The nationalization of electoral cycles in the United States: a wavelet analysis

To be sure, this diagnostic is less clear in what concerns congressional elections, considering the uncertainty about the actual size of incumbency effects, “presidential coattails”, or the causes behind the emergence of quality challengers in congressional elections. See, for example, Stokes (1967), Claggett et al. (1984), Kawato (1987), Brady et al. (2000), and Morgenstern et al. (2009). For a recent discussion of national forces in congressional elections, see Burden and Wichowsky (2010).

See also Archer and Shelley (1986), Shelley et al. (1996), and Heppen (2003).

It should be observed that there are different conventions in the definition of Fourier transform; with the above convention, ω is an angular (or radian) frequency. The relation to the more common Fourier frequency f is given by ω=2πf.

This means that \(\Vert\psi(t)\Vert ^{2}=\int_{-\infty}^{\infty}|\psi(t)|^{2}dt<\infty\). Usually, ψ(t) is normalized so that ∥ψ(t)∥=1.

There is also the Discrete Wavelet Transform (DWT), with which we will not deal. For an introduction to the DWT, the reader is referred to Crowley (2007).

The conjugate of a complex number, a+bi, is simply a−bi. If a number is real, then the conjugate is the number itself. Therefore, this distinction is relevant only in the case of a complex wavelet.

Although, strictly speaking, the Morlet function defined by (3) is not a “true” wavelet, since it has no zero mean, in practice, it can still be considered as a wavelet, provided that the parameter ω ₀ is sufficiently large (e.g., ω ₀>5). The case ω ₀=6 is the most common choice. This is so essentially because for this value, the wavelet can be considered to be a function of angular frequency ω ₀=6 (normal frequency f=6/2π≈1) and, when scaled by s, it becomes a function of frequency f≈1/s. Therefore, for this parameter choice conversion from wavelet scales (s) to frequencies is almost immediate.

Let μ _t and σ _t and μ _ω and σ _ω be the mean and standard deviation of the probability density functions defined by |ψ(t)|²/∥ψ(t)∥ and by |Ψ(ω)|²/∥|Ψ(ω)|∥², respectively, where Ψ(ω) is the Fourier transform of ψ(t). The most significant values of ψ(t) and Ψ(ω) are attained in the so-called Heisenberg box: H _ψ =[μ _t −σ _t ,μ _t +σ _t ]×[μ _ω −σ _ω ,μ _ω −σ _ω ]. We say that ψ is localized around the point (μ _t ,μ _ω ) with uncertainty given by σ _t σ _ω . The Heisenberg uncertainty principle establishes that σ _t σ _ω ≥1/2. Moreover, for the Morlet wavelet, one has that \(\sigma_{{t}}=\sigma_{\omega }=\frac{1}{\sqrt{2}}\), meaning that accuracy in time and in frequency are similar.

We have written a wavelet toolbox in Matlab. The toolbox, all of the data and codes can be at https://​sites.​google.​com/​site/​aguiarconraria/​joanasoares-wavelets. All of the data and codes used can be found there. The toolbox is quite user-friendly and comes with several examples and a quick user’s guide.

To perform significance tests of wavelet measures, we fit an AR(1) model and construct new samples by drawing errors from a Gaussian distribution with a variance equal to that of the estimated error terms. For each time series (or pair of time series) we perform the exercise 5000 times, and then extract the critical values.

As with other types of transforms, the CWT applied to a finite length time series suffers from border distortions due to the fact that the values of the transform at the beginning and the end of the time series involve missing values of the series which are artificially prescribed. In this area, the results have to be interpreted carefully.

All data used in this paper were provided by the American Presidency Project at UC Santa Barbara (http://​www.​presidency.​ucsb.​edu).

For a generalization to several variables, see Aguiar-Conraria and Soares (2013, forthcoming).

Smoothing is necessary; otherwise, just as with Fourier coherency, it would be identically one at all scales and times. Time and scale smoothing can be achieved, e.g., by convolution with appropriate windows.

In Sect. 5, when these results are discussed more deeply, we also discuss our estimation choices.

Note that we are comparing the wavelet transforms, and not the power spectrum. Therefore we are preserving the information given by the imaginary part.

For the mathematically oriented researcher: consider two complex vectors, a and b. Define the Hermitian inner product, 〈a,b〉 _C =a ^H b and the corresponding norm, \(\Vert{a}\Vert=\sqrt{\langle{a},{a}\rangle _{C}}\). The Hermitian angle, Θ _H , between a and b is obtained by the formula cos(Θ _H )=|〈a,b〉 _C |/(∥a∥∥b∥).

The only excluded states are Alaska, Arizona, Hawaii, New Mexico, and Oklahoma, as they only achieved statehood after 1896.

The only exception is for the 1912 presidential run. In that election, Theodore Roosevelt failed to receive the Republican nomination. Roosevelt created the Progressive party and ran for president, dividing the Republican electorate. For this individual election, we compare the votes of the Democratic candidate (Woodrow Wilson) with the total of the votes of the other two major contenders (William Taft and Roosevelt). A more recent case of a third party candidate receiving a large share of the vote (19 %) is Ross Perot in 1992. However, extant research shows that, rather than affecting one particular party, Perot voters would have been split almost evenly between the candidates of the two major parties (Alvarez and Nagler 1995). More generally, Merrill et al. (2008) show that the detection of cycles in US presidential elections is mostly insensitive to different ways of handling third-party candidates.

The online appendix, containing Table A1 and Figs. A1 and A2, is available at: http://​sites.​google.​com/​site/​aguiarconraria/​Online_​Appendix_​PUCH.​pdf.

This is so because the phase information is obtained from the imaginary part of a complex number. However, the wavelet power spectrum is the square of an absolute value and the absolute value transforms a complex number into a real number.

By “national,” we mean the aggregate electoral result of the states included in our sample. When we compute the distance between each state and the national aggregate defined in this way, we exclude that state from the aggregate.

With the exception of the largest cluster, which, as we have seen, includes all of the states that have an electoral cycle significantly synchronized with the national cycle.

It would be too fastidious to include the coherencies and phase-differences of 45 states here. Figure A2 (in the Appendix) shows the results for every state.

Similar conclusions would apply to other states in the first “core” group identified in the cluster analysis, like Michigan, Pennsylvania, Washington, New Jersey, Minnesota, and others.