A Smoothness Priors–State Space Modeling of Time Series with Trend and Seasonality

Abstract

A smoothness priors modeling of time series with trends and seasonalities is shown. An observed time series is decomposed into local polynomial trend, seasonal, globally stationary autoregressive and observation error components. Each component is characterized by an unknown variance–white noise perturbed difference equation constraint. The constraints or Bayesian smoothness priors are expressed in state space model form. Trading day factors are also incorporated in the model. A Kalman predictor yields the likelihood for the unknown variances (hyperparameters). Likelihoods are computed for different constraint order models in different subsets of constraint equation model classes. Akaike's minimum AIC procedure is used to select the best model fitted to the data within and between the alternative model classes. Smoothing is achieved by using a fixed-interval smoother algorithm. Examples are shown.

Key Words:

• Bayesian modeling
• Box-Jenkins
• Smoothing
• Seasonal adjustment
• Kalman filter
• Likelihood
• Trading day adjustment