  We introduce two new estimators of the bivariate Hurst exponent in the
power-law cross-correlations setting -- the cross-periodogram and local
$X$-Whittle estimators -- as generalizations of their univariate counterparts.
As the spectrum-based estimators are dependent on a part of the spectrum taken
into consideration during estimation, a simulation study showing performance of
the estimators under varying bandwidth parameter as well as correlation between
processes and their specification is provided as well. The newly introduced
estimators are less biased than the already existent averaged periodogram
estimator which, however, has slightly lower variance. The spectrum-based
estimators can serve as a good complement to the popular time domain
estimators.
