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

  • Phillips, P.C.B.

Under general conditions the sample covariance matrix of a vector martingale and its differences converges weakly to the matrix stochastic integral ∫ 01 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 ∫ 01 BdB′ + 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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Article provided by Cambridge University Press in its journal Econometric Theory.

Volume (Year): 4 (1988)
Issue (Month): 03 (December)
Pages: 528-533

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Handle: RePEc:cup:etheor:v:4:y:1988:i:03:p:528-533_01
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  1. 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 113-44, January.
  2. 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.
  3. Phillips, P C B, 1987. "Time Series Regression with a Unit Root," Econometrica, Econometric Society, vol. 55(2), pages 277-301, March.
  4. Peter C.B. Phillips, 1985. "Asymptotic Expansions in Nonstationary Vector Autoregressions," Cowles Foundation Discussion Papers 765, Cowles Foundation for Research in Economics, Yale University.
  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. Phillips, P.C.B., 1986. "Understanding spurious regressions in econometrics," Journal of Econometrics, Elsevier, vol. 33(3), pages 311-340, December.
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