THE APPLICATION OF RIDGE REGRESSION METHODS WHEN COMBINING FORECASTS

Forecasting of economic indicators with time series using one or another method or another but the only method leads to the situation that all the information contained in other forecasting methods is usually discarded. The information that is ignored may contain information that allows other features of the economic process to be assessed. Combining forecasts makes possible to take into account almost all the information contained in particular forecasts. In the article, we present the analysis of the application of the method of regression analysis, in particular, ridge regression for finding the weighting coefficients of the particular forecasts in the combined forecast. We compared the accuracy of prediction based on the ridge regression with other methods of combining predictions. The purpose of our research work was an analysis of the most common methods of combining forecasts — various modifications of Granger-Ramanathan methods and comparison with a new approach of combining forecasts based on the ridge regression for its use in practice. We used statistical methods of time series forecasting (the method of harmonic weights, adaptive exponential smoothing using a tracking signal, the method of simple exponential smoothing and the Box-Jenkins model), the method of constructing combined forecasts, as well as methods of regression analysis. As a result, we built the combined forecasts based on annual data for the period from 1950 to 2015 on the production in Russia of some products: steel, metallurgical coke, pulp, plywood, cement. We used the methods of Granger-Ramanathan (without restrictions and with restrictions on the sum of coefficients in partial predictions) and also the ∆-coefficients obtained by the ridge regression method. The forecasts constructed using the Granger-Ramanathan methods give the highest accuracy of the combined forecast. The method based on the ridge regression is less accurate, but better than the separate predictions. At the same time, the proposed method of calculating the weight coefficients on the basis of the ridge regression has a well- developed scheme of calculation and eliminates the negative weight coefficients in the combined forecast.

combining forecasts, time series, time series forecasting methods, Granger-Ramanathan methods

1. Granger C. W.J., Ramanathan R. Improved methods of combining forecasts. Journal of Forecasting. 1984;3(2):197–204. DOI: 10.1002/for.3980030207

2. Stock J. H., Watson M. W. Combination forecasts of output growth in a seven-country data set. Journal of Forecasting. 2004;23(6):405–430. DOI: 10.1002/for.928

3. Frenkel A. A., Volkova N. N., Surkov A. A., Romanyuk E. I. Comparative analysis of methods for constructing a combined forecast. Voprosy statistiki. 2017;(7): 17–27). (In Russ.).

4. Holden K., Peel D. A. An empirical investigation of combinations of economic forecasts. Journal of Forecasting. 1986;5(4):229–242. DOI: 10.1002/for.3980050404

5. Holden K., Peel D.A., Thomson J.L. Economic forecasting: An introduction. Cambridge, New York: Cambridge University Press; 1990. 213 p.

6. Mills T. C., Stepherson M. J. Forecasting contemporaneous aggregates and the combination of forecasts: The case of the U.K. monetary aggregates. Journal of Forecasting. 1985;4(3):273–281. DOI: 10.1002/ for.3980040304

7. Ershov E. B. About one method of combining private forecasts. In: Statistical analysis of economic time series and forecasting: Scientific notes on statistics. Vol. XXII–XXIII. Moscow: Nauka; 1973:87–105. (In Russ.).

8. Frenkel A. A., Volkova N. N., Surkov A. A., Romanyuk E. I. Step-by-step combining of individual forecasts based on the Granger-Ramanathan method. Voprosy statistiki. 2018;25(6):16–24. (In Russ.).

9. Lee T.-H. Combining forecasts with many predictors. In: Advances in economic forecasting. Kalamazoo, MI: W. E. Upjohn Institute for Employment Research; 2011:149–172. DOI: 10.17848/9780880993937.ch7 10. Exterkate P., Groenen P. J.F., Heij C., van Dijk D. Nonlinear forecasting with many predictors using

10. kernel ridge regression. International Journal of Forecasting. 2016;32(3):736–753. DOI: 10.1016/j.ijforecast.2015.11.017

11. Hoerl A. E. Application of ridge analysis to regression problems. Chemical Engineering Progress. 1962;58(3):54–59.

12. Hoerl A. E., Kennard R. W. Ridge regression: Aplications to nonorthogonal problems. Technometrics. 1970;12(1):69–82. DOI: 10.2307/1267352

13. Frenkel A.A., Volkova N.N., Sergienko Ya.V. Quantitative assessment of the impact of foreign economic activity on the dynamics of industrial production. Voprosy statistiki. 2014;(11):60–68. (In Russ.).

14. Frenkel A. A., Raiskaya N. N. Application of comb regression in statistical modeling. Ekonomika i matematicheskie metody = Economics and Mathematical Methods. 1985;21(4):715–725. (In Russ.).

15. Ferrar D. E., Glauber F. R. Multicollineary in regression analysis: The problem revisited. The Review Economics and Statistics. 1967;49(1):91–107. DOI: 10.2307/1937887