2008/10/01 by P. M. Robinson, Peter M. Robinson
Economics, Econometrics and Finance · Mathematics · #Complex Systems and Time Series Analysis #Financial Risk and Volatility Modeling #Monetary Policy and Economic Impact #math.ST #msc:62G20 #msc:62M09 #msc:62M10 #msc:62M15 #stat.TH
paper · pdf · doi:10.1214/07-aos545
published as Annals of Statistics 2008, Vol. 36, No. 5, 2508-2530 · Published in at http://dx.doi.org/10.1214/07-AOS545 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)
openalex publication_date 2008/10/01 · arxiv created 2008/11/06 · arxiv updated 2009/12/01 · openalex created_date 2021/02/01 · openalex updated_date 2026/08/01
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.