Local Whittle estimation in nonstationary and unit root cases
  Asymptotic properties of the local Whittle estimator in the nonstationary case (d>{1/2}) are explored. For {1/2}<d\leq 1, the estimator is shown to be consistent, and its limit distribution and the rate of convergence depend on the value of d. For d=1, the limit distribution is mixed normal.   For d>1 and when the process has a polynomial trend of order \alpha >{1/2}, the estimator is shown to be inconsistent and to converge in probability to unity. 
Peter C. B. Phillips and Katsumi Shimotsu