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Unit Root and Cointegrating Limit Theory When Initialization Is in the Infinite Past




It is well known that unit root limit distributions are sensitive to initial conditions in the distant past. If the distant past initialization is extended to the infinite past, the initial condition dominates the limit theory producing a faster rate of convergence, a limiting Cauchy distribution for the least squares coefficient and a limit normal distribution for the t ratio. This amounts to the tail of the unit root process wagging the dog of the unit root limit theory. These simple results apply in the case of a univariate autoregression with no intercept. The limit theory for vector unit root regression and cointegrating regression is affected but is no longer dominated by infinite past initializations. The latter contribute to the limiting distribution of the least squares estimator and produce a singularity in the limit theory, but do not change the principal rate of convergence. Usual cointegrating regression theory and inference continues to hold in spite of the degeneracy in the limit theory and is therefore robust to initial conditions that extend to the infinite past.

Suggested Citation

  • Peter C.B. Phillips & Tassos Magdalinos, 2008. "Unit Root and Cointegrating Limit Theory When Initialization Is in the Infinite Past," Cowles Foundation Discussion Papers 1655, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:1655

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    References listed on IDEAS

    1. Ulrich K. M¸ller & Graham Elliott, 2003. "Tests for Unit Roots and the Initial Condition," Econometrica, Econometric Society, vol. 71(4), pages 1269-1286, July.
    2. Abadir,Karim M. & Magnus,Jan R., 2005. "Matrix Algebra," Cambridge Books, Cambridge University Press, number 9780521537469, May.
    3. Harvey, David I. & Leybourne, Stephen J. & Taylor, A.M. Robert, 2009. "Unit Root Testing In Practice: Dealing With Uncertainty Over The Trend And Initial Condition," Econometric Theory, Cambridge University Press, vol. 25(03), pages 587-636, June.
    4. repec:cup:cbooks:9780521822893 is not listed on IDEAS
    5. Elliott, Graham & Muller, Ulrich K., 2006. "Minimizing the impact of the initial condition on testing for unit roots," Journal of Econometrics, Elsevier, vol. 135(1-2), pages 285-310.
    6. Phillips, Peter C.B. & Magdalinos, Tassos, 2008. "Limit Theory For Explosively Cointegrated Systems," Econometric Theory, Cambridge University Press, vol. 24(04), pages 865-887, August.
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    Cited by:

    1. Yu, Jun, 2012. "Bias in the estimation of the mean reversion parameter in continuous time models," Journal of Econometrics, Elsevier, vol. 169(1), pages 114-122.
    2. Roberto Esposti & Giulia Listorti, 2013. "Agricultural price transmission across space and commodities during price bubbles," Agricultural Economics, International Association of Agricultural Economists, vol. 44(1), pages 125-139, January.
    3. Peter C.B. Phillips & Ye Chen, "undated". "Restricted Likelihood Ratio Tests in Predictive Regression," Cowles Foundation Discussion Papers 1968, Cowles Foundation for Research in Economics, Yale University.
    4. Peter C. B. Phillips & Jun Yu, 2011. "Dating the timeline of financial bubbles during the subprime crisis," Quantitative Economics, Econometric Society, vol. 2(3), pages 455-491, November.
    5. Han, Chirok & Phillips, Peter C. B. & Sul, Donggyu, 2011. "Uniform Asymptotic Normality In Stationary And Unit Root Autoregression," Econometric Theory, Cambridge University Press, vol. 27(06), pages 1117-1151, December.
    6. Wang, Xiaohu & Yu, Jun, 2016. "Double asymptotics for explosive continuous time models," Journal of Econometrics, Elsevier, vol. 193(1), pages 35-53.
    7. Gutierrez, Luciano, 2011. "Bootstrapping asset price bubbles," Economic Modelling, Elsevier, vol. 28(6), pages 2488-2493.
    8. Niklas Ahlgren & Mikael Juselius, 2012. "Tests for cointegration rank and the initial condition," Empirical Economics, Springer, vol. 42(3), pages 667-691, June.
    9. Ye Chen & Jun Yu, 2011. "Optimal Jackknife for Discrete Time and Continuous Time Unit Root Models," Working Papers 12-2011, Singapore Management University, School of Economics.
    10. Lee, Ji Hyung & Phillips, Peter C.B., 2016. "Asset pricing with financial bubble risk," Journal of Empirical Finance, Elsevier, vol. 38(PB), pages 590-622.
    11. Jhih-Gang Chen & Biing-Shen Kuo, 2013. "Gaussian inference in general AR(1) models based on difference," Journal of Time Series Analysis, Wiley Blackwell, vol. 34(4), pages 447-453, July.
    12. Jarociński, Marek & Marcet, Albert, 2010. "Autoregressions in small samples, priors about observables and initial conditions," Working Paper Series 1263, European Central Bank.
    13. Skrobotov, Anton & Turuntseva, Marina, 2017. "Testing the Hypothesis of a Unit Root for Independent Panels," Working Papers 021707, Russian Presidential Academy of National Economy and Public Administration.
    14. Marek Jarocinski & Albert Marcet, 2014. "Contrasting Bayesian and Frequentist Approaches to Autoregressions: the Role of the Initial Condition," Working Papers 776, Barcelona Graduate School of Economics.
    15. Han, Chirok & Phillips, Peter C. B. & Sul, Donggyu, 2014. "X-Differencing And Dynamic Panel Model Estimation," Econometric Theory, Cambridge University Press, vol. 30(01), pages 201-251, February.
    16. Phillips, Peter C.B., 2010. "Bootstrapping I(1) data," Journal of Econometrics, Elsevier, vol. 158(2), pages 280-284, October.
    17. Phillips, Peter C.B. & Lee, Ji Hyung, 2016. "Robust econometric inference with mixed integrated and mildly explosive regressors," Journal of Econometrics, Elsevier, vol. 192(2), pages 433-450.
    18. Gutierrez, Luciano, 2011. "Looking for Rational Bubbles in Agricultural Commodity Markets," 2011 International Congress, August 30-September 2, 2011, Zurich, Switzerland 120377, European Association of Agricultural Economists.
    19. Peter C. B. Phillips, 2017. "Detecting Financial Collapse and Ballooning Sovereign Risk," Cowles Foundation Discussion Papers 3010, Cowles Foundation for Research in Economics, Yale University.
    20. Listorti, Giulia & Esposti, Roberto, 2012. "Horizontal Price Transmission in Agricultural Markets: Fundamental Concepts and Open Empirical Issues," Bio-based and Applied Economics Journal, Italian Association of Agricultural and Applied Economics (AIEAA), issue 1, April.
    21. Peter C.B. Phillips & Shu-Ping Shi, 2014. "Financial Bubble Implosion," Cowles Foundation Discussion Papers 1967, Cowles Foundation for Research in Economics, Yale University.
    22. repec:eee:econom:v:201:y:2017:i:2:p:400-416 is not listed on IDEAS
    23. Chen, Ye & Yu, Jun, 2015. "Optimal jackknife for unit root models," Statistics & Probability Letters, Elsevier, vol. 99(C), pages 135-142.

    More about this item


    Cauchy limit distribution; Cointegration; Distant past initialization; Infinite past initialization; Random orthonormalization; Singular limit theory;

    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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