Root--consistent estimation of weak fractional cointegration

Empirical evidence has emerged of the possibility of fractional cointegration such that the gap, β , between the integration order δ of observable time series and the integration order γ of cointegrating errors is less than 0.5. This includes circumstances when observables are stationary or asymptotically stationary with long memory so δ < 1 2 and when they are nonstationary so δ ⩾ 1 2 . This “weak cointegration” contrasts strongly with the traditional econometric prescription of unit-root observables and short memory cointegrating errors, where β = 1 . Asymptotic inferential theory also differs from this case and from other members of the class β > 1 2 , in particular n -consistent and asymptotically normal estimation of the cointegrating vector ν is possible when β < 1 2 , as we explore in a simple bivariate model. The estimate depends on γ and δ or, more realistically, on estimates of unknown γ and δ . These latter estimates need to be n -consistent, and the asymptotic distribution of the estimate of ν is sensitive to their precise form. We propose estimates of γ and δ 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.