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

Abstract

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.

An interpretable time series machine learning method for varying forecast and nowcast lengths in wastewater-based epidemiology

“…However, details are provided here to illustrate the implementation of the ARIMAX model in the context of WBE and for purposes of comparison with the TSML strategy. The ARIMAX model, without seasonality, is given by [ 2 ] as is the differenced response of order , is the differenced feature of order , is the regression coefficient for is an autoregressive moving average (ARMA) process satisfying , { is a white noise process satisfying , is the autoregressive operator, is the moving average operator, is the backshift operator satisfying …”

“…Even when ARIMAX outperforms the Naïve model, as with short-term and long-term nowcasts, the performance still lags behind that of the TSML method. While future research might be able to incorporate more refined feature selection strategies in ARIMAX, perhaps by leveraging temporal cross-validation strategies [ 22 ] or with theoretical approaches [ 2 ], the TSML method benefits from having a feature selection strategy that can be tailored to a specific prediction task and window without needing to meet specific model assumptions.…”

“…Dean and Dunsmuir [ 1 ] point out that pairs of autocorrelated series, such as COVID case numbers and observed viral load in wastewater, can exhibit spurious correlations. Additionally, variable selection for ARIMAX models can be sensitive to the presence of multicollinearity [ 2 ].…”

Adaptive food price forecasts enhance public information during rapid economic changes

“…While most of our estimated coefficients can be interpreted approximately as short-term predictive elasticities, the tendency of variables to follow recent trends ("dynamic dependencies") means that more rigorous evaluations improve policy projections 36 . Furthermore, a more comprehensive approach allows us to consider changes other than transitory (or single-month) shocks.…”

“…To select an optimal model, we evaluate many models (∼ 36 million for each series each month) from seasonal-autoregressive-integrated-moving-average-with-exogenous-variables (SARIMAX) models that permit the inclusion of past observations of food prices, lagged exogenous variables, and seasonality. Practitioners can use the sequential relationship between included exogenous variables and food prices (i.e., their "Granger causality") to provide empirically grounded explanations to policymakers and the public about observed food price changes rather than being limited to simple trend associations or draw explanations from external sources 35,36 .…”