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Unit Roots In White Noise

  • Onatski, Alexei
  • Uhlig, Harald

We show that the empirical distribution of the roots of the vector autoregression (VAR) of order p fitted to T observations of a general stationary or nonstationary process converges to the uniform distribution over the unit circle on the complex plane, when both T and p tend to infinity so that (ln T )/ p → 0 and p 3/ T → 0. In particular, even if the process is a white noise, nearly all roots of the estimated VAR will converge by absolute value to unity. For fixed p , we derive an asymptotic approximation to the expected empirical distribution of the estimated roots as T → ∞. The approximation is concentrated in a circular region in the complex plane for various data generating processes and sample sizes.

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Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 28 (2012)
Issue (Month): 03 (June)
Pages: 485-508

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Handle: RePEc:cup:etheor:v:28:y:2012:i:03:p:485-508_00
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  1. Clive W. J. Granger & Yongil Jeon, 2006. "Dynamics of Model Overfitting Measured in terms of Autoregressive Roots," Journal of Time Series Analysis, Wiley Blackwell, vol. 27(3), pages 347-365, 05.
  2. Ulrich Mueller & Mark W. Watson, 2006. "Testing Models of Low-Frequency Variability," NBER Working Papers 12671, National Bureau of Economic Research, Inc.
  3. Bent Nielsen & Heino Bohn Nielsen, 2008. "Properties of estimated characteristic roots," Economics Series Working Papers 2008-WO7, University of Oxford, Department of Economics.
  4. Lewis, Richard & Reinsel, Gregory C., 1985. "Prediction of multivariate time series by autoregressive model fitting," Journal of Multivariate Analysis, Elsevier, vol. 16(3), pages 393-411, June.
  5. Søren Johansen, 2003. "The asymptotic variance of the estimated roots in a cointegrated vector autoregressive model," Journal of Time Series Analysis, Wiley Blackwell, vol. 24(6), pages 663-678, November.
  6. Saikkonen, Pentti & Lütkepohl, HELMUT, 1996. "Infinite-Order Cointegrated Vector Autoregressive Processes," Econometric Theory, Cambridge University Press, vol. 12(05), pages 814-844, December.
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