Multiple local whittle estimation in stationary systems

TL;DR

This paper extends local Whittle estimation to multivariate stationary long-memory systems, addressing phase parameters, cointegration, and asymptotic properties, with implications for statistical inference and system extensions.

Contribution

It develops a novel framework for joint estimation of long-memory parameters, phase, and cointegration in multivariate stationary systems, including consistency and asymptotic normality results.

Findings

Establishes conditions for relevance of phase parameter $\

Provides joint asymptotic normality of estimates

Discusses implementation issues and potential extensions

Abstract

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 nonzero, 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 implemention 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.