April 2021 Estimation and inference in the presence of fractional d=1/2 and weakly nonstationary processes
James A. Duffy, Ioannis Kasparis
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Ann. Statist. 49(2): 1195-1217 (April 2021). DOI: 10.1214/20-AOS1998


We provide new limit theory for functionals of a general class of processes lying at the boundary between stationarity and nonstationarity—what we term weakly nonstationary processes (WNPs). This includes, as leading examples, fractional processes with d=1/2, and arrays of autoregressive processes with roots drifting slowly towards unity. We first apply the theory to study inference in parametric and nonparametric regression models involving WNPs as covariates. We then use these results to develop a new specification test for parametric regression models. By construction, our specification test statistic has a χ2 limiting distribution regardless of the form and extent of persistence of the regressor, implying that a practitioner can validly perform the test using a fixed critical value, while remaining agnostic about the mechanism generating the regressor. Simulation exercises confirm that the test controls size across a wide range of data generating processes, and outperforms a comparable test due to Wang and Phillips (Ann. Statist. 40 (2012) 727–758) against many alternatives.


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James A. Duffy. Ioannis Kasparis. "Estimation and inference in the presence of fractional d=1/2 and weakly nonstationary processes." Ann. Statist. 49 (2) 1195 - 1217, April 2021. https://doi.org/10.1214/20-AOS1998


Received: 1 January 2020; Published: April 2021
First available in Project Euclid: 2 April 2021

Digital Object Identifier: 10.1214/20-AOS1998

Primary: 62F12 , 62G08 , 62M10

Keywords: Fractional process , half unit root , kernel regression , mildly integrated process , specification testing , weakly nonstationary process

Rights: Copyright © 2021 Institute of Mathematical Statistics


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Vol.49 • No. 2 • April 2021
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