Unit root vector autoregression with volatility induced stationarity
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.
|Date of creation:||08 Jun 2012|
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- Frédérique Bec & Anders Rahbek & Neil Shephard, 2008.
"The ACR Model: A Multivariate Dynamic Mixture Autoregression,"
Oxford Bulletin of Economics and Statistics,
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"On Moment Conditions For Quasi-Maximum Likelihood Estimation Of Multivariate Arch Models,"
Cambridge University Press, vol. 29(03), pages 545-566, June.
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- Cavaliere, Giuseppe & Rahbek, Anders & Taylor, A.M. Robert, 2010.
"Cointegration Rank Testing Under Conditional Heteroskedasticity,"
Cambridge University Press, vol. 26(06), pages 1719-1760, December.
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- 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.
- Comte, F. & Lieberman, O., 2003. "Asymptotic theory for multivariate GARCH processes," Journal of Multivariate Analysis, Elsevier, vol. 84(1), pages 61-84, January.
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- Engle, Robert F. & Kroner, Kenneth F., 1995. "Multivariate Simultaneous Generalized ARCH," Econometric Theory, Cambridge University Press, vol. 11(01), pages 122-150, February.
- 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.
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