Javier Hidalgo

Published January 2005

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We consider the estimation of the location of the pole and memory parameter, ?⁰ and a respectively, of covariance stationary linear processes whose spectral density function f(?) satisfies f(?) ~ C|? - ?⁰|^-a in a neighbourhood of ?⁰. We define a consistent estimator of ?⁰ and derive its limit distribution Z_?⁰ . As in related optimization problems, when the true parameter value can lie on the boundary of the parameter space, we show that Z_?⁰ is distributed as a normal random variable when ?⁰ ? (0, p), whereas for ?⁰ = 0 or p, Z_?⁰ is a mixture of discrete and continuous random variables with weights equal to 1/2. More specifically, when ?⁰ = 0, Z_?⁰ 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 ?⁰. Thus, we reinforce and extend previous results with respect to the estimation of a when ?⁰ is assumed to be known a priori. A small Monte Carlo study is included to illustrate the finite sample performance of our estimators.