  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.
