Asymptotic properties of the local Whittle estimator in the nonstationary case (d>½) are explored. For ½<d≤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 α>½, the estimator is shown to be inconsistent and to converge in probability to unity.
"Local Whittle estimation in nonstationary and unit root cases." Ann. Statist. 32 (2) 656 - 692, April 2004. https://doi.org/10.1214/009053604000000139