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Threshold Autoregressions with a Unit Root

  • Bruce E. Hansen


    (Boston College)

  • Mehmet Caner

    (Koc University)

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.

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Paper provided by Boston College Department of Economics in its series Boston College Working Papers in Economics with number 381.

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Length: 34 pages
Date of creation: 01 Aug 1997
Date of revision:
Handle: RePEc:boc:bocoec:381
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  1. 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 473-81, November.
  2. Hansen, Bruce E, 1996. "Inference When a Nuisance Parameter Is Not Identified under the Null Hypothesis," Econometrica, Econometric Society, vol. 64(2), pages 413-30, March.
  3. Potter, Simon M, 1995. "A Nonlinear Approach to US GNP," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 10(2), pages 109-25, April-Jun.
  4. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
  5. 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 307-28, April.
  6. Beaudry, Paul & Koop, Gary, 1993. "Do recessions permanently change output?," Journal of Monetary Economics, Elsevier, vol. 31(2), pages 149-163, April.
  7. 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.
  8. Donald W.K. Andrews & Werner Ploberger, 1992. "Optimal Tests When a Nuisance Parameter Is Present Only Under the Alternative," Cowles Foundation Discussion Papers 1015, Cowles Foundation for Research in Economics, Yale University.
  9. 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.
  10. 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 357-84, March.
  11. Nathan S. Balke & Thomas B. Fomby, 1992. "Threshold cointegration," Research Paper 9209, Federal Reserve Bank of Dallas.
    • 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 627-45, August.
  12. 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.
  13. 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 419-29, July-Aug..
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