Forecasting Compositional Time Series: A State Space Approach

A framework for the forecasting of composite time series, such as market shares, is proposed. Based on Gaussian multi-series innovations state space models, it relies on the log-ratio function to transform the observed shares (proportions) onto the real line. The models possess an unrestricted covariance matrix, but also have certain structural elements that are common to all series, which is proved to be both necessary and sufficient to ensure that the predictions of shares are invariant to the choice of base series. The framework includes a computationally efficient maximum likelihood approach to estimation, relying on exponential smoothing methods, which can be adapted to handle series that start late or finish early (new or withdrawn products). Simulated joint prediction distributions provide approximations to the required prediction distributions of individual shares and the associated quantities of interest. The approach is illustrated on US automobile market share data for the period 1961–2013.