UCM Procedure

A UCM decomposes the response series into components such as trend, seasons, cycles, and the regression effects due to predictor series. The following model shows a possible scenario:

The terms , and represent the trend, seasonal, and cyclical components, respectively. In fact the model can contain multiple seasons and cycles, and the seasons can be of different types. For simplicity of discussion the preceding model contains only one of each of these components. The regression term, , includes contribution of regression variables with fixed regression coefficients. A model can also contain regression variables that have time-varying regression coefficients or that have a nonlinear or a transfer-function relationship with the dependent series (see Incorporating Predictors of Different Types). The disturbance term , also called the irregular component, is usually assumed to be Gaussian white noise. In some cases it is useful to model the irregular component as a stationary ARMA process. For additional information, see the section Modeling the Irregular Component.

By controlling the presence or absence of various terms and by choosing the proper flavor of the included terms, the UCMs can generate a rich variety of time series patterns. A UCM can be applied to variables after transforming them by transforms such as log and difference.

The components , and model structurally different aspects of the time series. For example, the trend models the natural tendency of the series in the absence of any other perturbing effects such as seasonality, cyclical components, and the effects of exogenous variables, while the seasonal component models the correction to the level due to the seasonal effects. These components are assumed to be statistically independent of each other and independent of the irregular component. All of the component models can be thought of as stochastic generalizations of the relevant deterministic patterns in time. This way the deterministic cases emerge as special cases of the stochastic models. The different models available for these unobserved components are discussed next.