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Limit Theory for Explosively Cointegrated Systems

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Abstract

A limit theory is developed for multivariate regression in an explosive cointegrated system. The asymptotic behavior of the least squares estimator of the cointegrating coefficients is found to depend upon the precise relationship between the explosive regressors. When the eigenvalues of the autoregressive matrix are distinct, the centered least squares estimator has an exponential rate of convergence and a mixed normal limit distribution. No central limit theory is applicable here and Gaussian innovations are assumed. On the other hand, when some regressors exhibit common explosive behavior, a different mixed normal limiting distribution is derived with rate of convergence reduced to n^0.5. In the latter case, mixed normality applies without any distributional assumptions on the innovation errors by virtue of a Lindeberg type central limit theorem. Conventional statistical inference procedures are valid in this case, the stationary convergence rate dominating the behavior of the least squares estimator.

Suggested Citation

  • Peter C.B. Phillips & Tassos Magdalinos, 2007. "Limit Theory for Explosively Cointegrated Systems," Cowles Foundation Discussion Papers 1614, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1614
    Note: CFP 1244
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    File URL: http://cowles.yale.edu/sites/default/files/files/pub/d16/d1614.pdf
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    Cited by:

    1. Li, Jian & Chavas, Jean-Paul & Etienne, Xiaoli & Li, Chongguang, 2016. "Commodity Price Bubbles and Macroeconomics: Evidence from Chinese Agricultural Markets," 2016 Annual Meeting, July 31-August 2, 2016, Boston, Massachusetts 235068, Agricultural and Applied Economics Association.
    2. Wang, Xiaohu & Yu, Jun, 2015. "Limit theory for an explosive autoregressive process," Economics Letters, Elsevier, vol. 126(C), pages 176-180.
    3. Peter C. B. Phillips & Sainan Jin, 2014. "Testing the Martingale Hypothesis," Journal of Business & Economic Statistics, Taylor & Francis Journals, vol. 32(4), pages 537-554, October.
    4. B. Nielsen, 2009. "Test for cointegration rank in general vector autoregressions," Economics Papers 2009-W10, Economics Group, Nuffield College, University of Oxford.
    5. White Halbert & Granger Clive W.J., 2011. "Consideration of Trends in Time Series," Journal of Time Series Econometrics, De Gruyter, vol. 3(1), pages 1-40, February.
    6. Phillips, Peter C.B. & Li, Degui & Gao, Jiti, 2017. "Estimating smooth structural change in cointegration models," Journal of Econometrics, Elsevier, vol. 196(1), pages 180-195.
    7. Phillips, Peter C.B. & Magdalinos, Tassos, 2009. "Unit Root And Cointegrating Limit Theory When Initialization Is In The Infinite Past," Econometric Theory, Cambridge University Press, vol. 25(06), pages 1682-1715, December.
    8. Bent Nielsen, 2008. "Singular vector autoregressions with deterministic terms: Strong consistency and lag order determination," Economics Series Working Papers 2008-W14, University of Oxford, Department of Economics.
    9. Norbert Christopeit & Michael Massmann, 2013. "Estimating Structural Parameters in Regression Models with Adaptive Learning," Tinbergen Institute Discussion Papers 13-111/III, Tinbergen Institute.
    10. repec:kap:compec:v:51:y:2018:i:3:d:10.1007_s10614-016-9628-6 is not listed on IDEAS
    11. HORIE, Tetsushi & YAMAMOTO, Yohei, 2016. "Testing for Speculative Bubbles in Large-Dimensional Financial Panel Data Sets," Discussion Papers 2016-04, Graduate School of Economics, Hitotsubashi University.
    12. Phillips, Peter C.B. & Magdalinos, Tassos, 2013. "Inconsistent Var Regression With Common Explosive Roots," Econometric Theory, Cambridge University Press, vol. 29(04), pages 808-837, August.
    13. Peter C.B. Phillips & Ji Hyung Lee, 2012. "VARs with Mixed Roots Near Unity," Cowles Foundation Discussion Papers 1845, Cowles Foundation for Research in Economics, Yale University.

    More about this item

    Keywords

    Central limit theory; Exposive cointegration; Explosive process; Mixed normality;

    JEL classification:

    • C22 - Mathematical and Quantitative Methods - - Single Equation Models; Single Variables - - - Time-Series Models; Dynamic Quantile Regressions; Dynamic Treatment Effect Models; Diffusion Processes

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