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Econometrics Paper
Semiparametric Estimation for Stationary Processes whose Spectra have an Unknown Pole
Javier Hidalgo
January 2005
Paper No' EM/2005/481:
Full Paper (pdf)

JEL Classification: C14; G22

Tags: spectral density estimation; long memory processes; gaussian processes

We consider the estimation of the location of the pole and memory parameter, ?<sup>0</sup> and a respectively, of covariance stationary linear processes whose spectral density function f(?) satisfies f(?) ~ C|? - ?<sup>0</sup>|<sup>-a</sup> in a neighbourhood of ?<sup>0</sup>. We define a consistent estimator of ?<sup>0</sup> and derive its limit distribution Z<sub>?<sup>0</sup></sub> . As in related optimization problems, when the true parameter value can lie on the boundary of the parameter space, we show that Z<sub>?<sup>0</sup></sub> is distributed as a normal random variable when ?<sup>0</sup> ? (0, p), whereas for ?<sup>0</sup> = 0 or p, Z<sub>?<sup>0</sup></sub> is a mixture of discrete and continuous random variables with weights equal to 1/2. More specifically, when ?<sup>0</sup> = 0, Z<sub>?<sup>0</sup></sub> is distributed as a normal random variable truncated at zero. Moreover, we describe and examine a two-step estimator of the memory parameter a, showing that neither its limit distribution nor its rate of convergence is affected by the estimation of ?<sup>0</sup>. Thus, we reinforce and extend previous results with respect to the estimation of a when ?<sup>0</sup> is assumed to be known a priori. A small Monte Carlo study is included to illustrate the finite sample performance of our estimators.