This paper develops a wavelet (spectral) approach to testing the presence of a unit root in a stochastic process. The wavelet approach is appealing, since it is based directly on the different behavior of the spectra of a unit root process and that of a short memory stationary process. By decomposing the variance (energy) of the underlying process into the variance of its low frequency components and that of its high frequency components via the discrete wavelet transformation (DWT), we design unit root tests against near unit root alternatives. Since DWT is an energy preserving transformation and able to disbalance energy across high and low frequency components of a series, it is possible to isolate the most persistent component of a series in a small number of scaling coefficients. We demonstrate the size and power properties of our tests through Monte Carlo simulations.
This is a substantially shortened version of the paper “Unit root and cointegration tests with wavelets.” We are grateful to Pentti Saikkonen and two anonymous referees for detailed comments on the early paper, which have helped improve the presentation of the results in the current paper. We also thank Stelios Bekiros, Buz Brock, Russell Davidson, Cees Diks, Cars Hommes, Peter Kennedy, Benoit Perron, Hashem Pesaran, James MacKinnon, James Ramsey, Alessio Sancetta, Mototsugu Shintani, and Zhijie Xiao for helpful discussions. All errors belong to the authors. Part of the work in this paper was done when Fan visited the Department of Economics at Simon Fraser University, whose hospitality and support are acknowledged. Yanqin Fan is grateful to the National Science Foundation for research support. Ramo Gençay is grateful to the Natural Sciences and Engineering Research Council of Canada and the Social Sciences and Humanities Research Council of Canada for research support. Gençay is also grateful for the feedback of seminar participants at CESG, CIREQ, IRMACS, University of Amsterdam, University of British Columbia, Cambridge University, University College Dublin, and New York University.
The software for this paper is available at www.sfu.ca/∼rgencay.