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Weak Convergence to the Matrix Stochastic Integral BdB

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Abstract

The asymptotic theory of regression with integrated processes of the ARIMA type frequently involves weak convergence to stochastic integrals of the form integral_{0}^{1}WdW, where W(r) is standard Brownian motion. In multiple regressions and vector autoregressions with vector ARIMA processes the theory involves weak convergence to matrix stochastic integrals of the form integral_{0}^{1}BdB', where B(r) is vector Brownian motion with non scalar covariance matrix. This paper studies the weak convergence of sample covariance matrices to integral_{0}^{1}BdB' under quite general conditions. The theory is applied to vector autoregressions with integrated processes.

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File URL: http://cowles.econ.yale.edu/P/cd/d07b/d0796.pdf
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Bibliographic Info

Paper provided by Cowles Foundation for Research in Economics, Yale University in its series Cowles Foundation Discussion Papers with number 796.

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Length: 18 pages
Date of creation: Jul 1986
Date of revision:
Publication status: Published in Journal of Multivariate Analysis (February 1988), 24(2): 252-264
Handle: RePEc:cwl:cwldpp:796

Note: CFP 697.
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Postal: Cowles Foundation, Yale University, Box 208281, New Haven, CT 06520-8281 USA

Related research

Keywords: Integrated process; invariance principle; near integrated time series; stochastic integral; vector autoregression; weak convergence;

References

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  1. Peter C.B. Phillips, 1985. "Time Series Regression with a Unit Root," Cowles Foundation Discussion Papers 740R, Cowles Foundation for Research in Economics, Yale University, revised Feb 1986.
  2. Peter C.B. Phillips & Steven N. Durlauf, 1985. "Multiple Time Series Regression with Integrated Processes," Cowles Foundation Discussion Papers 768, Cowles Foundation for Research in Economics, Yale University.
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Cited by:
  1. Westerlund, Joakim, 2005. "Panel Cointegration Tests of the Fisher Hypothesis," Working Papers 2005:10, Lund University, Department of Economics.
  2. Peter C.B. Phillips & Sam Ouliaris, 1986. "Testing for Cointegration Using Principal Component Measures," Cowles Foundation Discussion Papers 809R, Cowles Foundation for Research in Economics, Yale University, revised Jul 1987.
  3. Peter C.B. Phillips & Steven N. Durlauf, 1985. "Multiple Time Series Regression with Integrated Processes," Cowles Foundation Discussion Papers 768, Cowles Foundation for Research in Economics, Yale University.
  4. Joseph Beaulieu, J. & Miron, Jeffrey A., 1993. "Seasonal unit roots in aggregate U.S. data," Journal of Econometrics, Elsevier, vol. 55(1-2), pages 305-328.
  5. Peter C.B. Phillips, 1986. "Regression Theory for Near-Integrated Time Series," Cowles Foundation Discussion Papers 781R, Cowles Foundation for Research in Economics, Yale University, revised Jan 1987.

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