Periodicity

Periodic variation is a trademark of time that has long mystified ordinary people, inspired philosophers, and frustrated mathematicians. Through thousands of years, famines have alternated with prosperity, war with peace, ice ages with warming trends, among others, in a way that hinted at the existence of cycles. A cycle is a pattern that is “continually repeated with little variation in a periodic and fairly regular fashion” (Granger, 1989, p. 93). With the help of trigonometric functions, the cyclical component of a time series ztmay be expressed as follows:

where A represents the amplitude, f the frequency, and ϕ the initial phase of the periodic variation, with ut denoting white-noise error. The amplitude indicates how widely the series swings during a cycle (the distance from peak to valley), whereas the frequency f measures the speed of the cyclical movement:

where P refers to the period of a cycle, the number of time units it takes for a cycle to be completed. Hence a 4-year cycle would operate at a frequency of 0.25 (cycles per unit time). The shortest cycle that can be detected with any given data would have a period of two time units. It would correspond to a frequency of 0.5, which is the fastest possible one. Any cycle that is done in fewer than two time units is not detectable. The longer the cycle—that is, the more time units it takes—the lower the frequency. A time series without a cycle would register a frequency of zero, which corresponds to an infinite period.

SPECTRAL ANALYSIS provides the major tool for identifying cyclical components in time series (Gottman, 1981; Jenkins & Watts, 1968). The typical plot of a spectral density function features spectral density along the vertical axis and frequency (ranging from 0 to 0.5) along the horizontal axis. A time series with a 4-year cycle (a year being the time unit of the series) would show a density peak at a frequency of 1/4 or 0.25. The problem is that few time series of interest are generated in such perfectly cyclical fashion that their spectral density functions would be able to tell us so.

Aside from strictly seasonal phenomena, many apparent cycles do not repeat themselves with constant regularity. Economic recessions, for example, do not occur every 7 years. Nor do they hurt the same each time. Although boom and bust still alternate, the period of this “cycle” is not fixed. Nor is its “amplitude.” That lack of constancy undermines the utility of deterministic models such as sine waves. As an alternative, Yule (1927/1971) proposed the use of probabilistic models for time series with cycles whose period and amplitude are irregular. His pioneering idea was to fashion a second-order autoregressive model to mimic the periodic, albeit irregular, fluctuations of sunspot observations over a span of 175 years. Yule's AR(2) model had the following form:

With only two parameters and lags going no further than two, this model succeeded in estimating the periodicity of the sunspot fluctuations (10.6 years). The model also provided estimates of the random disturbances that continually affected the amplitude and phase of those fluctuations. Box-Jenkins modeling has extended the use of probability models for identifying and estimating periodic components in time series, with a special set of models for seasonal fluctuations.

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