Polynomial Trend Models

Abstract

Polynomial models find wide use in time series and forecasting as they do in other branches of applied statistics, such as static regression and experimental design. Their use in time series modelling is to provide simple yet flexible forms to describe the local trend components, where by trend we mean smooth variation in time. Relative to the sampling interval of the series and the required forecast horizons, such trends can frequently be well approximated by low order polynomial functions of time. Indeed a first or second-order polynomial model alone is often quite adequate for short-term forecasting, and very widely applicable when combined via superposition with other components such as seasonality and regression. In Chapter 2 we introduced polynomial DLMs with the simplest, yet most widely used, case of the first-order models. The next in terms of both complexity and applicability are the second-order models, also sometimes referred to as linear growth models, that much of this chapter is concerned with. Higher order polynomial models are also discussed for completeness, although it is rare that polynomials of order greater than three are required for practice. The structure of polynomial models was discussed in Harrison (1965, 1967), and theoretical aspects explored in Godolphin and Harrison (1975). See also Abraham and Ledolter (1983, Chapter 3).