CONSISTENCY OF THE AVERAGED CROSS-PERIODOGRAM IN LONG MEMORY SERIES BY IGNACIO N. LOBATO University of Iowa First version received September 1995 Abstract. Several aspects of inference with long memory series in a multivariate framework are examined. The main result of this paper is to prove the consistency of the averaged cross-periodogram evaluated in a degenerating neighbourhood of zero frequency. We also illustrate several applications of that result and consider some specification issues. Keywords. Long memory; semiparametric inference; specification. 1. INTRODUCTION Inference issues with long memory time series have been considered increasingly in the literature recently. This analysis has usually been carried out following a parametric approach (Fox and Taqqu, 1986; Dahlhaus, 1989; Sowell, 1992), but an alternative approach (Geweke and Porter-Hudak, 1983; Robinson, 1994a, b; 1995a, b), called semiparametric in Robinson (1994a), has also been employed. This approach can be justified as follows. Consider a univariate covariance stationary process x t and assume that its spectral density function (SDF hereafter) exists, denoted by f (λ). Then x t exhibits long memory if f (λ) Cλ 1 2 H as λ 0 , for 1 2  H  1, with C a positive constant (this could be relaxed to a slowly varying function, see Robinson (1994a)) where the symbol means that the ratio of both sides tends to 1 as λ 0 . Notice that this condition only characterizes f (λ) close to zero frequency and nothing is specified about the medium- and short-term behaviour of the process. As H measures the degree of long memory (the higher the H the longer the memory), its robust estimation is of great importance. The parametric approach assumes a full parametric model in which H is just another parameter. So, in this framework a misspecification of the short- or medium-term component of the model will lead to an inconsistent estimate of H. This motivates an interest in finding robust inference procedures for H. It is intended to do this in the semiparametric approach by introducing a bandwidth parameter m so that, in the estimation of H, only periodograms evaluated up to frequency λ m ( 2π mn), with λ m 0, will be used ( n denotes the sample size). Robinson (1994a) proposed an estimate based on the average period- 0143-9782/97/02 137–155 JOURNAL OF TIME SERIES ANALYSIS Vol. 18, No. 2 1997 Blackwell Publishers Ltd., 108 Cowley Road, Oxford OX4 1JF, UK and 350 Main Street, Malden, MA 02148, USA.