A Methodology for Calculating the Contribution of Exogenous Variables to ARIMAX Predictions

Autoregressive integrated moving average with exogenous variables (ARIMAX) is a prevailing model in time series forecasting, yet little attention has been paid to explain the predictions of ARIMAX, which is essential for understanding business behavior and making decisions. Here we argue that the regression coefficients of exogenous variables are not sufficient to measure their contribution to the predictions due to the dynamic nature of the stochastic process in ARIMAX models. In this work, we propose an approach that explains the predictions of ARIMAX in a comprehensive way, by evaluating the effect of the changes in all past values of exogenous variables on the current level of observations, and taking a linear transformation to eliminate the contribution dependence. Through numerical experiments on several datasets, we demonstrate its excellent permformance, good consistency with intuition and much better identification of influential exogenous variables compared with the direct use of regression coefficients.