Semiparametric Modeling of Seasonal Time Series

Often, in practice, one may regard an observed time series as being composed of a function that is smooth over years, with additive seasonal effects. As a modification, we formulate a particular multiplicative model that expresses the observed data as a yearly trend function with additive amplitude-modulated seasonal factors. Without smoothness restrictions on the yearly trend and modulation components, the least squares solutions for the seasonal components are shown to be proportional to the eigenvector corresponding to the maximum eigen value of the within-season covariance matrix. If the trend and seasonal modulations are modeled as smooth splines, we give the comparable estimators for the smooth functions and the seasonal factors. We show consistency for the trend, modulation and seasonal factors as well as asymptotic normality for the seasonal estimates. Model selection, fitting and forecasting are considered for a quarterly earnings series that exhibits extreme nonlinear and nonstationary behavior. We compare the results with those obtained using a competing nonstationary multiplicative ARIMA model