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

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  • Phillips, P.C.B.

Abstract

Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral ∫01BdB′, 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 ∫01BdB′ + Λ and involves a constant matrix Λ 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.

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  • Phillips, P.C.B., 1988. "Weak Convergence of Sample Covariance Matrices to Stochastic Integrals Via Martingale Approximations," Econometric Theory, Cambridge University Press, vol. 4(3), pages 528-533, December.
    • Handle: RePEc:cup:etheor:v:4:y:1988:i:03:p:528-533_01
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    References listed on IDEAS

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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(1), pages 95-131, April.
    2. Phillips, P. C. B., 1987. "Asymptotic Expansions in Nonstationary Vector Autoregressions," Econometric Theory, Cambridge University Press, vol. 3(1), pages 45-68, February.
    3. Park, Joon Y. & Phillips, Peter C.B., 1988. "Statistical Inference in Regressions with Integrated Processes: Part 1," Econometric Theory, Cambridge University Press, vol. 4(3), pages 468-497, 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. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December.
    6. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
    7. 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.
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