Semiparametric Estimation of Fractional Cointegrating Subspaces

We consider a common components model for multivariate fractional cointegration, in which the s ¸ 1 components have different memory parameters. The cointegrating rank is allowed to exceed 1. The true cointegrating vectors can be decomposed into orthogonal fractional cointegrating subspaces suchthat vectors from distinct subspaces yield cointegrating errors with distinct memory parameters, denoted by dk, for k = 1; : : : ; s. We estimate each cointegrating subspace separately using appropriate sets ofeigenvectors of an averaged periodogram matrix of tapered, differenced observations. The averaging uses the first m Fourier frequencies, with m fixed. We will show that any vector in the k th estimatedcointegrating subspace is, with high probability, close to the k th true cointegrating subspace, in the sensethat the angle between the estimated cointegrating vector and the true cointegrating subspace convergesin probability to zero. This angle is Op(ni®k ), where n is the sample size and ®k is the shortest distance between the memory parameters corresponding to the given and adjacent subspaces. We show that the cointegrating residuals corresponding to an estimated cointegrating vector can be used to obtain a consistent and asymptotically normal estimate of the memory parameter for the given cointegrating subspace, using a univariate Gaussian semiparametric estimator with a bandwidth that tends to 1 more slowly than n. We also show how these memory parameter estimates can be used to test for fractional cointegration and to consistently identify the cointegrating subspaces.