The Continuous Wavelet Transform: A Primer

Wavelet analysis is becoming more popular in the Economics discipline. Until recently, most works have made use of tools associated with the Discrete Wavelet Transform. However, after 2005, there has been a growing body of work in Economics and Finance that makes use of the Continuous Wavelet Transform tools. In this article, we give a self-contained summary on the most relevant theoretical results associated with the Continuous Wavelet Transform, the Cross-Wavelet Transform, the Wavelet Coherency and the Wavelet Phase-Difference. We describe how the transforms are usually implemented in practice and provide some examples. We also introduce the Economists to a new class of analytic wavelets, the Generalized Morse Wavelets, which have some desirable properties and provide an alternative to the Morlet Wavelet. Finally, we provide a user friendly toolbox which will allow any researcher to replicate our results and to use it in his/her own research.Economic cycles; ContinuousWavelet Transform, Cross-Wavelet Transform, Wavelet Coherency, Wavelet Phase-Difference; The Great Moderation.

Using cross-wavelets to decompose the time-frequency relation between oil and the macroeconomy

A large body of empirical literature has suggested that oil price shocks have an important effect on economic activity. But in most of the literature the analysis is exclusively done in the time domain. However, interesting relations exist at different frequencies. We use (cross) wavelet analysis to uncover some of these relations, estimating the spectral characteristics of the time-series as a function of time. Our analysis suggests that the volatility of both the inflation rate and the output growth rate started to decrease in the decades of 1950 and 1960, suggesting that the great moderation started then,but that it was temporarily interrupted due to the oils crisis of the 1970s, whose effects extend until the mid 1980s. We also show that while at business cycle frequencies oil prices lead industrial production, in the very long run production increases lead oil price increases. The exception to this long-run relation occurred between the mid 1970s and mid 1980s. Our analysis also suggests that monetary policy became much more eficient after 1980 to deal with the inflationary pressures of oil shocks.Business cycles, time-frequency analysis, non-stationary time series, wavelets, cross wavelets, wavelet coherency.

Using Wavelets to decompose time-frequency economic relations

Economic agents simultaneously operate at different horizons. Many economic processes are the result of the actions of several agents with different term objectives. Therefore, economic time-series is a combination of components operating on different frequencies. Several questions about the data are connected to the understanding of the time-series behavior at different frequencies. While Fourier analysis is not appropriate to study the cyclical nature of economic time-series, because these are rarely stationary, wavelet analysis performs the estimation of the spectral characteristics of a time-series as a function of time. In spite of all its advantages, wavelets are hardly ever used in economics. The purpose of this paper is to show that cross wavelet analysis can be used to directly study the interactions different time-series in the time-frequency domain. We use wavelets to analyze the impact of interest rate price changes on some macroeconomic variables: Industrial Production, Inflation and the monetary aggregates M1 and M2. Specifically, three tools are utilized: the wavelet power spectrum, wavelet coherency and wavelet phase-difference. These instruments illustrate how the use of wavelets may help to unravel economic time-frequency relations that would otherwise remain hidden.Monetary policy, time-frequency analysis, non-stationary time series, wavelets, cross wavelets, wavelet coherency.

Cycles in Politics: Wavelet Analysis of Political Time-Series

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.

Synchronism in Electoral Cycles: How United are the United States?

The role of national, sectional, state, and local forces in driving electoral outcomes in the United States has remained a matter of considerable indeterminacy in the American politics literature. In what concerns House elections, different approaches and methods have yielded widely divergent results. In what concerns presidential elections, considerable doubts remain about the timing and the plausible causes of a long-term trend towards homogeneity. In this paper, we take a new look at the nationalization of politics in the United States. We are particularly interested in the dynamic nationalization in presidential elections, i.e., the extent to which swings and shifts from one election to the next have been similar across states and whether or not that similarity has increased through time. We treat this problem as one of similarity or dissimilarity — and convergence or divergence of — electoral cycles, and use wavelets analysis in order to ascertain the degree to which the national and state election cycles have been synchronized and the degree to which that synchronization has increased or decreased. We determine, first, the states where electoral change has been more in sync with the national cycle and clusters of states defined in terms of the mutual synchronization of their own electoral cycles. Second, we analyze how the degree of synchronization of electoral cycles in the states has changed through time, answering questions as to when, to what extent, and where has the tendency towards a “universality of political trends” in presidential elections been more strongly felt. We present evidence strongly in favor of an increase in the dynamic nationalization of presidential elections taking place in the 1950s, showing that alternative interpretations concerning the historical turning point in this respect are not supported by empirical evidence.

The yield curve and the macro-economy across time and frequencies

This paper assesses the relation between the yield curve and the main macroeconomic variables in the U.S. between early 1960s and 2009 across time and frequencies, using wavelet analyses. The shape of the yield curve is modelled by latent factors corresponding to its level, slope and curvature, estimated by maximum likelihood with the Kalman filter. The macroeconomic variables measure econmic activity, unemployment, inflation and the fed funds rate. The cross wavelet tools employed - coherency and phase difference - , the set of variables and the length of the sample, allow for a thorough appraisal of the time- variation and structural breaks in the direction,intensity,synchronization and periodicity of the relation between the yield curve and the macro-economy. Our evidence establishes a number of new stylized facts on the yield curve-macro relation; and sheds light on several results found in the literature, which could not have been achieved with analyses conducted strictly in the time-domain(as most of the literature)or purely in the frequency-domain.Macro-finance; Yield curve; Kalman filter; Continuous wavelet transform;Wavelet coherency;Phase-difference

The yield curve and the macro-economy across time and frequencies

This paper assesses the relation between the yield curve and the main macroeconomic variables in the U.S. between early 1960s and 2010 across time and frequencies, using wavelet analyses. The shape of the yield curve is modelled by latent factors corresponding to its level, slope and curvature, estimated by maximum likelihood with the Kalman filter. The macroeconomic variables measure economic activity, unemployment, inflation and the fed funds rate. The cross wavelet tools employed — coherency and phase difference —, the set of variables and the length of the sample, allow for a thorough appraisal of the timevariation and structural breaks in the direction, intensity, synchronization and periodicity of the relation between the yield curve and the macro-economy. Our evidence establishes a number of new stylized facts on the yield curve-macro relation; and sheds light on several results found in the literature, which could not have been achieved with analyses conducted strictly in the time-domain (as most of the literature) or purely in the frequency-domain.Macro-finance; Yield curve; Kalman filter; Continuous wavelet transform; Wavelet coherency; Phase-difference.

Synchronization of Economic Sentiment Cycles in the Euro Area: a time-frequency analysis

We use wavelet tools and Economic Sentiment Indicators to study the synchronization of economic cycles in the Euro Area. We assess the time-varying and frequency-varying pattern of business cycles synchronization in the Area and test the impact of the creation of the European Monetary Union in 1999. Among several results, we find that (a) the EMU is associated with a significant increase in synchronization of economic sentiment in the Euro Area; (b) the hard-peg of its currency to the Euro led to a comparable synchronization of Denmark's economic sentiment after 1999, differently from what happened in the case of the UK.Business cycle synchronization; Economic sentiment; Euro Area; Continuous wavelet transform; Wavelet coherency; Wavelet distance; Phase-difference.