Chapter 10 Explanatory variables in ADAM

In real life, the need for explanatory variables arises when some external factors influence the response variable, which cannot be ignored and impact the final forecasts and their accuracy. Examples of such variables in the demand forecasting context include price changes, promotional activities, temperature etc. In some cases, the changes in these factors would not substantially impact the demand, but this does not apply universally to the problem. If we omit this information from the model, this will be damaging for both point forecasts and prediction intervals (see discussion in Chapter 12 of Svetunkov, 2022a).

While the inclusion of explanatory variables in the context of ARIMA models is a relatively well-studied topic (for example, this was discussed by Box and Jenkins, 1976), in the case of ETS, there is only a Chapter 9 in Hyndman et al. (2008) and a handful of papers. Koehler et al. (2012) discuss the mechanism 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 the series, it will seriously impact the final forecast. However, if it appears either in the middle or at the beginning of the 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 outliers in Koehler et al. (2012) and including explanatory variables in ETSX in terms of how the model is formulated in these two situations. Kourentzes and Petropoulos (2016) used ETSX successfully for promotional modelling, demonstrating that it outperforms the conventional ETS in terms of point forecasts accuracy in cases when promotions happen.

The state-space model (7.1) can be easily extended by including additional components and explanatory variables. This chapter discusses the main aspects of ADAM with explanatory variables, how it is formulated, and how the more advanced models can be built upon it. Furthermore, the parameters for these additional components can either be fixed (static) or change over time (dynamic). We discuss both in the following sections. We also show that the stability and forecastability conditions, discussed in Section 5.4 for the pure additive ETS model, will be different in the case of the ETSX model and that the classical definitions should be updated to cater for the introduction of the explanatory variables. We also briefly discuss the inclusion of categorical variables in the ETSX model and show that the seasonal ETS models can be considered special cases of ADAM ETSX in some situations.

Furthermore, we will use the terms “deterministic” and “stochastic” explanatory variables to denote the situations when the values of these variables are known in advance or can be controlled by us. An example of the former would be the price of a product or a promotion that we decide to have. An example of the latter would be the temperature, which we cannot control and thus can be considered as a stochastic variable.

As a final note, we will carry out the discussion of the topic on the example of ADAM ETSX, keeping in mind that the same principles will hold for ADAM ARIMAX because the two are formulated in the same way. The more general dynamic model (encompassing ETS and/or ARIMA) with explanatory variables is called “ADAMX” in this and further chapters.


• Box, G., Jenkins, G., 1976. Time series analysis: forecasting and control. Holden-day, Oakland, California.
• Hyndman, R.J., Koehler, A.B., Ord, J.K., Snyder, R.D., 2008. Forecasting with Exponential Smoothing. Springer Berlin Heidelberg.
• 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.
• Kourentzes, N., Petropoulos, F., 2016. Forecasting with multivariate temporal aggregation: The case of promotional modelling. International Journal of Production Economics. 181, 145–153.
• Svetunkov, I., 2022a. Statistics for business analytics. (version: 31.03.2022)