We prove the consistency of the averaged periodogram estimator (APE) in
two new cases. First, we prove that the APE is consistent for negative
memory parameters, after suitable tapering. Second, we prove that the
APE is consistent for a power law in the cross-spectrum and therefore
for a power law in the coherency, provided that sufficiently many
frequencies are used in estimation. Simulation evidence suggests that
the lower bound on the number of frequencies is a necessary condition
for consistency. For a Taylor series approximation to the estimator of
the power law in the cross-spectrum, we consider the rate of
convergence, and obtain a central limit theorem under suitable
regularity conditions.J.P. Morgan Chase and Co. and New York UniversityStatistics Working Papers Serie