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Testing against smooth stochastic trends


  • Jukka Nyblom

    (Department of Statistics, University of Joensuu, PO Box 111, FIN-80101 Joensuu, Finland)

  • Andrew Harvey

    (Faculty of Economics and Politics, University of Cambridge, Sidgwick Avenue, Cambridge, CB3 9DD, UK)


A trend estimated from an unobserved components model tends to be smoother when it is modelled as an integrated random walk rather than a random walk with drift. This article derives a test of the null hypothesis that the trend is deterministic against the alternative that it is an integrated random walk. It is assumed that the other component in the model is normally distributed white noise. Critical values are tabulated, the asymptotic distribution is derived and the performance of the test is compared with the test against a trend specified as a random walk with drift. The test is extended to allow for serially correlated and evolving seasonal components. When there is a stationary process containing a single autoregressive unit root close to one, a bounds test can be applied. In the case of a first-order autoregressive disturbance, it is shown that a consistent test can still be obtained by carrying out estimation of the nuisance parameters under the null hypothesis. The overall conclusion is that the most effective test against an integrated random walk is a parametric one based on the random walk plus drift test statistic, constructed from innovations, with the nuisance parameters estimated in the unrestricted model. Copyright © 2001 John Wiley & Sons, Ltd.

Suggested Citation

  • Jukka Nyblom & Andrew Harvey, 2001. "Testing against smooth stochastic trends," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(3), pages 415-429.
  • Handle: RePEc:jae:japmet:v:16:y:2001:i:3:p:415-429

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    References listed on IDEAS

    1. Nyblom, Jukka & Harvey, Andrew, 2000. "Tests Of Common Stochastic Trends," Econometric Theory, Cambridge University Press, vol. 16(02), pages 176-199, April.
    2. Gersch, Will & Kitagawa, Genshiro, 1983. "The Prediction of Time Series with Trends and Seasonalities," Journal of Business & Economic Statistics, American Statistical Association, vol. 1(3), pages 253-264, July.
    3. Kwiatkowski, Denis & Phillips, Peter C. B. & Schmidt, Peter & Shin, Yongcheol, 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, Elsevier, vol. 54(1-3), pages 159-178.
    4. Hodrick, Robert J & Prescott, Edward C, 1997. "Postwar U.S. Business Cycles: An Empirical Investigation," Journal of Money, Credit and Banking, Blackwell Publishing, vol. 29(1), pages 1-16, February.
    5. Nyblom, Jukka, 2001. "Invariant Tests for Covariance Structures in Multivariate Linear Model," Journal of Multivariate Analysis, Elsevier, vol. 76(2), pages 294-315, February.
    6. Sargan, J D & Bhargava, Alok, 1983. "Maximum Likelihood Estimation of Regression Models with First Order Moving Average Errors When the Root Lies on the Unit Circle," Econometrica, Econometric Society, vol. 51(3), pages 799-820, May.
    7. Leybourne, S J & McCabe, B P M, 1994. "A Consistent Test for a Unit Root," Journal of Business & Economic Statistics, American Statistical Association, vol. 12(2), pages 157-166, April.
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    Cited by:

    1. Guilhem Bentoglio & Jacky Fayolle & Matthieu Lemoine, 2002. "Unity and Plurality of the European Cycle," Sciences Po publications N°2002-3, Sciences Po.
    2. repec:spo:wpecon:info:hdl:2441/2133 is not listed on IDEAS
    3. Fabio Busetti & Andrew Harvey, 2011. "When is a Copula Constant? A Test for Changing Relationships," Journal of Financial Econometrics, Society for Financial Econometrics, vol. 9(1), pages 106-131, Winter.
    4. DeRossi, G. & Harvey, A., 2006. "Time-Varying Quantiles," Cambridge Working Papers in Economics 0649, Faculty of Economics, University of Cambridge.
    5. Christopher Klein & Shea Slonaker, 2010. "Chart Turnover and Sales in the Recorded Music Industry: 1990–2005," Review of Industrial Organization, Springer;The Industrial Organization Society, vol. 36(4), pages 351-372, June.
    6. Busettti, F. & Harvey, A., 2007. "Tests of time-invariance," Cambridge Working Papers in Economics 0701, Faculty of Economics, University of Cambridge.
    7. Ferrara, L. & Koopman, S J., 2010. "Common business and housing market cycles in the Euro area from a multivariate decomposition," Working papers 275, Banque de France.
    8. Xiaoshan Chen & Terence Mills, 2012. "Measuring the Euro area output gap using a multivariate unobserved components model containing phase shifts," Empirical Economics, Springer, vol. 43(2), pages 671-692, October.
    9. Xiaoshan Chen & Ronald MacDonald, 2010. "Revisiting the Dollar-Euro Permanent Equilibrium Exchange Rate: Evidence from Multivariate Unobserved Components Models," Working Papers 2010_16, Business School - Economics, University of Glasgow.
    10. Siem Jan Koopman & Marius Ooms & André Lucas & Kees van Montfort & Victor van der Geest, 2008. "Estimating systematic continuous-time trends in recidivism using a non-Gaussian panel data model," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 62(1), pages 104-130.
    11. Fabio Busetti & Andrew Harvey, 2010. "Tests of strict stationarity based on quantile indicators," Journal of Time Series Analysis, Wiley Blackwell, vol. 31(6), pages 435-450, November.
    12. Vujić Sunčica & Koopman Siem Jan & Commandeur J.F., 2012. "Economic Trends and Cycles in Crime: A Study for England and Wales," Journal of Economics and Statistics (Jahrbuecher fuer Nationaloekonomie und Statistik), De Gruyter, vol. 232(6), pages 652-677, December.
    13. Charles S. Bos & Siem Jan Koopman & Marius Ooms, 2007. "Long memory modelling of inflation with stochastic variance and structural breaks," CREATES Research Papers 2007-44, Department of Economics and Business Economics, Aarhus University.
    14. Xiaoyi Mu and Haichun Ye, 2015. "Small Trends and Big Cycles in Crude Oil Prices," The Energy Journal, International Association for Energy Economics, vol. 0(Number 1).

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