Spectral Analysis for Bivariate Time Series with Long Memory - (Now published in 'Econometric Theory',12 (1997)pp.773-792.)

This paper provides limit theorems for special density matrix estimators and functionals of it for a bivariate co variance stationary process whose spectral density matrix has singularities not only at the origin but possibly at some other frequencies, and thus applies to time series exhibiting long memory. In particular, we show that the consistency and asymptotic normality of the spectral density matrix estimator at a frequency, say ? which hold for weakly dependent time series, continue to hold for long memory processes when ? lies outside any arbitrary neighbourhood of the singularities. Specifically, we show that for the standard properties of spectral density matrix estimators to hold, only local smoothness of the spectral density matrix is required in a neighbourhood of the frequency in which we are interested. Therefore, we are able to relax, among other conditions, the absolute summability of the autocovariance function and of the fourth order cumulants or summability conditions on mixing coefficients, assumed in such a literature, which imply that the spectral density matrix is globally smooth and bounded.