Abstract
Business cycles are defined as fluctuations of economic activity between 2 and 8 years. Since market economies are complex and evolving systems subject to internal changes and to exogenous shocks, the statistical properties of economic fluctuations are likely to change over time. Frequency-based phenomena with time-varying features are better studied using time-frequency methods. Wavelet methods are preferable to Fourier analysis because of their optimal time-frequency resolution properties and ability to address the nonstationary features of economic fluctuations. The usefulness of wavelet techniques for business cycle analysis is illustrated by the application of discrete and continuous wavelet tools to certain aspects of the business cycle: output volatility moderation and the cyclical behavior of monetary aggregates.
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Notes
- 1.
The only exceptions are the early 1990s recession, when the growth rate decreased by 2%, and the Great Recession in 2009.
- 2.
Whereas some scholars generally believed that business activity was subject to cycles of 7–10 years [see Schumpeter 1939; Hansen 1951], the strong empirical evidence for the United States provided by Burns and Mitchell [1946] and later by NBER and CIBCR studies for several developed countries (France, Germany, and the United Kingdom) was supportive of the main business cycle being a relatively short or “minor” cycle, generally varying from 3 to 4 years (i.e., a Kitchin cycle). Indeed, the observed timing of fluctuations of the early business cycles in the NBER chronology conformed well to the cycle length popularized as the Kitchin cycle by Schumpeter [1939]. Kitchin [1923] distinguishes between minor cycles averaging 3 and a half years and major cycles (or trade cycles) which are an aggregation of two or three minor cycles. Similar results in terms of average duration for economic fluctuations are provided by Hansen [1951], who identified 27 minor cycles between 1837 and 1937 for the United States. All of these results were supportive of the rejection of the notion of a regular pattern of economic activity of roughly 7–10 years’ duration [Matthews 1959].
- 3.
Time domain analysis tells us everything about the value of a signal at a specific location, but little about its frequency content. Frequency domain analysis tells us everything about the frequency content of a signal, but not when these frequencies occur.
- 4.
Frequency domain methods introduced in the early 1960s [e.g., Hannan 1963; Nerlove 1964; Granger and Hatanaka 1964] were used to uncover key characteristics of economic time series such as the typical spectral shape of an economic variable [Granger 1966], and to run regression analysis in the frequency domain, for example, band spectrum regressions [Engle 1974].
- 5.
After the initial studies undertaken by Ramsey and his coauthors, applications in economics and finance have increased significantly in recent years [Gallegati and Semmler 2014].
- 6.
- 7.
Dividing by 1/√2j ensures that the energy of the wavelet family will remain the same at different scales.
- 8.
The translation parameter is matched to the scale parameter in the sense that as the wavelet basis functions broaden, their translation steps become correspondingly larger.
- 9.
Better business practices (“good practices”), such as better inventory management techniques, more sophisticated financial markets, or expanding international trade flows, are also considered potential explanations for the decline in output volatility. However, since the decline in variance reflecting better business practices is expected to occur primarily at relatively high frequencies, we are forced to exclude this explanation from our analysis.
- 10.
The boundary method used in the wavelet decomposition is the reflection boundary condition. In this method, an extension is made by pasting a reflected (time-reversed) version of the original signal of length T at its end and then applying a periodic boundary condition on the first T elements of the reconstructed signal (and omitting the remaining T elements).
- 11.
The 4-level decomposition extracts the smooth component S4 (not shown in Table 1), which captures oscillations with a period longer than 32 years that correspond to the very low-frequency components of the signal.
- 12.
The only exception is the temporary increase of output volatility during the oil price shocks of the 1970s.
- 13.
Real Business Cycle models are a class of macroeconomic models that attribute fluctuations in business cycles to real shocks such as technological or productivity shocks, rather than to nominal or monetary shocks.
- 14.
Nominal money growth tends to fall in the late stages of an expansionary phase as banks become increasingly restrained in their ability to create deposits by the availability of reserves. Real money balances would typically decline before an economic downturn, as the increase in prices usually picks up late in the cycle.
- 15.
Dynamic Stochastic General Equilibrium models represent the current macroeconomic framework adopted by Central Banks for their economic and monetary policy analyses.
- 16.
The continuous and discrete scale parameters s and j are linked by the following relationship: s = s0j, so that when the computation is done octave by octave, that is, s0 = 2, the scaling parameter in the discrete case is s = 2j. When the Nyquist sampling rule is used the link between the translation parameters u and k is: u = k u0 s0j, so that when u0 = 1 and s0 = 2 the translation parameter in the discrete case is u = k 2j.
- 17.
All practical implementations of the CWT use a near-continuous discretization.
- 18.
The series real M2 is an inflation-adjusted version of the M2 money supply, which includes currency, demand deposits, other checkable deposits, travelers’ checks, savings deposits, small denomination time deposits, and balances in money market mutual funds. The TCB-CEI is the composite coincident indicator developed by the Conference Board for measuring and dating business cycles in the aggregate economy using Employees on Non-Agricultural Payrolls, Index of Industrial Production, Real Personal Income less Transfer Payments, and Real Manufacturing and Trade Sales as individual components [The Conference Board 2000]. Monthly data from the TCB dataset are used for the period 1959:1–2017:4.
- 19.
Wavelet coherence coefficient values range between 0 and 1, so that values close to zero indicate weak correlation at a given frequency, while values close to one imply strong correlation between the two series considered.
- 20.
Nominal money growth tends to fall in the late stages of an expansionary phase as banks become increasingly restrained in their ability to create deposits by the availability of reserves. Real money balances would typically decline before an economic downturn, as the increase in prices usually picks up late in the cycle [Levanon et al. 2011].
- 21.
- 22.
Friedman and Kuttner [1992] were among the first to document that by the early 1990s the relationship between M2 and GDP had weakened.
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Gallegati, M. (2023). Timescale Methods in Economics: Wavelet Analysis of Business Cycle Fluctuations. In: Booß-Bavnbek, B., Hesselbjerg Christensen, J., Richardson, K., Vallès Codina, O. (eds) Multiplicity of Time Scales in Complex Systems. Mathematics Online First Collections. Springer, Cham. https://doi.org/10.1007/16618_2022_40
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