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Unit root vector autoregression with volatility induced stationarity

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  • Anders Rahbek

    (Department of Economics)

  • Heino Bohn Nielsen

    (Department of Economics)

Abstract

We propose a discrete-time multivariate model where lagged levels of the process enter both the conditional mean and the conditional variance. This way we allow for the empirically observed persistence in time series such as interest rates, often implying unit-roots, while at the same time maintain stationarity despite such unit-roots. Specifically, the model bridges vector autoregressions and multivariate ARCH models in which residuals are replaced by levels lagged. An empirical illustration using recent US term structure data is given in which the individual interest rates have unit roots, have no finite first-order moments, but remain strictly stationary and ergodic, while they co-move in the sense that their spread has no unit root. The model thus allows for volatility induced stationarity, and the paper shows conditions under which the multivariate process is strictly stationary and geometrically ergodic. Interestingly, these conditions include the case of unit roots and a reduced rank structure in the conditional mean, known from linear co-integration to imply non-stationarity. Asymptotic theory of the maximum likelihood estimators for a particular structured case (so-called self-exciting) is provided, and it is shown that vT-convergence to Gaussian distributions apply despite unit roots as well as absence of finite first and higher order moments. Monte Carlo simulations confirm the usefulness of the asymptotics in finite samples.

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Bibliographic Info

Paper provided by University of Copenhagen. Department of Economics in its series Discussion Papers with number 12-02.

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Length: 36 pages
Date of creation: 08 Jun 2012
Date of revision:
Handle: RePEc:kud:kuiedp:1202

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Keywords: Vector Autoregression; Unit-Root; Reduced Rank; Volatility Induced Stationarity; Term Structure; Double Autoregression;

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References

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  1. BAUWENS, Luc & LAURENT, Sébastien & ROMBOUTS, Jeroen VK, . "Multivariate GARCH models: a survey," CORE Discussion Papers RP -1847, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  2. Jensen, S ren Tolver & Rahbek, Anders, 2007. "On The Law Of Large Numbers For (Geometrically) Ergodic Markov Chains," Econometric Theory, Cambridge University Press, vol. 23(04), pages 761-766, August.
  3. Comte, F. & Lieberman, O., 2003. "Asymptotic theory for multivariate GARCH processes," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 61-84, January.
  4. Giuseppe Cavaliere & Anders Rahbek & Taylor A.M.Robert, 2011. "Bootstrap determination of the co-integration rank in VAR models," Quaderni di Dipartimento 9, Department of Statistics, University of Bologna.
  5. Frédérique Bec & Anders Rahbek, 2004. "Vector equilibrium correction models with non-linear discontinuous adjustments," Econometrics Journal, Royal Economic Society, vol. 7(2), pages 628-651, December.
  6. Jensen, S ren Tolver & Rahbek, Anders, 2004. "Asymptotic Inference For Nonstationary Garch," Econometric Theory, Cambridge University Press, vol. 20(06), pages 1203-1226, December.
  7. Jeantheau, Thierry, 1998. "Strong Consistency Of Estimators For Multivariate Arch Models," Econometric Theory, Cambridge University Press, vol. 14(01), pages 70-86, February.
  8. Giuseppe Cavaliere & Anders Rahbek & A.M.Robert Taylor, 2009. "Co-integration Rank Testing under Conditional Heteroskedasticity," CREATES Research Papers 2009-22, School of Economics and Management, University of Aarhus.
  9. Engle, Robert F. & Kroner, Kenneth F., 1995. "Multivariate Simultaneous Generalized ARCH," Econometric Theory, Cambridge University Press, vol. 11(01), pages 122-150, February.
  10. Christian Gourieroux & Alain Monfort & Vassilis Polimenis, 2002. "Affine Term Structure Models," Working Papers 2002-49, Centre de Recherche en Economie et Statistique.
  11. Marco Avarucci & Eric Beutner & Paolo Zaffaroni, 2012. "On moment conditions for quasi-maximum likelihood estimation of multivariate ARCH models," DSS Empirical Economics and Econometrics Working Papers Series 2012/1, Centre for Empirical Economics and Econometrics, Department of Statistics, "Sapienza" University of Rome.
  12. Frédérique Bec & Anders Rahbek & Neil Shephard, 2008. "The ACR model: a multivariate dynamic mixture autoregression," THEMA Working Papers 2008-11, THEMA (THéorie Economique, Modélisation et Applications), Université de Cergy-Pontoise.
  13. Theis Lange & Anders Rahbek & S�ren Tolver Jensen, 2011. "Estimation and Asymptotic Inference in the AR-ARCH Model," Econometric Reviews, Taylor & Francis Journals, vol. 30(2), pages 129-153.
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