Method for Constructing Vector Autoregressions of Any Complexity

Vector autoregressions are one of the rapidly developing areas of many areas of modern science. They are actively used in modeling and forecasting various economic processes, most often in modeling the stock market and retail prices. Their most important advantage is the ability to consider the simultaneous influence of modeled indicators not only from their past values, but also from the past values of other indicators interrelated with them. The main problem why vector autoregressions are not actively used in practice (as they deserve) is the “curse of dimensionality”, which consists in a quadratic increase in the number of model coefficients depending on the increase in the dimension of the modeled vector. This circumstance leads to the fact that researchers in various fields of modern science are forced to limit the dimension of the vector, including only the most important ones in the model, or reducing the order of autoregression. Attempts to overcome the “curse of dimensionality” by using special mathematical methods result in a significant complication of the mathematical apparatus for constructing vector autoregressions, which does not contribute to the expansion of the practice of using vector autoregressions. The article proposes to use for this purpose a step-by-step decomposition method for constructing vector autoregressions of any dimension, which makes the process of constructing these models simple and accessible to any researcher. To test the possibility of using this method in practice, data series on the dynamics of eight main industry indices of the Moscow Exchange were used. At the same time, it was decided to construct a large vector autoregression of the order of p = 10. Using the least squares method, a total of 648 unknown coefficients of this model were estimated. The verification of this model was confirmed by simple autoregressions.

1. Aliaskarova Zh.A., Asadulaev A.B., Pashkus V.Yu. (2020). Forecasting the dynamics of investments in fixed capital and gross added value based on VAR and VECM models. Problems of Modern Economics, no. 4 (76), pp. 41–45 (in Russian).

2. Balasanyan S.Sh., Gevorgyan E.M. (2016). Comparative analysis of regression methods and group accounting of arguments in modeling mineral processing processes. Bulletin of the Tomsk Polytechnic University. Geo Assets Engineering, vol. 327, no. 4, pp. 23–34 (in Russian).

3. Belov V.V., Chistyakova V.I. (2008). Modeling and forecasting business processes using algorithms for self-organization of formal descriptions. Business Informatics, no. 4 (06), pp. 37–45 (in Russian).

4. Geets V.M., Klebanova T.S. et al. (2005). Models and methods of socio-economic forecasting: textbook. Kharkov: VD “INZHEK”. 396 p. (in Ukrainian).

5. Gelrud Ya.D., Ugryumov E.A., Rybak V.L. (2018). Vector model of autoregression of production activity indicators of a construction enterprise. Bulletin of SUSU. Series: Computational Mathematics and Computer Science, vol. 7, no. 3, pp. 19–30 (in Russian).

6. Dorokhov E.V. (2008). Statistical approach to the study of forecasting the RTS index based on vector autoregression and cointegration methods. Finance and Business, no. 1, pp. 85–110 (in Russian).

7. Dyachkov M.Yu. (2017). Inductive modeling of objects and phenomena by the method of group accounting of arguments: shortcomings and ways to eliminate them. RUDN Journal. Series: Mathematics. Computer science. Physics, vol. 25, no. 4, pp. 323–330 (in Russian).

8. Zubarev A.V., Kirillova M.A. (2023). Construction of the GVAR model for the Russian economy. HSE Economic Journal, no. 1, pp. 9–32 (in Russian).

9. Ivakhnenko A.G., Muller J.A. (1985). Self-organization of predictive models. Kiev: Technique; Berlin: FEB Verlag Technik, 1985. 223 p. (in Russian).

10. Malov D.N. (2019). Assessing the investment attractiveness of companies based on the VAR (vector autoregression) and ARIMA models, taking into account risks. Innovations and Investments, no. 1, pp. 152–159 (in Russian).

11. Mamatova N. (2015). Application of a vector autoregression model for the analysis of electricity consumption. Mathematical Models of Economics: Collection of Scientific Papers (NRU HSE), no. 4, pp. 15–19. (in Russian).

12. Mekhovich S.A., Akhiezer E.B., Dunaevskaya O.I. (2014). Economic and mathematical model of zoning of industrial enterprises. Energy Saving, Power Engineering, Energy Audit, no. 8 (126), pp. 39–49 (in Russian).

13. Petrov K.E., Deineko A.A., Chalaya O.V., Panferova I.Yu. (2020). Method for ranking alternatives during the procedure of collective expert assessment. Radioelectronics, Informatics, Management, no. 2, pp. 84–94 (in Russian).

14. Poghosyan K. (2015). Alternative models for forecasting the main macroeconomic indicators in Armenia. Quantile, no. 13, pp. 25–39 (in Russian).

15. Salmanov O.N., Zaernyuk V.M., Lopatina O.A. (2016). Establishing the influence of monetary policy using the vector autoregression method. Finance and Credit, no. 28, pp. 2–17 (in Russian).

16. Svetunkov S.G., Bazhenova M.P., Lukash E.V. (2022). Prospects for the use of vector autoregressions in economic forecasting. Modern Economics: Problems and Solutions, no. 6 (150), pp. 44–57 (in Russian).

17. Sukhanova E.I., Shirnaeva S.Yu. (2014). Forecasting indicators of stabilization processes in the Russian economy based on vector autoregression models. Fundamental Research, no. 9, pp. 1590–1595 (in Russian).

18. Shimanovsky D.V., Tretyakova E.A. (2020). Modeling socio-ecological-economic relationships as a way to assess the sustainable development of regions of the Russian Federation. Perm University Herald. Series: Economics, no. 3 (15), pp. 369–384 (in Russian). DOI: 10.17072/1994-9960-2020-3-369-384

19. Ahelegbey D.F., Billio M., Casarin R. (2016). Special issue on recent developments in financial econometrics. Annals of Economics and Statistics, no. 123/124, pp. 333–361. DOI: 10.15609/annaeconstat2009.123-124.0333

20. Barrett A. (2021). Forecasting the Prices of Cryptocurrencies using a Novel Parameter Optimization of VARIMA Models. Chapman: Chapman University Digital Commons. 277 p.

21. Carolyn N.N., Sherris M. (2020). Modeling mortality with a Bayesian vector autoregression. Mathematics and Economics, no. 94, pp. 40–57.

22. Chandra S.R., Al-Deek H. (2009). Predictions of Freeway Traffic Speeds and Volumes Using Vector Autoregressive Models. Journal of Intelligent Transportation Systems, vol. 13, no. 2, pp. 53–72.

23. Fitrianti H., Belwawin S.M., Riyana M., Amin R. (2019). Climate modeling using vector moving average autoregressive. IOP Conference Series: Earth and Environmental Science. Jaipur, pp. 28–39.

24. Garcia-Martos C., Rodriguez J., Sanchez M.J. (2013). Modelling and forecasting fossil fuels, CO2 and electricity prices and their volatilities. Applied Energy, vol. 101, pp. 363–375.

25. Jusmawati M.H., Penerapan V. (2020). Model Vector Autoregressive Integrated Moving Av-erage dalam Peramalan Laju Inflasi dan Suku Bunga di Indonesia. EIGEN mathematics journal, 2020, december, no. 3 (2), pp. 73–82.

26. Kilian L., Lütkepohl H. (2017). Structural Vector Autoregressive Аnalysis. Cambridge: Cambridge University Press. 735 p. DOI: https://doi.org/10.1017/9781108164818

27. Lusia D.A., Ambarwati A. (2018). Multivariate Forecasting Using Hybrid VARIMA-Neural Network in JCI Case, Proceeding // International Symposium on Advanced Intelligent Informatics: Revolutionize Intelligent Informatics Spectrum for Humanity. Yogyakarta. Pp. 11–14.

28. Lütkepohl H. (2005). New Introduction to Multiple Time Series Analysis. Berlin: Springer. 764 p.

29. Meimela A., Lestari S.S. et al. (2021). Modeling of

30. COVID-19 in Indonesia using vector autoregressive integrated moving average. Journal of Physics: Conference Series, pp. 55–79.

31. Olson D.R., Riedel T.O. et al. (2021). Time series analysis of wintertime O3 and NOx formation using vector autoregressions. Atmospheric Environment, vol. 259, pp. 218–232.

32. Ord K., Fildes R., Kourentzes N. (2017). Principles of business forecasting. Wessex: Wessex Press, Inc. 544 p.

33. Rusyana A., Tatsara N., Balqis R., Rahmi S. (2020). Application of Clustering and VARIMA for Rainfall Prediction. IOP Conference Series: Materials Science and Engineering, pp. 428–438.

34. Schmidhuber J. (2015). Deep learning in neural networks: An overview. Neural Networks, no. 61, pp. 85–117.

35. Thelin E.P., Raj R. et al. (2020). Comparison of high versus low frequency cerebral physiology for cerebrovascular reactivity assessment in traumatic brain injury: a multi-center pilot study. Journal of Clinical Monitoring and Computing, no. 34 (5), pp. 971–994.

36. Yi Zh., Chuntian Ch. et al. (2021). Multivariate probabilistic forecasting and its performance’s impacts on long-term dispatch of hydro-wind hybrid systems. Applied Energy, vol. 283. pp. 116–243.

37. Xu Bin, Boqiang Lin. (2016). What cause a surge in China's CO2 emissions? A dynamic vector autoregression analysis. Journal of Cleaner Production, vol. 143, pp. 17–26.

38. Zeiler F.A., Ercole A., Cabeleira M. (2020). Evaluation of the relationship between slow-waves of intracranial pressure, mean arterial pressure and brain tissue oxygen in TBI: a CENTER-TBI exploratory analysis. Journal of Clinical Monitoring and Computing, no. 35 (4), pp. 781–799.

39. Zhang Cheng, Liao Huchang, Luo Li, Xu Zeshui (2021). Low-carbon tourism destination selection by a thermodynamic feature-based method. Journal of the Operational Research Society, June.