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Weak Convergence of Sample Covariance Matrices to Stochastic Integrals via Martingale Approximations

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Abstract

Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral from zero to one of BdB; where B is vector Brownian motion. For strictly stationary and ergodic sequences, rather than martingale differences, a similar result obtains. In this case, the limit is the same with a constant matrix, of bias terms whose magnitude depends on the serial correlation properties of the sequence. This note gives a simple proof of the result using martingale approximations.

Suggested Citation

  • Peter C.B. Phillips, 1987. "Weak Convergence of Sample Covariance Matrices to Stochastic Integrals via Martingale Approximations," Cowles Foundation Discussion Papers 846, Cowles Foundation for Research in Economics, Yale University.
  • Handle: RePEc:cwl:cwldpp:846
    Note: CFP 716.
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    1. Park, Joon Y. & Phillips, Peter C.B., 1989. "Statistical Inference in Regressions with Integrated Processes: Part 2," Econometric Theory, Cambridge University Press, vol. 5(01), pages 95-131, April.
    2. Phillips, P. C. B., 1987. "Asymptotic Expansions in Nonstationary Vector Autoregressions," Econometric Theory, Cambridge University Press, vol. 3(01), pages 45-68, February.
    3. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December.
    4. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    5. Park, Joon Y. & Phillips, Peter C.B., 1988. "Statistical Inference in Regressions with Integrated Processes: Part 1," Econometric Theory, Cambridge University Press, vol. 4(03), pages 468-497, December.
    6. Sims, Christopher A & Stock, James H & Watson, Mark W, 1990. "Inference in Linear Time Series Models with Some Unit Roots," Econometrica, Econometric Society, vol. 58(1), pages 113-144, January.
    7. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
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    Cited by:

    1. Minxian, Yang, 1998. "System estimators of cointegrating matrix in absence of normalising information," Journal of Econometrics, Elsevier, vol. 85(2), pages 317-337, August.
    2. Gregory, Allan W. & Hansen, Bruce E., 1996. "Residual-based tests for cointegration in models with regime shifts," Journal of Econometrics, Elsevier, vol. 70(1), pages 99-126, January.
    3. Jansson, Michael, 2005. "Point optimal tests of the null hypothesis of cointegration," Journal of Econometrics, Elsevier, vol. 124(1), pages 187-201, January.
    4. D. Marinucci & P. M. Robinson, 2001. "Finite sample improvements in statistical inference with I(1) processes," Journal of Applied Econometrics, John Wiley & Sons, Ltd., vol. 16(3), pages 431-444.
    5. Mueller, Ulrich, 2008. "An Alternative Sense of Asymptotic Efficiency," MPRA Paper 7741, University Library of Munich, Germany.
    6. Arai, Yoichi, 2016. "Testing For Linearity In Regressions With I(1) Processes," Hitotsubashi Journal of Economics, Hitotsubashi University, vol. 57(1), pages 111-138, June.
    7. Michael Jansson & Marcelo J. Moreira, 2006. "Optimal Inference in Regression Models with Nearly Integrated Regressors," Econometrica, Econometric Society, vol. 74(3), pages 681-714, May.
    8. Peter C. B. Phillips & Hyungsik R. Moon, 1999. "Linear Regression Limit Theory for Nonstationary Panel Data," Econometrica, Econometric Society, vol. 67(5), pages 1057-1112, September.
    9. Tanaka, Katsuto, 2011. "Linear Nonstationary Models : A Review of the Work of Professor P.C.B. Phillips," Discussion Papers 2011-05, Graduate School of Economics, Hitotsubashi University.
    10. Liang, Hanying & Phillips, Peter C.B. & Wang, Hanchao & Wang, Qiying, 2016. "Weak Convergence To Stochastic Integrals For Econometric Applications," Econometric Theory, Cambridge University Press, vol. 32(06), pages 1349-1375, December.
    11. Chihwa Kao & Lorenzo Trapani & Giovanni Urga, 2012. "Testing for Breaks in Cointegrated Panels," Center for Policy Research Working Papers 135, Center for Policy Research, Maxwell School, Syracuse University.
    12. Phillips, P C B, 1991. "Optimal Inference in Cointegrated Systems," Econometrica, Econometric Society, vol. 59(2), pages 283-306, March.
    13. Seo Byeongseon, 2011. "Nonparametric Testing for Linearity in Cointegrated Error-Correction Models," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 15(2), pages 1-28, March.
    14. Trapani, Lorenzo, 2013. "On bootstrapping panel factor series," Journal of Econometrics, Elsevier, vol. 172(1), pages 127-141.
    15. Francesc Marmol & Juan J. Dolado, 1999. "Asymptotic Inference for Nonstationary Fractionally Integrated Processes," Computing in Economics and Finance 1999 513, Society for Computational Economics.
    16. Demetrescu Matei, 2009. "Panel Unit Root Testing with Nonlinear Instruments for Infinite-Order Autoregressive Processes," Journal of Time Series Econometrics, De Gruyter, vol. 1(2), pages 1-30, December.
    17. Dilem Yildirim & Ralf Becker & Denise R Osborn, 2009. "Bootstrap Unit Root Tests for Nonlinear Threshold Models," The School of Economics Discussion Paper Series 0915, Economics, The University of Manchester.
    18. Phillips, Peter C.B., 1995. "Robust Nonstationary Regression," Econometric Theory, Cambridge University Press, vol. 11(05), pages 912-951, October.
    19. Vogelsang, Timothy J. & Wagner, Martin, 2014. "Integrated modified OLS estimation and fixed-b inference for cointegrating regressions," Journal of Econometrics, Elsevier, vol. 178(2), pages 741-760.
    20. Jansson, Michael, 2004. "Stationarity Testing With Covariates," Econometric Theory, Cambridge University Press, vol. 20(01), pages 56-94, February.
    21. Uwe Hassler & Francesc Marmol & Carlos Velasco, 2008. "Fractional cointegration in the presence of linear trends," Journal of Time Series Analysis, Wiley Blackwell, vol. 29(6), pages 1088-1103, November.
    22. Pentti Saikkonen & Rickard Sandberg, 2016. "Testing for a Unit Root in Noncausal Autoregressive Models," Journal of Time Series Analysis, Wiley Blackwell, vol. 37(1), pages 99-125, January.
    23. Gregory, Allan W. & Hansen, Bruce E., 1996. "Residual-based tests for cointegration in models with regime shifts," Journal of Econometrics, Elsevier, vol. 70(1), pages 99-126, January.
    24. Farzad Sabzikar & Qiying Wang & Peter C.B. Phillips, 2018. "Asymptotic Theory for Near Integrated Process Driven by Tempered Linear Process," Cowles Foundation Discussion Papers 2131, Cowles Foundation for Research in Economics, Yale University.

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