Journal of Analytical and Applied Computer Sciences | Volume 2 Issue 2 | Pages: 13-23 | Doi : 10.67914/jaacs/rsa/2.2/13-23

Review Article

OPEN ACCESS | Published on : 15-Sep-2026

Using the Autoregressive Model with External Variables ARX for Forecasting

Hadeel Edrees Abd Al-Hameed
• Department of Statistics and Informatics, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq.


• Heyam A.A. Hayawi
• Department of Statistics and Informatics, College of Computer Science and Mathematics, University of Mosul, Mosul, Iraq.

Abstract

Time series forecasting models have witnessed a remarkable development over recent decades as a result of the increasing need to improve the accuracy of forecasting in many scientific and applied fields. The subjective Autoregressive model with external variables (ARX) is one of the important statistical models that combines the impact of historical values of the variable under study and the impact of related external variables, making it an effective tool for forecasting in the fields of Economics, Energy, Environment, Engineering, the health sector and others. This reference article aims to provide a comprehensive scientific review of the ARX model by reviewing its theoretical foundations, mathematical formulation, statistical assumptions, mechanisms for estimating its parameters, and diagnostic procedures, in addition to analyzing published studies that have dealt with its applications in forecasting in recent years, with a critical comparison between it and the most prominent traditional and modern forecasting models. The study adopted the method of systematic review of the scientific literature by analyzing the studies published in the scientific databases, classifying them according to the applied fields, methodologies of model building, methods of performance evaluation, focusing on modern research trends. The results of the review showed that the ARX model achieves high levels of accuracy when external variables with appropriate explanatory ability are available, and it is characterized by Ease of interpretation and low computational complexity compared to many modern models, while its performance declines in cases of non-linear relationships, poor selection of external variables, or the presence of problems such as multiple linear correlation and anomalous values. The review also revealed the existence of research gaps represented by the limited studies that dealt with the development of ARX hybrid models, the lack of research on durable versions of the model, the weak use of machine learning and deep learning techniques to improve its performance, in addition to the scarcity of comprehensive comparisons using multiple databases and unified evaluation metrics. Based on this, the study recommends directing future research towards the development of hybrid models combining ARX, machine learning and deep learning technologies, taking advantage of modern variable selection methods, and developing models capable of dealing with nonlinear and high-dimensional data and real-time forecasting in dynamic environments, thereby contributing to improving prediction accuracy and increasing the efficiency of Model applications in various fields.

Keywords

Autoregressive model with external variables, time series, forecasting, external variables, literature review, evaluation of models, machine learning, hybrid models

References

• Åström, K. J., & Bohlin, T. (1965). Numerical identification of linear dynamic systems from normal operating records. IFAC Symposium on Self-Adaptive Systems.

  Åström, K. J., & Eykhoff, P. (1971). System identification: A survey. Automatica, 7(2), 123–162.

  Billings, S. A. (2013). Nonlinear system identification: NARMAX methods in the time, frequency, and spatio-temporal domains. Wiley.

  Box, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2016). Time series analysis: Forecasting and control (5th ed.). Wiley.

  Box, G. E. P., Jenkins, G. M., Reinsel, G. C., & Ljung, G. M. (2015). Time series analysis: Forecasting and control (5th ed.). Wiley.

  Breiman, L. (2001). Random forests. Machine Learning, 45(1), 5–32. https://doi.org/10.1023/A:1010933404324

  Brockwell, P. J., & Davis, R. A. (2016). Introduction to time series and forecasting (3rd ed.). Springer.

  Chen, T., & Guestrin, C. (2016). XGBoost: A scalable tree boosting system. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 785–794. https://doi.org/10.1145/2939672.2939785

  Cooper, A., Simpson, D., Kennedy, L., Forbes, C., & Vehtari, A. (2023). Cross-validatory model selection for Bayesian autoregressives with exogenous regressors. arXiv:2301.08276.

  DelSole, T., Tippett, M. K., & Johnson, N. C. (2025). Diagnosing errors in climate forecast models using forced autoregressive models. Journal of Advances in Modeling Earth Systems, 17, e2024MS004926.

  Diebold, F. X., & Mariano, R. S. (1995). Comparing predictive accuracy. Journal of Business & Economic Statistics, 13(3), 253–263.

  Franses, P. H. (2016). A note on the mean absolute scaled error. International Journal of Forecasting, 32(1), 20–22.

  Galrinho, M., Everitt, N., & Hjalmarsson, H. (2016). ARX modeling of unstable linear systems. arXiv:1603.04184.

  Goodfellow, I., Bengio, Y., & Courville, A. (2016). Deep learning. MIT Press.

  Hamilton, J. D. (1994). Time series analysis. Princeton University Press.

  Hochreiter, S., & Schmidhuber, J. (1997). Long short-term memory. Neural Computation, 9(8), 1735–1780. https://doi.org/10.1162/neco.1997.9.8.1735

  Hsiao, C. (1986). Tests of Granger causality by the selection of the orders of a bivariate autoregressive model. Economics Letters, 22(2–3), 223–227. https://doi.org/10.1016/0165-1765(86)90236-3

  Hyndman, R. J., & Athanasopoulos, G. (2021). Forecasting: Principles and practice (3rd ed.). OTexts.

  Hyndman, R. J., & Koehler, A. B. (2006). Another look at measures of forecast accuracy. International Journal of Forecasting, 22(4), 679–688. https://doi.org/10.1016/j.ijforecast.2006.03.001

  Jenkins, G. M. (2014). Autoregressive–integrated moving average (ARIMA) models. In Wiley StatsRef: Statistics Reference Online. Wiley. https://doi.org/10.1002/9781118445112.stat03472

  Kang, H. (1989). The optimal lag selection and transfer function analysis in Granger causality tests. Journal of Economic Dynamics and Control, 13(2), 151–169. https://doi.org/10.1016/0165-1889(89)90015-8

  Ljung, L. (1999). System identification: Theory for the user (2nd ed.). Prentice Hall.

  Ljung, L., & Söderström, T. (1983). Theory and practice of recursive identification. MIT Press.

  Makridakis, S., & Hibon, M. (1997). ARMA models and the Box–Jenkins methodology. Journal of Forecasting, 16(3), 147–163.

  Makridakis, S., Spiliotis, E., & Assimakopoulos, V. (2022). The M5 accuracy competition: Results, findings and conclusions. International Journal of Forecasting.

  Malakouti, M. (2026). Machine learning for renewable energy forecasting: Advances, challenges, and future directions. Sustainable Energy Research.

  MathWorks. (n.d.). ARX model estimation documentation.

  Maurya, D., Tangirala, A. K., & Narasimhan, S. (2020). ARX model identification using generalized spectral decomposition. arXiv:2008.04779.

  Nounou, M. N. (2006). Multiscale ARX process modeling. Journal of Process Control, 16(3), 255–268.

  Oldewurtel, F., Parisio, A., Jones, C. N., Gyalistras, D., Gwerder, M., Stauch, V., Lehmann, B., & Morari, M. (2012). Use of model predictive control and weather forecasts for energy efficient building climate control. Energy and Buildings, 45, 15–27.

  Petropoulos, F., Apiletti, D., Assimakopoulos, V., et al. (2022). Forecasting: Theory and practice. International Journal of Forecasting.

  Qin, S. J., Lin, W., & Ljung, L. (2005). A novel subspace identification approach with enforced causal models. Automatica, 41(12), 2043–2053.

  Ribeiro, M. T., Singh, S., & Guestrin, C. (2016). “Why should I trust you?”: Explaining the predictions of any classifier. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 1135–1144. https://doi.org/10.1145/2939672.2939778

  Shojaie, A., & Fox, E. B. (2022). Granger causality: A review and recent advances. Annual Review of Statistics and Its Application, 9, 289–319.

  Söderström, T., & Stoica, P. (1989). System identification. Prentice Hall.

  van Randenborgh, J., & Schulze Darup, M. (2025). MPC for aquifer thermal energy storage systems using ARX models. arXiv.

  Vaswani, A., Shazeer, N., Parmar, N., et al. (2017). Attention is all you need. Advances in Neural Information Processing Systems, 30, 5998–6008.

  Weyer, E. (2000). Finite sample properties of system identification of ARX models under mixing conditions. Automatica, 36(9), 1291–1299.

  Ying, J., Dong, X., Li, B., & Tian, Z. (2024). Auto-regressive model with exogenous input (ARX)-based traffic-flow prediction. arXiv:2401.07762.