Semiparametric inference in multivariate fractionally cointegrated systems

A semiparametric multivariate fractionally cointegrated system is considered, integration orders possibly being unknown and I ( 0 ) unobservable inputs having nonparametric spectral density. Two estimates of the vector of cointegrating parameters ν are considered. One involves inverse spectral weighting and the other is unweighted but uses a spectral estimate at frequency zero. Both corresponding Wald statistics for testing linear restrictions on ν are shown to have a standard null χ 2 limit distribution under quite general conditions. Notably, this outcome is irrespective of whether cointegrating relations are “strong” (when the difference between integration orders of observables and cointegrating errors exceeds 1/2), or “weak” (when that difference is less than 1/2), or when both cases are involved. Finite-sample properties are examined in a Monte Carlo study and an empirical example is presented.