Estimation and Testing in a Perturbed Multivariate Long Memory Framework

We propose a semiparametric multivariate estimator and a multivariate score-type testing procedure under a perturbed multivariate fractional process. The estimator is based on the periodogram and uses a local Whittle criterion function which is generalised by an additional constant to capture the perturbation given in the long memory process. Explicitly addressing the noise term when approximating the spectral density near the origin results in a bias reduction, but at the cost of an increase in the asymptotic variance of the estimator. Further, we introduce a multivariate testing procedure to detect spurious long memory under a perturbed fractional framework. The test statistic is based on the weighted sum of the partial derivatives of the multivariate local Whittle with noise estimator. We show consistency of the test against the alternatives of smooth trend and random level shift processes. In addition, we prove consistency and asymptotic normality of the local Whittle estimator and we derive the limiting distribution of the test. An empirical example on the squared returns and the realised volatilities from the BEL 20, S&P BSE SENSEX, and the Spanish IBEX is conducted, and shows the usefulness of the procedures.