Central banks have different objectives in the short and long run. Governments operate simultaneously at different timescales. Many economic processes are the result of the actions of several agents, who have different term objectives. Therefore, a macroeconomic time series is a combination of components operating on different frequencies. Several questions about economic time series are connected to the understanding of the behavior of key variables at different frequencies over time, but this type of information is difficult to uncover using pure time-domain or pure frequency-domain methods.To our knowledge, for the first time in an economic setup, we use cross-wavelet tools to show that the relation between monetary policy variables and macroeconomic variables has changed and evolved with time. These changes are not homogeneous across the different frequencies.

A body of work using the continuous wavelet transform has been growing. We provide a self-contained summary on its most relevant theoretical results, describe how such transforms can be implemented in practice, and generalize the concept of simple coherency to partial wavelet coherency and multiple wavelet coherency, moving beyond bivariate analysis. We also describe a family of wavelets, which emerges as an alternative to the popular Morlet wavelet, the generalized Morse wavelets. A user-friendly toolbox, with examples, is attached to this paper. ∞ −∞ |x(t)| 2 dt is usually referred to as the energy of x, the space L 2 (R) is known as the space of finite energy functions.We use the convention g(t) ↔ G(ω) to denote a Fourier pair, that is, we denote by the corresponding capital letter the Fourier transform of a given function. Hence, if ψ(t) ∈ L 2 (R), (ω) will denote its Fourier transform, here defined as (ω) := ∞ −∞ ψ(t)e −iωt dt. With this definition, ω is an angular (or radian) frequency. The relation to the more common Fourier frequency is given by f = ω 2π .

We use (cross) wavelet analysis to decompose the time-frequency effects of oil price changes on the macroeconomy. We argue that the relation between oil prices and industrial production is not clear-cut. There are periods and frequencies where the causality runs from one variable to the other and vice-versa, justifying some instability in the empirical evidence about the macroeconomic effects of oil price shocks. We also show that the volatility of both the inflation rate and the industrial output growth rate started to decrease in the decades of 1950 and 1960.

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a b s t r a c tWe assess the relation between the yield curve and the macroeconomy in the U.S. between 1961 and 2011. We add to the standard parametric macro-finance models, as we uncover evidence simultaneously on the time and frequency domains. We model the shape of the yield curve by latent factors corresponding to its level, slope and curvature. The macroeconomic variables measure real activity, inflation and monetary policy. The tools of wavelet analysis, the set of variables and the length of the sample allow for a thorough appraisal of the time-variation in the direction, intensity, synchronization and periodicity of the yield curve-macroeconomy relation.

" "E Es st ti im ma at ti in ng g t th he e T Ta ay yl lo or r R Ru ul le e i in n t th he e T Ti im me e-F Fr re eq qu ue en nc cy y D Do om ma ai in n" " L Lu uí ís s A Ag gu ui ia ar r-C Co on nr ra ar ri ia a M Ma an nu ue el l M M.. F F.. M Ma ar rt ti in ns s M Ma ar ri ia a J Jo oa an na a S So oa ar re es s NIPE WP 18/ 2014 " "E Es st ti im ma at ti in ng g t th he e T Ta ay yl lo or r R Ru ul le e i in n t th he e T Ti im me e-F Fr re eq qu ue en nc cy y D Do om ma ai in n" " L Lu uí ís s A Ag gu ui ia ar r-C Co on nr ra ar ri ia a M Ma an nu ue el l M M.. F F.. M Ma ar rt ti in ns s M Ma ar ri ia a J Jo oa an na a S So oa ar re es s N NI IP PE E *

The literature on quaternionic polynomials and, in particular, on methods for determining and classifying their zero-sets, is fast developing and reveals a growing interest on this subject. In contrast, polynomials defined over the algebra of coquaternions have received very little attention from researchers. One of the few exceptions is the very recent paper by Janovská and Opfer [Electronic Transactions on Numerical Analysis, Volume 46, pp. 55-70, 2017], where, among other results, we can find a first attempt to prove that a unilateral coquaternionic polynomial of degree n has, at most, n(2n − 1) zeros.In this paper we present a full proof of the referred result, using a totally different and, from our point of view, much simpler approach. Also, we give a complete characterization of the zero-sets of such polynomials and present a new result giving conditions which guarantee the existence of a special type of zeros. An algorithm to compute and classify all the zeros of a coquaternionic polynomial is proposed and several numerical examples are carefully constructed.

Po ol li it ti ic cs s: : W Wa av ve el le et t A An na al ly ys si is s o of f P Po ol li it ti ic ca al l T Ti im me e--S Se er ri ie es s" " L Lu uí ís s A Ag gu ui ia ar r--C Co on nr ra ar ri ia a P Pe ed dr ro o C C. . M Ma ag ga al lh hã ãe es s M Ma ar ri ia a J Jo oa an na a S So oa ar re es s NIPE WP 25/ 2011 " "C Cy yc cl le es s i in n P Po ol li it ti ic cs s: : W Wa av ve el le et t A An na al ly ys si is s o of f P Po ol li it ti ic ca al l T Ti im me e--S Se er ri ie es s" " L Lu uí ís s A Ag gu ui ia ar r--C Co on nr ra ar ri ia a P Pe ed dr ro o C C. . M Ma ag ga al lh hã ãe es s M Ma ar ri ia a J Jo oa an na a S So oa ar re es s Abstract Spectral analysis and ARMA models have been the most common weapons of choice for the detection of cycles in political time-series. Controversies about cycles, however, tend to revolve about an issue that both techniques are badly equipped to address: the possibility of irregular cycles without fixed periodicity throughout the entire time-series. This has led to two main consequences. On the one hand, proponents of cyclical theories have often dismissed established statistical techniques. On the other hand, proponents of established techniques have dismissed the possibility of cycles without fixed periodicity. Wavelets allow the detection of transient and coexisting cycles and structural breaks in periodicity. In this paper, we present the tools of wavelet analysis and apply them to the study to two lingering puzzles in the political science literature: the existence to cycles in election returns in the United States and in the severity of major power wars. 1 3 The paper proceeds as follows. The next section uses empirical and constructed numerical examples to illustrate the use of ARMA models and spectral analysis in the detection of cyclesand their shared inability to deal with irregular and transient cycles. We will show that this inability does not extend to wavelet analysis. Following that section, we present the three basic tools in wavelet analysis: the wavelet power spectrum, cross-wavelets, and phase-differences.Equiped with these tools, we will develop two applications to real world political time-series data. The first is the analysis of presidential and congressional election returns in the UnitedStates from 1854 to 2008. The second application of wavelet analysis concerns the severity of major power wars from 1495 to 1975, using well-established data in the literature (Levy 1983;Goldstein 1988). In the last section, we sketch a research agenda in political science where the use of wavelet analysis may shed light on important empirical and theoretical discussions. In the appendix, we describe how to computationaly implement the wavelet tools.