Semiparametric Estimation of Fractional Cointegration

A semiparametric bivariate fractionally cointegrated system is considered, integration orders possibly being unknown and I (0) unobservable inputs having nonparametric spectral density. Two kinds of estimate of the cointegrating parameter ν are considered, one involving inverse spectral weighting and the other, unweighted statistics with a spectral estimate at frequency zero. We establish under quite general conditions the asymptotic distributional properties of the estimates of ν, both in case of “strong cointegration” (when the difference between integration orders of observables and cointegrating errors exceeds 1/2) and in case of “weak cointegration” (when that difference is less than 1/2), which includes the case of (asymptotically) stationary observables. Across both cases, the same Wald test statistic has the same standard null χ2 limit distribution, irrespective of whether integration orders are known or estimated. The regularity conditions include unprimitive ones on the integration orders and spectral density estimates, but we check these under more primitive conditions on particular estimates. Finite-sample properties are examined in a Monte Carlo study.