Discontinuity of Fully Extended (Local) Whittle Estimation

We consider the fully extended local Whittle estimator of the fractional order of integration d proposed by Abadir, Distaso and Giraitis (2007), and the extended parametric Whittle estimator suggested by Shao (2010). They are valid under stationarity as well as nonstationarity: a priori knowledge whether the true d_0 < 1/2 or not is not required. Experimentally, we observe a lack of continuity of the objective functions at d = 1/2 that has not been reported before. It results in a pile-up of the estimates at d = 1/2 when the true value is in a neighbourhood to this half point. Consequently, studentized test statistics may be heavily oversized