IDEAS home Printed from https://ideas.repec.org/p/boc/bocoec/381.html
   My bibliography  Save this paper

Threshold Autoregressions with a Unit Root

Author

Listed:
  • Bruce E. Hansen

    () (Boston College)

  • Mehmet Caner

    (Koc University)

Abstract

This paper develops an asymptotic theory of inference for a two-regime threshold autoregressive (TAR) model with an autoregressive root which is local-to-unity. We find that the asymptotic null distribution of the Wald test for a threshold is non-standard and mildly dependent on the local-to-unity coefficient. We also study the asymptotic null distribution of the Wald test for an autoregressive unit root, and find that it is non-standard and dependent on the presence of a threshold effect. These tests and distribution theory allow for the joint consideration of non-linearity (thresholds) and non-stationarity (unit roots). Our limit theory is based on a new set of tools which combines unit root asymptotics with empirical process methods. We work with a particular two-parameter empirical processes which converges weakly to a two-parameter Brownian motion. Our limit distributions involve stochastic integrals with respect to this two-parameter process. This theory is entirely new and may find applications in other contexts. We illustrate the methods with an application to the U.S. monthly unemployment rate. We find strong evidence of a threshold effect. The point estimates suggest that in about 80% of the observations, the regression function is close to a driftless I(1) process, and in the other 20% of the observations, the regression function is mean-reverting with an unconditional mean of 5%. While the conventional ADF test for a unit root is quite insignificant, our TAR unit root test is arguably significant, with an asymptotic p-value of 3.5%, suggesting that the unemployment rate follows a stationary TAR process.

Suggested Citation

  • Bruce E. Hansen & Mehmet Caner, 1997. "Threshold Autoregressions with a Unit Root," Boston College Working Papers in Economics 381, Boston College Department of Economics.
  • Handle: RePEc:boc:bocoec:381
    as

    Download full text from publisher

    File URL: http://fmwww.bc.edu/EC-P/wp381.pdf
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    as
    1. Balke, Nathan S & Fomby, Thomas B, 1997. "Threshold Cointegration," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 38(3), pages 627-645, August.
    2. Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, vol. 64(2), pages 413-430, March.
    3. Hansen, Bruce E, 1997. "Approximate Asymptotic P Values for Structural-Change Tests," Journal of Business & Economic Statistics, American Statistical Association, vol. 15(1), pages 60-67, January.
    4. Hansen, Bruce E., 2000. "Testing for structural change in conditional models," Journal of Econometrics, Elsevier, vol. 97(1), pages 93-115, July.
    5. Bai, Jushan, 1996. "Testing for Parameter Constancy in Linear Regressions: An Empirical Distribution Function Approach," Econometrica, Econometric Society, vol. 64(3), pages 597-622, May.
    6. Beaudry, Paul & Koop, Gary, 1993. "Do recessions permanently change output?," Journal of Monetary Economics, Elsevier, vol. 31(2), pages 149-163, April.
    7. Potter, Simon M, 1995. "A Nonlinear Approach to US GNP," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 10(2), pages 109-125, April-Jun.
    8. Galbraith, John W, 1996. "Credit Rationing and Threshold Effects in the Relation between Money and Output," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 11(4), pages 419-429, July-Aug..
    9. Andrews, Donald W K & Ploberger, Werner, 1994. "Optimal Tests When a Nuisance Parameter Is Present Only under the Alternative," Econometrica, Econometric Society, vol. 62(6), pages 1383-1414, November.
    10. Neftci, Salih N, 1984. "Are Economic Time Series Asymmetric over the Business Cycle?," Journal of Political Economy, University of Chicago Press, vol. 92(2), pages 307-328, April.
    11. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    12. Hamilton, James D, 1989. "A New Approach to the Economic Analysis of Nonstationary Time Series and the Business Cycle," Econometrica, Econometric Society, vol. 57(2), pages 357-384, March.
    13. Pippenger, Michael K & Goering, Gregory E, 1993. "A Note on the Empirical Power of Unit Root Tests under Threshold Processes," Oxford Bulletin of Economics and Statistics, Department of Economics, University of Oxford, vol. 55(4), pages 473-481, November.
    14. Robert B. Davies, 2002. "Hypothesis testing when a nuisance parameter is present only under the alternative: Linear model case," Biometrika, Biometrika Trust, vol. 89(2), pages 484-489, June.
    Full references (including those not matched with items on IDEAS)

    More about this item

    Keywords

    threshold autoregression; unit root; empirical process methods;

    JEL classification:

    • C12 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Hypothesis Testing: General
    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:boc:bocoec:381. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Christopher F Baum) or (Christopher F. Baum). General contact details of provider: http://edirc.repec.org/data/debocus.html .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.