We consider six variants of (local) Whittle estimators of the fractional order of integration d. They follow a limiting normal distribution under stationarity as well as under (a certain degree of) nonstationarity. Experimentally, we observe a lack of continuity of the objective functions of the two fully extended versions at \(d=1/2\) that has not been reported before. It results in a pileup of the estimates at \(d=1/2\) when the true value is in a neighborhood to this half point. Consequently, studentized test statistics may be heavily oversized. The other four versions suffer from size distortions, too, although of a different pattern and to a different extent.

中文翻译：

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长记忆和非平稳性下的Whittle型估计

我们考虑积分d的（局部）Whittle估计量的六个变体。它们在平稳性以及（一定程度的）非平稳性下遵循有限的正态分布。从实验上，我们观察到两个完全扩展版本的目标函数在\（d = 1/2 \）处缺乏连续性，这一点以前没有报道过。当真实值在该半点附近时，将导致\（d = 1/2 \）处的估计值堆积。因此，学生化的测试统计数据可能会严重超标。其他四个版本也遭受尺寸失真，尽管它们的图案不同且程度不同。