Forecasting daily time series using periodic unobserved components time series models

We explore a periodic analysis in the context of unobserved components time series models that decompose time series into components of interest such as trend, seasonal and irregular. Periodic time series models allow dynamic characteristics such as autocovariances to depend on the period of the year, month, week or day. In the standard multivariate approach one can interpret periodic time series modelling as a simultaneous analysis of a set of, traditionally, yearly time series where each series is related to a particular season, and the time index is in years. The periodic analysis in this paper applies to a monthly vector time series related to each day of the month. Particular focus is on 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. We take a statistical state space approach to estimate our model. In this way we can identify stochastic unobserved components and we can deal with irregularly spaced daily time series. We extend existing algorithms to compute observation weights for forecasting based on state space models with regressor variables. Our methods are illustrated by an application to a time series of clearly periodic daily Dutch tax revenues. The dimension of our periodic unobserved components model is relatively large as we allow the time series for each day of the month to be subject to a changing seasonal pattern. Nevertheless, even with only five years of data we find that the increased periodic flexibility can help in simulated out-of-sample forecasting for two extra years of data.