Semiparametric Estimation of Fractional Cointegration

A semiparametric bivariate fractionally cointegrated system is considered, integrationorders possibly being unknown and I (0) unobservable inputs having nonparametricspectral density. Two kinds of estimate of the cointegrating parameter ? are considered,one involving inverse spectral weighting and the other, unweighted statistics with a spectralestimate at frequency zero. We establish under quite general conditions the asymptoticdistributional properties of the estimates of ?, both in case of "strong cointegration" (whenthe difference between integration orders of observables and cointegrating errors exceeds1/2) and in case of "weak cointegration" (when that difference is less than 1/2), whichincludes the case of (asymptotically) stationary observables. Across both cases, the sameWald test statistic has the same standard null ?2 limit distribution, irrespective of whetherintegration orders are known or estimated. The regularity conditions include unprimitiveones on the integration orders and spectral density estimates, but we check these undermore primitive conditions on particular estimates. Finite-sample properties are examined ina Monte Carlo study.Fractional cointegration, semiparametric model, unknown integration orders.