Moving from univariate to bivariate jointly dependent long-memory time series introduces a phase parameter (γ), at the frequency of principal interest, zero; for short-memory series γ=0 automatically. The latter case has also been stressed under long memory, along with the “fractional differencing” case γ=(δ2−δ1)π/2, where δ1, δ2 are the memory parameters of the two series. We develop time domain conditions under which these are and are not relevant, and relate the consequent properties of cross-autocovariances to ones of the (possibly bilateral) moving average representation which, with martingale difference innovations of arbitrary dimension, is used in asymptotic theory for local Whittle parameter estimates depending on a single smoothing number. Incorporating also a regression parameter (β) which, when nonzero, indicates cointegration, the consistency proof of these implicitly defined estimates is nonstandard due to the β estimate converging faster than the others. We also establish joint asymptotic normality of the estimates, and indicate how this outcome can apply in statistical inference on several questions of interest. Issues of implemention are discussed, along with implications of knowing β and of correct or incorrect specification of γ, and possible extensions to higher-dimensional systems and nonstationary series.
"Multiple local whittle estimation in stationary systems." Ann. Statist. 36 (5) 2508 - 2530, October 2008. https://doi.org/10.1214/07-AOS545