A model for seasonal changes in time series regression relationships: With an application in psychiatry

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Abstract
This paper proposes a two‐stage linear model for the regression relationships between a time series and a set of time‐dependent covariates when the relationships change according to a seasonal pattern. The first stage consists of a piecewise constant autoregression model for these relationships. The second stage models the piecewise coefficients with harmonic equations to examine for seasonal trends. This approach models the seasonal trends in time series regression relationships in the presence of non‐constant autoregression, non‐constant residual variance, and stochastic variation in the regression effects across time. Seasonal affective disorder )SAD( time series data illustrate the methodology. I model the seasonally changing regression relationships between the change in daily sunlight )from the previous day( and depressive mood and between the change in daily temperature and depressive mood for a SAD patient, and I employ an empirical Bayes method to obtain efficient month‐specific estimates for these relationships.