Semiparametric estimation for stationary processes whose spectra have an unknown pole

Abstract

We consider the estimation of the location of the pole and memory parameter, λ⁰ and α, respectively, of covariance stationary linear processes whose spectral density function f(λ) satisfies f(λ)∼C|λ−λ⁰|^−α in a neighborhood of λ⁰. We define a consistent estimator of λ⁰ and derive its limit distribution Z_λ⁰. As in related optimization problems, when the true parameter value can lie on the boundary of the parameter space, we show that Z_λ⁰ is distributed as a normal random variable when λ⁰∈(0,π), whereas for λ⁰=0 or π, Z_λ⁰ is a mixture of discrete and continuous random variables with weights equal to 1/2. More specifically, when λ⁰=0, Z_λ⁰ is distributed as a normal random variable truncated at zero. Moreover, we describe and examine a two-step estimator of the memory parameter α, showing that neither its limit distribution nor its rate of convergence is affected by the estimation of λ⁰. Thus, we reinforce and extend previous results with respect to the estimation of α when λ⁰ is assumed to be known a priori. A small Monte Carlo study is included to illustrate the finite sample performance of our estimators.