Business Cycles in the Contemporary World pp 15-32 | Cite as

# The Analysis of Cyclical Dynamics

## Abstract

According to the preceding argumentation an important first step in a project to investigate economic fluctuations is the thorough determination of stylized facts. When dealing with business cycles, it seems natural to analyze the time series at stake first by means of a general analysis of variance to quantitatively assess their relative volatility and contribution to the constitution of some aggregate series and second to investigate their cyclical properties in the frequency domain by means of spectral analysis. Given that the latter method seems an ideal tool for studying cyclical phenomena, one might wonder why it has not been used more in economics. In my opinion the answer lies in four problems that spectral analysis faces when confronted with economic data:

First, it can be applied only to stationary time series. Because most of economic variables contain a trend component, this must be effectively removed. Failure to do this leads to the “typical spectral shape,” reported by Granger (1966), where most of the mass of the spectrum is skewed in a “hump-shaped” fashion to the low frequency range. The following section suggests adequate strategies as well as recently established detrending technologies to overcome this problem.

## Keywords

Spectral Density Function Short Time Series Relative Volatility Schwarz Criterion Aggregate Series## Preview

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## Notes

- 1.Van den Bos (1971) noted — already thirty years ago: “In the classical approach to calculation of power spectra the sample autocorrelation function is multiplied by some lag window and then transformed. The window is applied to avoid leakage from other parts of the spectrum. On the other hand, the window limits the spectral resolution,” p. 493.Google Scholar
- 2.See the recent contributions of Croux, Forni and Reichlin (1999), Forni and Reichlin (1999), Forni et al. (1999) and Forni et al. (2000).Google Scholar
- 3.Another recent reference in the unit root discussion is Diebold and Senhadji (1996) who note: “There is no doubt that unit-root tests do suffer from low power in many situations of interest … Our results suggest that U.S. aggregate output is not likely to be difference-stationary: the dominant autoregressive root is close to, but less than, unity … There is simply no substitute for serious, case-by-case, analysis,” p. 1297.Google Scholar
- 4.In my opinion the most intuitive outline of the HP-filter’s logic is given in Prescott (1986). A survey of the statistical justifications of the HP-filter is provided by Reeves et al. (1996). For a detailed specification of the ‘digital’ or ‘bandpass’ filters, BK-and MBK-filter, the exact differences in their derivation and their effects on series with different frequencies (i.e. annual, quarterly and monthly), the reader is referred to the thorough presentation and discussion in Woitek (1998). A comparison of unfiltered, HP-and BK-filtered series, having a peak in their spectrum at business cycle frequencies, can be found in Guay and St-Amant (1997).Google Scholar
- 5.Obviously, the gain function peaks for the HP (100)-filter in the frequency range corresponding to period lengths of about 15 years. The BK-and MBK-filter’s gain functions take on values below or equal to the HP-filter’s one and peak at a periodicity of about 7 years. The not displayed gain of the logD-detrending procedure would equal a sigmoid close to linear function peaking at a frequency corresponding to a period length beyond two years, i.e. > π, see Woitek (1996, Appendix A).Google Scholar
- 6.The fact that the conventional BK-filter does not necessarily prevent the time series analyst from eventually producing spurious cycles has been stressed recently in a simulation study by Schenk-Hoppé (2000). In contrast to this claim, Den Haan and Sumner (2001) show that the BK-filter isolates the desired set of frequencies not only when the series are stationary but also when they are first or second-order integrated processes.Google Scholar
- 7.A survey of similar volatility measures is given by McKenzie (1999).Google Scholar
- 8.Besides these conventional terms, Marple (1987) also calls it
*autocorrelation sequence*.Google Scholar - 9.The argumentation and formulae outlined in the present subsection widely follow Koopmans (1995, section 3.3).Google Scholar
- 10.Though the matter of choosing the right method of estimation played a decisive role in earlier methodological work on ME-(or AR) spectra conducted at SEMECON, todays computer power allows all forms of estimation methods ranging from OLS to full information maximum likelihood.Google Scholar
- 11.In the literature of the engineering sciences AR models are sometimes referred to as
*all-pole(s) models*. Therefore the ME-spectral estimation method is sometimes denoted all-pole(s) method; see for example, van den Bos (1971) or more recently, Press et al. (1992, section 13.7).Google Scholar - 12.Similarly. Ulrych and Bishop (1975) call these set-to-zero values “data that lie outside of the parameter space,” p. 55. This is also the reason why these methods sometimes are termed
*direct methods or all-zero models*: see Press et al. (1992). p. 573.Google Scholar - 13.An excellent graphical representation of this argument is given in Marple (1987). Figure 7.4. p. 201.Google Scholar
- 14.The derivation of the rate is rather complex, implying
*inter alia*the Weierstrass approximation theorem and the exploitation of several properties of the Toeplitz matrix, so that it is here just presented. The interested reader is referred to the more thorough outline and derivation in Süssmuth (2002b).Google Scholar - 15.The following argumentation summarizes main parts of Granger and Newbold (1986, section 1.5 — 1.7).Google Scholar
- 16.A summary of the elementary mathematics applied in the argumentation of this section, can be found for example in Hamilton (1994, Appendix A).Google Scholar
- 17.The corresponding null would be a mod = 0, implying that the cyclic component plays no role.Google Scholar
- 18.See for example Maddala and Kim (1998) for a survey of the recent literature on cointegration.Google Scholar
- 19.Namely, the cospectrum, cf. the immediately following presentation.Google Scholar
- 20.Which Croux, Forni and Reichlin (2000) entitle “cohesion.”Google Scholar
- 21.Reiter (1995) accurately summarizes: “The bivariate spectral parameters are ideal tools for investigating the comovements of series over the business cycle. For example, a good leading indicator is a series having a positive lead relative to the reference series, which can be seen from the phase spectrum, and a high coherency.” p. 25.Google Scholar