The cyclical components of time series data have been typically examined with the use of spectral analysis or ARMA models. While spectral analysis allows direct estimation of which frequencies play relevant roles in explaining time series variance, ARMA models are a time domain approach that also allows the indirect detection of those cycles. What they also share, however, is both an assumption of stationarity and of the time invariance of the cycles they uncover. Unfortunately, many economic and political time-series are, in fact, noisy, complex and strongly non-stationary. And most importantly, it is probably unwise to assume, especially over prolonged periods of time, that the underlying processes generating the time series data we observe are themselves time invariant. Wavelet analysis helps overcoming these problems in the analysis of the cyclical components of a time series and of the frequencies that explain its variance. It performs the estimation of the spectral characteristics of a time-series as a function of time, revealing how the different periodic components of the time-series change over time. In this paper, we present three tools that, to our knowledge, have not yet been used by political scientists - the wavelet power spectrum, the cross-wavelet coherency and the phase difference - as well as a metric to compare different wavelet spectra. We apply these tools to the study of presidential election cycles in the United States.