599–608.10.1093/biomet/71.3.599CrossRefGoogle Scholar
Schmidt, P. & Phillips, P.C.B. (1992) LM tests for a unit root in the presence of deterministic trends. Oxford Bulletin of Economics and Statistics 54, 257–287.CrossRefGoogle Scholar
Vogelsang, T.J. (1998) Trend function hypothesis testing in the presence of serial correlation. Econometrica 66, 123–148.10.2307/2998543CrossRefGoogle Scholar
West, K.D. (1988) Asymptotic normality, when regressors have a unit root. Econometrica 56, 1397–1417.10.2307/1913104CrossRefGoogle Scholar
Xiao, Z. & Phillips, P.C.B. (1998) An ADF coefficient test for a unit root in ARMA models of unknown order with empirical applications to the US economy. Econometrics Journal 1, 27–43.CrossRefGoogle Scholar