Whittle-type estimation under long memory and nonstationarity
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Whittle-type estimation under long memory and nonstationarity Ying Lun Cheung1 · Uwe Hassler2 Received: 7 November 2018 / Accepted: 16 October 2019 © Springer-Verlag GmbH Germany, part of Springer Nature 2019
Abstract 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. Keywords Fractionally integrated time series · Discontinuity · Test distortion
1 Introduction Long memory and nonstationarity are considered as stylized facts in many empirical time series from a variety of fields. For surveys from economics and finance over political sciences to hydrology, see e.g., Baillie (1996), Box-Steffensmeier and Tomlinson (2000), and Montanari (2003), respectively. Fractional integration of order d is the most widely used model to capture long memory and nonstationarity. Estimated values of d mostly vary between 0 and 1, with d = 1/2 separating the reign of stationarity from nonstationarity. The properties of the so-called (local) Whittle estimator depend on the true d0 , which is a nuisance, since this parameter is not known a priori; see
An earlier version was presented at the International Conference on Time Series and Forecasting (ITISE 2018) in Granada, September 2018. The authors thank Mehdi Hosseinkouchack and Carlos Velasco for helpful comments. They are further grateful to two anonymous referees for many useful comments.
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Uwe Hassler [email protected]
1
Capital University of Economics and Business, Beijing, China
2
Statistics and Econometric Methods, Goethe University Frankfurt, Theodor-W.-Adorno-Platz 4, 60323 Frankfurt, Germany
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Y. L. Cheung, U. Hassler
Robinson (1995), Velasco (1999), and Velasco and Robinson (2000). This is not the case with the exact local Whittle estimator proposed by Shimotsu and Phillips (2005) and the tapered local Whittle estimator by Velasco (1999); the former, however, is burdened by the necessity of mean estimation (see Shimotsu (2010)), while the latter is plagued by variance inflation. Similarly, the fully nonstationarity-extended (local or parametric) Whittle estimators that have been proposed by Abadir et al. (2007) and Shao (2010), respectively, do not depend on the true d0 , although the half point is not covered by their theory. Abadir et al. (2007, p. 1366) “illustrate how our objective function is well behaved” by presenting plots of three objective functions for three different true values d0 , revealing “a smooth function with a unique minimi
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