Root-N-Consistent Estimation of Weak Fractional Cointegration

Empirical evidence has emerged of the possibility of fractional cointegration such that the gap, amp;#946;, between the integration order amp;#948; of observable time series, and the integration order amp;#947; of cointegrating errors, is less than 0.5. This includes circumstances when observables are stationary or asymptotically stationary with long memory (so amp;#948; lt; 1/2), and when they are nonstationary (so amp;#948; 1/2). This quot;weak cointegrationquot; contrasts strongly with the traditional econometric prescription of unit root observables and short memory cointegrating errors, where amp;#946; = 1. Asymptotic inferential theory also differs from this case, and from other members of the class amp;#946; gt; 1/2, in particular amp;#8805; consistent - n and asymptotically normal estimation of the cointegrating vector amp;#957; is possible when amp;#946; lt; 1/2, as we explore in a simple bivariate model. The estimate depends on amp;#947; and amp;#948; or, more realistically, on estimates of unknown amp;#947; and amp;#948;. These latter estimates need to be consistent - n , and the asymptotic distribution of the estimate of amp;#957; is sensitive to their precise form. We propose estimates of amp;#947; and amp;#948; 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