Temporal and spatial Taylor's law: application to Japanese subnational mortality rates

Taylor's law is a widely observed empirical pattern that relates the variances to the means of population densities. We present four extensions of the classical Taylor's law (TL): (1) a cubic extension of the linear TL describes the mean–variance relationship of human mortality at subnational levels well; (2) in a time series, long-run variance measures not only variance but also autocovariance, and it is a more suitable measure than variance alone to capture temporal/spatial correlation; (3) an extension of the classical equally weighted spatial variance takes account of synchrony and proximity; (4) robust linear regression estimators of TL parameters reduce vulnerability to outliers. Applying the proposed methods to age-specific Japanese subnational death rates from 1975 to 2018, we study temporal and spatial variations, compare different coefficient estimators, and interpret the implications. We apply a clustering algorithm to the estimated TL coefficients and find that cluster memberships are strongly related to prefectural gross domestic product. The time series of spatial TL coefficients has a decreasing trend that confirms the narrowing gap between rural and urban mortality in Japan.