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TESTING FOR STRICT STATIONARITY VIA THE DISCRETE FOURIER TRANSFORM

Published online by Cambridge University Press:  28 October 2022

Zhonghao Fu,
Shang Gao,
Liangjun Su and
Xia Wang
Zhonghao Fu
Affiliation:
Fudan University and Shanghai Institute of International Finance and Economics
Shang Gao
Affiliation:
Fudan University
Liangjun Su
Affiliation:
Tsinghua University
Xia Wang*
Affiliation:
Renmin University of China
*
Address correspondence to Xia Wang, School of Economics, Renmin University of China, Beijing, China; e-mail: wxia@ruc.edu.cn.
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Abstract

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This paper proposes a model-free test for the strict stationarity of a potentially vector-valued time series using the discrete Fourier transform (DFT) approach. We show that the DFT of a residual process based on the empirical characteristic function weakly converges to a zero spectrum in the frequency domain for a strictly stationary time series and a nonzero spectrum otherwise. The proposed test is powerful against various types of nonstationarity including deterministic trends and smooth or abrupt structural changes. It does not require smoothed nonparametric estimation and, thus, can detect the Pitman sequence of local alternatives at the parametric rate $T^{-1/2}$, faster than all existing nonparametric tests. We also design a class of derivative tests based on the characteristic function to test the stationarity in various moments. Monte Carlo studies demonstrate that our test has reasonarble size and excellent power. Our empirical application of exchange rates strongly suggests that both nominal and real exchange rate returns are nonstationary, which the augmented Dickey–Fuller and Kwiatkowski–Phillips–Schmidt–Shin tests may overlook.

Information

Type
ARTICLES
Information
Econometric Theory , Volume 40 , Issue 3 , June 2024, pp. 511 - 557
Copyright
© The Author(s), 2022. Published by Cambridge University Press

Footnotes

Fu acknowledges financial support from the National Science Foundation of China (NSFC) under Nos. 71903032 and 72121002. Gao acknowledges financial support from the NSFC under No. 71973030. Su gratefully acknowledges the NSFC for financial support under No. 72133002. Wang acknowledges financial support from the NSFC under No. 71873151.

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