Root-n-Consistent Estimation of Weak Fractional Cointegration

Empirical evidence has emerged of the possibility of fractional cointegration such that the gap, beta , between the integration order delta of the observables and the integration order gamma of the cointegrating errors is less than 0.5. This includes circumstances both when the observables are stationary or asymptotically stationary with long memory (so delta is less than 0.5) and when they are nonstationary (so delta is greater or equal than 0.5). We call this weak cointegration, and it contrasts strongly with the traditional econometric prescription of unit root observables and short memory cointegrating errors, where beta equals one. Asymptotic inferential theory also differs from this case, and from other members of the class beta greater than 0.5, in particular root-n-consistent and asymptotically normal estimation of the cointegrating vector is possible when beta is less than 0.5, as we explore in a simple bivariate model. The estimate depends on gamma and delta or, more realistically, on estimates of unknown gamma and delta. These latter estimates need to be root-n-consistent, and the asymptotic distribution of the estimate of the cointegrating vector is sensitive to their precise form. We propose estimates of gamma and delta that are computationally relatively convenient, relying on only univariate nonlinear optimization. Finite sample performance of the methods is examined by means of Monte Carlo simulations, and several applications to empirical data included.