So far, we have focused our discussion on the location of a model (e.g. conditional mean, point forecasts), neglecting the fact that in some situations, the variance of a model might exhibit some time-varying patterns. In statistics, this is called “heteroscedasticity” (see discussion in Section 14.6) and implies that the variance of the residuals of a model is not constant. In some cases, we might get away with the multiplicative model, which takes care of heteroscedasticity caused by changing the level of data if the variance is proportional to its value. But there might be situations where variance changes due to some external factors, not necessarily available to the analyst. In this situation, it should be captured separately using a different model. Hereafter the original model producing conditional mean will be called location model, while the model for the variance will be called scale model.
In this chapter, we discuss the scale model for ADAM ETS/ARIMA/Regression, the model that allows capturing time-varying variance and using it for forecasting. We discuss how this model is formulated, how it can be estimated and then move to the discussion of its relation to such models as ARCH and GARCH. This chapter is inspired by the GAMLSS model, which models the scale of distribution using functions of explanatory variables. We build upon that by introducing the dynamic element in the model.