  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.
