Weak convergence to a matrix stochastic integral with stable processes

Abstract

This paper generalizes the univariate results of Chan and Tran (1989, Econometric Theory 5, 354-362) and Phillips (1990, Econometric Theory 6, 44-62) to multivariate time series. We develop the limit theory for the least-squares estimate of a VAR(1) for a random walk with independent and identically distributed errors and for I(1) processes with weakly dependent errors whose distributions are in the domain of attraction of a stable law. The limit laws are represented by functionals of a stable process. A semiparametric correction is used in order to asymptotically eliminate the ''bias'' term in the limit law. These results are also an extension of the multivariate limit theory for square-integrable disturbances derived by Phillips and Durlauf (1986, Review of Economic Studies 53, 473-495). Potential applications include tests for multivariate unit roots and cointegration.