Structural Analysis of Time Series Using the SAS/ETS® UCM Procedure

This article introduces the SAS/ETS UCM procedure, which uses structural models to analyze time series data. Structural models provide regression-like decomposition of the response series into latent components (such as trend, seasonal, or other periodic components) and linear and nonlinear regression effects. Apart from the series forecasts, structural modeling provides estimates of these unobserved components; these estimates are very useful in practical decision making. In SAS® 9.2 the UCM procedure contains several new features: incorporation of linear and nonlinear regression effects with time-varying coefficients, approximation of long and complex seasonal patterns by using splines and trigonometric polynomials, detection of structural change, and additional ODS graphics. A few real-life examples illustrate the functionality of the UCM procedure. INTRODUCTION An important problem in time series analysis concerns analyzing observations that are collected on a numeric response variable at regular time intervals, such as monthly or quarterly. In addition to the response variable, observations on some predictor variables might also be available. Usually the goals of time series analysis are: Forecast the future values of the response series along with some measure of the forecast error. Study the impact of predictors on the response series. Decompose the response series into constituent factors such as trend, periodic effects, and regressor effects, and be able to estimate and forecast these constituent factors. Discover unusual observations or structural changes in the response variable’s historic pattern. Interpolate the missing response values. Obtain an estimate of the response series after noise is removed. You can use a variety of statistical techniques for achieving some or all of the preceding goals. The structural time series models, also called the unobserved components models (UCMs), constitute a large and flexible class of models that has proved very useful for these purposes. Structural models assume a regression-like decomposition of the response series into latent components (such as trend, seasonal, or other periodic components) and regression effects that can be linear, nonlinear, time-varying, and time-invariant. The latent components, which follow suitable stochastic models, are a key feature of the structural models; they provide a customizable set of patterns to capture the salient movements of the response series. The goal is to include appropriate latent components and regressors and to choose suitable stochastic models of the selected components so that the resulting model explains the observed series well. The UCM procedure implements many of the commonly needed UCMs and provides a rich framework for structural modeling. Harvey (1989) is a good reference for UCM-based time series modeling. Traditionally, the integrated autoregressive moving average (ARIMA) models (Box and Jenkins 1976) and, to a limited extent, the exponential smoothing models have been the main tools in the analysis of this type of time series data. It is fair to say that the UCMs capture the versatility of the ARIMA models while possessing the interpretability of the smoothing models. A thorough discussion of the correspondence between the ARIMA models and the UCMs, and the relative merits of UCM and ARIMA modeling, is given in Harvey (1989). The UCMs are also very similar to another set of models, called the dynamic models, that are popular in the Bayesian time series literature (West and Harrison 1999). The remaining sections of this paper are organized as follows: the second section has a simple illustrative example to motivate the main ideas behind structural modeling, the third section provides a brief description of the different UCMs that can be specified, the fourth section describes the available diagnostic tools, and the last section provides two additional examples.