Multiple Local Whittle Estimation in Stationary Systems

35 Pages Posted: 21 Jul 2008

See all articles by Peter M. Robinson

Peter M. Robinson

London School of Economics & Political Science (LSE) - Department of Economics; National Bureau of Economic Research (NBER)

Date Written: October 2007


Moving from univariate to bivariate jointly dependent long memory time series introduces a phase parameter (gamma), at the frequency of principal interest, zero; for short memory series gamma = 0 automatically. The latter case has also been stressed under long memory, along with the "fractional differencing" case gamma = (delta_2 - delta_1)pi/2; where delta_1, delta_2 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 (beta) which, when non-zero, indicates cointegration, the consistency proof of these implicitly-defined estimates is nonstandard due to the beta 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 implementation are discussed, along with implications of knowing beta and of correct or incorrect specification of gamma, and possible extensions to higher-dimensional systems and nonstationary series.

JEL Classification: C32

Suggested Citation

Robinson, Peter M., Multiple Local Whittle Estimation in Stationary Systems (October 2007). LSE STICERD Research Paper No. EM525, Available at SSRN:

Peter M. Robinson (Contact Author)

London School of Economics & Political Science (LSE) - Department of Economics ( email )

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