Multivariate analysis of long memory series in the frequency domain.

This thesis examines some statistical procedures in the frequency domain to analyze long-memory series. We define a long-memory series and review part of the literature. Then we proceed by analyzing different estimation procedures for H, the parameter that characterizes the existence of long-memory. Parametric estimates have as a main drawback that they can lead to inconsistent estimates of H if the parametric model is misspecified. Therefore we focus on semiparametric estimates in the frequency domain. In our case, semiparametric means that we only need to assume a parametric model for the spectral density in a neighbourhood of zero frequency. We focus mainly on a multivariate framework. First we analyze estimates based on the average periodogram. We prove the consistency of the average cross-periodogram for the cumulative cross-spectrum. We also establish the asymptotic distribution in the scalar case. Then we focus on an implicit estimate based on a discrete approximation of the Gaussian likelihood in a neighbourhood of zero frequency. We prove the consistency and asymptotic normality of this estimate. Based on this estimate we establish a Lagrange multiplier test for weak dependence. We finish with an application of these methods to financial data.