Cointegration of nonstationary time series is considered in a fractional context. Both the observable series and the cointegrating error can be fractional processes. The familiar situation in which the respective integration orders are 1 and 0 is nested, but these values have typically been assumed known. We allow one or more of them to be unknown real values, in which case Robinson and Marinucci (1997,2001) have justified least squares estimates of the cointegrating vector, as well as narrow-band frequencydomain estimates, which may be less biased. While consistent, these estimates do not always have optimal convergence rates, and they have non-standard limit distributional behaviour. We consider estimates formulated in the frequency domain, that consequently allow for a wide variety of (parametric) autocorrelation in the short memory input series, as well as time-domain estimates based on autoregressive transformation. Both can be interpreted as approximating generalized least squares and Gaussian maximum likelihood estimates. The estimates share the same limiting distribution, having mixed normal asymptotics (yielding Wald test statistics with 2 c null limit distributions), irrespective of whether the integration orders are known or unknown, subject in the latter case to their estimation with adequate rates of convergence. The parameters describing the short memory stationary input series are n -consistently estimable, but the assumptions imposed on these series are much more general than ones of autoregressive moving average type. A Monte Carlo study of finite-sample performance and an empirical application to testing the PPP hypothesis are included. ACKNOWLEDGMENTSWe are grateful for the comments of a Co-Editor and referees, which have led to significant improvements.

We consider the estimation of parametric fractional time series models in which not only is the memory parameter unknown, but one may not know whether it lies in the stationary/invertible region or the nonstationary or noninvertible regions. In these circumstances, a proof of consistency (which is a prerequisite for proving asymptotic normality) can be difficult owing to nonuniform convergence of the objective function over a large admissible parameter space. In particular, this is the case for the conditional sum of squares estimate, which can be expected to be asymptotically efficient under Gaussianity. Without the latter assumption, we establish consistency and asymptotic normality for this estimate in case of a quite general univariate model. For a multivariate model, we establish asymptotic normality of a one-step estimate based on an initial √ n-consistent estimate.

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a b s t r a c tA 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.

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