On some problems in discrete wavelet analysis of bivariate spectra with an application to business cycle synchronization in the euro zone

The paper considers some of the problems emerging from discrete wavelet analysis of popular bivariate spectral quantities like the coherence and phase spectra and the frequency-dependent time delay. The approach taken here, introduced by Whitcher and Craigmile (2004), is based on the maximal overlap discrete Hilbert wavelet transform (MODHWT). Firstly, we point at a deficiency in the implementation of the MODHWT and suggest using a modified implementation scheme resembling the one applied in the context of the dual-tree complex wavelet transform of Kingsbury (see Selesnick et al., 2005). Secondly, via a broad set of simulation experiments we examine small and large sample properties of two wavelet estimators of the scale-dependent time delay. The estimators are: the wavelet cross-correlator and the wavelet phase angle-based estimator. Our results provide some practical guidelines for empirical examination of short- and medium-term lead-lag relations for octave frequency bands. Besides, we show how the MODHWT-based wavelet quantities can serve to approximate the Fourier bivariate spectra and discuss certain issues connected with building confidence intervals for them. The discrete wavelet analysis of coherence and phase angle is illustrated with a scale-dependent examination of business cycle synchronization between 11 euro zone member countries. The study is supplemented with wavelet analysis of variance and covariance of the euro zone business cycles. The empirical examination underlines good localization properties and high computational efficiency of the wavelet transformations applied, and provides new arguments in favour of the endogeneity hypothesis of the optimum currency area criteria as well as a wavelet evidence on dating the Great Moderation in the euro zone.