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Chapter 10 Explanatory variables in ADAM

Having the state space model (5.1) allows easily extending the model to include additional components and explanatory variables. Furthremore, parameters for these additional components can either be fixed or change over time. The model becomes more complicated in the latter case and more difficult to estimate, but nonetheless still potentially useful.

In practice, the need for explanatory variables arises, when there are some external factors influencing the response variable, which cannot be ignored and impact the final forecasts. Examples of such variables in demand forecasting context include price changes, promotional activities, temperature etc. In some cases the changes in these factors would not have a substantial impact on the demand, but in the others they would be essential for improving the accuracy.

While inclusion of explanatory variables in context of ARIMA models is relatively well studied idea, in case of ETS, there is only a handful of papers on the topic. One of such papers is Koehler et al. (2012), which discusses the mechanism of outlier detection and approximation of outliers via an ETSX model (ETS with explanatory variables). The authors show that if an outlier appears at the end of series, then it will have a serious impact on the final forecast. However, if it appears either in the middle or in the beginning of series, the impact on the final forecast is typically negligible. This is relevant to our discussion, because there is a direct link between dealing with outlier in Koehler et al. (2012) and including explanatory variables in ETSX in terms of how the ETS model is formulated in these two cases.

In this chapter we discuss the main aspects of ADAM with explanatory variables, how it is formulated and how the more advanced models can be built upon it. We will do the discussions on the example of ADAM ETSX, keeping in mind that the same principles will hold for ADAM ARIMAX as well, because the two are formulated in the same way. The more general dynamic model with explanatory variables is called “ADAMX” in this and further chapters.


• Koehler, A.B., Snyder, R.D., Ord, J.K., Beaumont, A., 2012. A study of outliers in the exponential smoothing approach to forecasting. International Journal of Forecasting. 28, 477–484.