Graphical Techniques for Selecting Explanatory Variables for Time Series Data By J. M. MARRIOTT NottinghamTrent University, UK and A. N. PETTITT{ Queensland University ofTechnology, Brisbane, Australia [Received January 1996. Revised November 1996] SUMMARY Bayesian model building techniques are developed for data with a strong time series structure and possi- bly exogenous explanatory variables that have strong explanatory and predictive power. The emphasis is on finding whether there are any explanatory variables that might be used for modelling if the data have a strong time series structure that should also be included. We use a time series model that is linear in past observations and that can capture both stochastic and deterministic trend, seasonality and serial correla- tion. We propose the plotting of absolute predictive error against predictive standard deviation. A series of such plots is utilized to determine which of several nested and non-nested models is optimal in terms of minimizing the dispersion of the predictive distribution and restricting predictive outliers. We apply the techniques to modelling monthly counts of fatal road crashes in Australia where economic, consumption and weather variables are available and we find that three such variables should be included in addition to the time series filter. The approach leads to graphical techniques to determine strengths of relationships between the dependent variable and covariates and to detect model inadequacy as well as determining useful numerical summaries. Keywords: Bayesian prediction; Graphical techniques; Model selection; T|me series filter 1. Introduction Time series techniques have been developed extensively over the last 30 years, but emphasis on the understanding of the role of explanatory variables has not been at the forefront. Interest has typically focused on forecasting models such as the autoregressive moving average type of model (Box and Jenkins, 1970) and more recently autoregressive conditionally heteroscedastic models, and on structural models which involve explicit random terms for trend and seasonal components (Harvey, 1989; West and Harrison, 1989). An extensive review of linear processes involving explanatory variables was provided by Hannan and Deistler (1988). The model that we propose to use here takes the form Y t  filter  covariates  noise where the ®lter is a `time series ®lter' and is designed to capture stochastic and {Address for correspondence: School of Mathematics, Queensland University of Technology, GPO Box 2434, Brisbane 4001, Australia. E-mail: a.pettitt@qut.edu.au & 1997 Royal Statistical Society 0035^9254/97/46253 Appl. Statist. (1997) 46, No. 3, pp. 253^264