Unit Root and Cointegration Tests with Wavelets∗

This paper develops a wavelet (spectral) approach to test 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 which have substantial power 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. Our tests utilize the wavelet coefficients of the coarsest scale. We demonstrate the size and power properties of our tests through Monte Carlo simulations, and apply them to financial time series. A generalization of our unit root tests to the maximum overlap DWT (MODWT) and to residual based tests for cointegration are also provided.