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Abstract:
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Related long memory time series x(t), y(t) may have different or identical memory parameters. In the latter case the possibility exists of domination by a single common component, such that an instantaneous linear combination y(t)-ßx(t) of the series has shorter memory than x(t), y(t). A semiparametric version of the problem presents four unknown parameters of interest: two memory parameters, a phase parameter, and ß. We estimate them jointly by optimizing a local Whittle function that entails a single smoothing number, covering also the case where ß=0 and y(t) and z(t) have different memory parameters. The consistency proof is non-standard in that the ß estimate converges faster than the others. Joint asymptotic normality is also established. Misspecification of the phase can slow convergence of estimates of ß, and cause inconsistency of memory parameter estimates.
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