Natural Fertility, Population Cycles and the Spectral Analysis of Births and Marriages

Birth and marriage series are viewed as simultaneous mixed auto-regressive and moving average processes with known coefficients, subject to random rates of nuptiality and marital fertility. The theoretical spectra and cross-spectrum are calculated under various hypotheses and compared to empirical estimates for historical birth and marriage series. We reject the “natural fertility” hypothesis and conclude that nuptiality and marital fertility were equally variable and very highly correlated. We also use the model to explain how the age structure of reproduction and nuptiality and duration structure of marital fertility act as filters which transform white noise variation into 30-year cycles in series of births and marriages.