The Seasons They Are A-Changin': A Century of Definitions and a Way Forward
(with Gary Cornwall)
What is seasonality? To date, there is not yet a consensus definition, as over a dozen unique definitions of seasonality can be found in the economics and statistics literature. This lack of consistency complicates the identification of, and adjustment for, seasonal effects in economic time series. In this paper, we propose a new definition of seasonality as well as how to detect and adjust for it, ultimately aiming to build consensus around a more unified approach. We first review the literature and identify common themes across definitions with an eye towards formulating a foundational definition – expressed in both plain language and mathematical terms – that can be used by academics, data providers, policymakers, and practitioners alike. We then propose to classify a series as seasonal if, among all its possible repeating cycles, the ones associated with the fundamental periodicity are demonstrably larger than the others. In other words, we define seasonality as a measure of relative peak dominance in the spectral density function, following Granger (1978). Next, we develop a new methodology for detecting seasonality and performing seasonal adjustment that follows directly from our definition and leverages properties of the sample periodogram, the empirical counterpart to the spectral density function. We illustrate the performance of our test statistic through simulation studies and find that it is well-sized and has good power. Lastly, we introduce Stochastic Spectral Imputation (SSI), an adjustment procedure that outperforms standard methods currently used by national statistical offices when applied to prominent macroeconomic time series.