Semiparametric Estimation of Multivariate Fractional Cointegration

Abstract

We consider the semiparametric estimation of fractional cointegration in a multivariate process of cointegrating rank r > 0. We estimate the cointegrating relationships by the eigenvectors corresponding to the r smallest eigenvalues of an averaged periodogram matrix of tapered, differenced observations. The number of frequencies m used in the periodogram average is held fixed as the sample size grows. We first show that the averaged periodogram matrix converges in distribution to a singular matrix whose null eigenvectors span the space of cointegrating vectors. We then show that the angle between the estimated cointegrating vectors and the space of true cointegrating vectors is Op(ndu−d), where d and du are the memory parameters of the observations and cointegrating errors. The proposed estimator is invariant to the labeling of the component series and thus does not require that one of the variables be specified as a dependent variable. We determine the rate of convergence of the r smallest eigenvalues of the periodogram matrix and present a criterion that allows for consistent estimation of r. Finally, we apply our methodology to the analysis of fractional cointegration in interest rates.