Forecasting daily time series using periodic unobserved components time series models

A periodic time series analysis is explored in the context of unobserved components time series models that include stochastic time functions for trend, seasonal and irregular effects. Periodic time series models allow dynamic characteristics (autocovariances) to depend on the period of the year, month, week or day. In the standard multivariate approach one can interpret a periodic time series analysis as a simultaneous treatment of typically yearly time series where each series is related to a particular season. Here, the periodic analysis applies to a vector of monthly time series related to each day of the month. Particular focus is on the forecasting performance and therefore on the underlying periodic forecast function, defined by the in-sample observation weights for producing (multi-step) forecasts. These weight patterns facilitate the interpretation of periodic model extensions. A statistical state space approach is used to estimate the model and allows for irregularly spaced observations in daily time series. Recent algorithms are adopted for the computation of observation weights for forecasting based on state space models with regressor variables. The methodology is illustrated for daily Dutch tax revenues that appear to have periodic dynamic properties. The dimension of our periodic unobserved components model is relatively large as we allow each element (day) of the vector of monthly time series to have a changing seasonal pattern. Nevertheless, even with only five years of data we find that the increased periodic flexibility can help in out-of-sample forecasting for two extra years of data.