Local Whittle estimation of fractional integration and some of its variants

Asymptotic properties of the local Whittle estimator are studied under a model of fractional integration that provides a uniform representation of the data for any value of d. For d∈(-12,12), the estimator is shown to be consistent and asymptotically normally distributed. For d∈[-1,-12), the estimator converges either to the true parameter value or to zero, depending on the number of frequencies used in estimation. This behavior manifests itself as a positive bias in finite samples and the inconsistency affects the asymptotic properties of the 'differencing/adding-one-back' estimator that has been suggested for use. The relationship between several variants of the LW estimator are discussed. It is shown that the LW estimator, the 'differencing/adding-one-back' estimator and the modified LW estimator (Cowles Foundation Discussion Paper no. 1265, Yale University, 2000) can all be viewed as approximations to an exact LW estimator (Cowles Foundation Discussion Paper no. 1367, Yale University, 2003) that uses an exact expression for the model in frequency domain form. The approximations are valid over specific domains of d, like the stationary region (-12,12) and the nonstationary region (12,32), whereas the exact LW estimator is consistent and asymptotically N(0,14) for all values of d.