Multiple Local Whittle Estimation in StationarySystems

Abstract

Moving from univariate to bivariate jointly dependent long memory time series introduces a phase parameter (?), at the frequency of principal interest, zero; for shortmemory series ? = 0 automatically. The latter case has also been stressed under longmemory, along with the 'fractional differencing' case ( ) / 2; 2 1 ? = d - d p where 1 2 d , dare the memory parameters of the two series. We develop time domain conditionsunder which these are and are not relevant, and relate the consequent properties ofcross-autocovariances to ones of the (possibly bilateral) moving averagerepresentation which, with martingale difference innovations of arbitrary dimension,is used in asymptotic theory for local Whittle parameter estimates depending on asingle smoothing number. Incorporating also a regression parameter (ß) which, whennon-zero, indicates cointegration, the consistency proof of these implicitly-definedestimates is nonstandard due to the ß estimate converging faster than the others. Wealso establish joint asymptotic normality of the estimates, and indicate how thisoutcome can apply in statistical inference on several questions of interest. Issues ofimplementation are discussed, along with implications of knowing ß and of correct orincorrect specification of ? , and possible extensions to higher-dimensional systemsand nonstationary series.Long memory, phase, cointegration, semiparametricestimation, consistency, asymptotic normality.