COWLES FOUNDATION FOR RESEARCH IN ECONOMICS
AT YALE UNIVERSITY

Box 208281
New Haven, CT 06520-8281

COWLES FOUNDATION DISCUSSION PAPER NO. 796

"Weak Convergence to the Matrix Stocastic Integral /₀¹BdB'"

Peter C.B. Phillips

July 1986

The asymptotic theory of regression with integrated processes of the ARIMA type frequently involves weak convergence to stochastic integrals of the form /₀¹WdW, 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 /₀¹BdB', where B(r) is vector Brownian motion with non scalar covariance matrix. This paper studies the weak convergence of sample covariance matrices to /₀¹BdB' under quite general conditions. The theory is applied to vector autoregressions with integrated processes.

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