Frequentist Averaging

  • Felix ChanEmail author
  • Laurent Pauwels
  • Sylvia Soltyk
Part of the Advanced Studies in Theoretical and Applied Econometrics book series (ASTA, volume 52)


This chapter summarises the recent approaches to optimal forecast combination from a frequentist perspective. The availability of big data leads to the development of many different models of the same macroeconomic variables. The challenge is to seek the best way to combine all relevant information from big data to create optimal forecast. Forecast combination provides one plausible approach. This chapter discusses the practical aspects of combining forecasts optimally and theoretical properties of the combination both for point forecasts and density forecasts. Specifically, the chapter derives the asymptotic distributions of the estimated optimal weight under two of the most popular forecasting criteria: Mean Squared Forecast Error and Mean Absolute Deviation. This chapter also revisits the insights of the so-called forecast combination puzzle, which shows that in practice a simple average of forecasts outperforms more complex weighting strategies. These theoretical results help address the puzzle by providing a mean to test statistically the difference between the estimated optimal weight and the simple average. The optimal weights obtained from minimising the Kullback–Leibler Information Criterion (KLIC) are discussed in the context of density forecast combination. This chapter also proposes a novel Generalized Method of Moments approach for density forecast combination. The connection between the proposed approach and the conventional approach by minimising KLIC is also investigated in some details.


  1. Aiolfi, A., Capistrán, C., & Timmermann, A. (2010). A simple explanation of the forecast combination puzzle. Aarhus: Center for Research in Econometric Analysis of Time Series (CREATS), Aarhus University.Google Scholar
  2. Bates, J. M., & Granger, C. W. J. (1969). The combination of forecasts. Operational Research Quarterly, 20, 451–468.CrossRefGoogle Scholar
  3. Buckland, S., Burnham, K., & Augustin, N. (1997). Model selection: An integral part of inference. Biometrics, 53, 603–618.CrossRefGoogle Scholar
  4. Busetti, F. (2017). Quantile aggregation of density forecasts. Oxford Bulletin of Economics and Statistics, 79(4), 495–512. Scholar
  5. Chan, F., & James, A. (2011). Application of forecast combination in volatility modelling. In F. Chan, D. Marinova, & R. Anderssen (Eds.), Modsim2011, 19th international congress on modelling and simulation (pp. 1610–1616). Canberra: Modelling, Simulation Society of Australia, and New Zealand. Retrieved from Google Scholar
  6. Chan, F., & Pauwels, L. L. (2018). Some theoretical results on forecast combinations. International Journal of Forecasting, 34(1), 64–74.CrossRefGoogle Scholar
  7. Chan, F., & Pauwels, L. L. (2019). Equivalence of optimal forecast combinations under affine constraints. Working paper.Google Scholar
  8. Chatfield, C. (1993). Calculating interval forecasts. Journal of Business and Economic Statistics, 11, 121–135.Google Scholar
  9. Christoffersen, P. F. (1998). Evaluating interval forecasts. International Economic Review, 39, 841–862.CrossRefGoogle Scholar
  10. Claeskens, G., & Hjort, N. L. (2003). The focused information criterion. Journal of the American Statistical Association, 98(464), 900–916. Scholar
  11. Claeskens, G., & Hjort, N. L. (2008). Model selection and model averaging. Cambridge: Cambridge University Press.CrossRefGoogle Scholar
  12. Claeskens, G., Magnus, J. R., Vasnev, A. L., & Wang, W. (2016). A simple theoretical explanation of the forecast combination puzzle. International Journal of Forecasting, 32(3), 754–62.CrossRefGoogle Scholar
  13. Clark, T. E., & West, K. D. (2006). Using out-of-sample mean squared prediction errors to test the martingale difference hypothesis. Journal of Econometrics, 135(1–2), 155–186. CrossRefGoogle Scholar
  14. Clark, T. E., & West, K. D. (2007). Approximately normal tests for equal predictive accuracy in nested models. Journal of Econometrics, 138(1), 291–311. CrossRefGoogle Scholar
  15. Clemen, R. T. (1989). Combining forecasts: A review and annotated bibliography. International Journal of Forecasting, 5, 559–583.CrossRefGoogle Scholar
  16. Cook, M. B. (1951). Bi-variate k-statistics and cumulants of their joint sampling distribution. Biometrika, 38(1), 179–195.CrossRefGoogle Scholar
  17. Corradi, V., & Swanson, N. R. (2006). Predictive density evaluation. In Handbook of economic forecasting (Chap. 5, Vol. 1, pp. 135–196).
  18. Diebold, F. X. (1989). Forecast combination and encompassing: Reconciling two divergent literatures. International Journal of Forecasting, 5, 589–592.CrossRefGoogle Scholar
  19. Diebold, F. X., Gunther, T. A., & Tay, A. S. (1998). Evaluating interval forecasts with applications to financial risk management. International Economic Review, 39, 863–883.CrossRefGoogle Scholar
  20. Diebold, F. X., & Lopez, J. A. (1996). Forecast evaluation and combination. Cambridge: National Bureau of Economic Research (NBER).CrossRefGoogle Scholar
  21. Diebold, F. X., & Pauly, P. (1987). Structural change and the combination of forecasts. Journal of Forecasting, 6, 21–40.CrossRefGoogle Scholar
  22. Elliott, G. (2011). Averaging and the optimal combination of forecasts. San Diego: University of California.Google Scholar
  23. Elliott, G., Gargano, A., & Timmermann, A. (2013). Complete subset regressions. Journal of Econometrics, 177, 357–373. Scholar
  24. Elliott, G., & Timmermann, A. (2004). Optimal forecast combinations under general loss functions and forecast error distributions. Journal of Econometrics, 122, 47–79.CrossRefGoogle Scholar
  25. Genre, V., Kenny, G., Meyler, A., & Timmermann, A. (2013). Combining expert forecasts: Can anything beat the simple average? International Journal of Forecasting, 29, 108–121.CrossRefGoogle Scholar
  26. Geweke, J., & Amisano, G. (2011). Optimal prediction pools. Journal of Econometrics, 164, 130–141. Scholar
  27. Granger, C. J. W., & Ramanathan, R. (1984). Improved methods of combining forecasts. Journal of Forecasting, 3, 197–204.CrossRefGoogle Scholar
  28. Hall, S. G., & Mitchell, J. (2007). Combining density forecasts. International Journal of Forecasting, 23, 1–13.CrossRefGoogle Scholar
  29. Hansen, B. E. (2007). Leasts squares model averaging. Econometrica, 75(4), 1175–1189.CrossRefGoogle Scholar
  30. Hansen, B. E. (2008). Least squares forecast averaging. Journal of Econometrics, 1146, 342–350.CrossRefGoogle Scholar
  31. Hansen, B. E. (2014). Model averaging, asymptotic risk, and regressor groups. Quantitative Economics, 5(3), 495–530. Scholar
  32. Hansen, B. E., & Racine, J. (2012). Jackknife model averaging. Journal of Econometrics, 167, 38–46.CrossRefGoogle Scholar
  33. Hansen, L. (1982). Large sample properties of generalized method of moments estimators. Econometrica, 50, 1029–1054.CrossRefGoogle Scholar
  34. Harvey, D., Leybourne, S., & Newbold, P. (1997). Testing the equality of prediction mean squared errors. International Journal of Forecasting, 13(2), 281–291. Scholar
  35. Hendry, D. F., & Clements, M. P. (2004). Pooling of forecasts. Econometrics Journal, 7, 1–31.CrossRefGoogle Scholar
  36. Hjort, N. L., & Claeskens, G. (2003). Frequentist model average estimators. Journal of the American Statistical Association, 98(464), 879–899. arXiv:1011.1669v3.CrossRefGoogle Scholar
  37. Hsiao, C., & Wan, S. (2014). Is there an optimal forecast combination? Journal of Econometrics, 178, 294–309.CrossRefGoogle Scholar
  38. Iwashita, T., & Siotani, M. (1994). Asymptotic distributions of functions of a sample covariance matrix under the elliptical distribution. Canadian Journal of Statistics, 22(2), 273–283.CrossRefGoogle Scholar
  39. Jore, A. S., Mitchell, J., & Vahey, S. P. (2010). Combining forecast densities from VARs with instabilities. Journal of Applied Econometrics, 25, 621–634.CrossRefGoogle Scholar
  40. Kapetanios, G., Mitchell, J., Price, S., & Fawcett, N. (2015). Generalised density forecast combinations. Journal of Econometrics, 188, 150–165. Scholar
  41. Kascha, C., & Ravazzolo, F. (2010). Combining inflation density forecasts. Journal of Forecasting, 29, 231–250.CrossRefGoogle Scholar
  42. Knight, K. (1998). Limiting distributions for L1 regression estimators under general conditions. The Annals of Statistics, 26(2), 755–770.CrossRefGoogle Scholar
  43. Knight, K. (2001). Limiting distributions of linear programming estimators. Extremes, 4(2), 87–103. Retrieved from{%}7B{%}5C{%}{%}7D3A1013991808181.CrossRefGoogle Scholar
  44. Laplace, P. (1818). Deuxième supplément a la théorie analytique des probabilitiés. Paris: Courcier.Google Scholar
  45. Liu, C. A., & Kuo, B. S. (2016). Model averaging in predictive regressions. The Econometrics Journal, 19, 203–231. Scholar
  46. Marcellino, M. (2004). Forecast pooling for European macroeconomic variables. Oxford Bulletin of Economics and Statistics, 66, 91–112.CrossRefGoogle Scholar
  47. Mitchell, J., & Hall, S. G. (2005). Evaluating, comparing and combining density forecasts using the KLIC with an application to the bank of England and NIESR ‘fan’ charts of inflation. Oxford Bulletin of Economics and Statistics, 67, 995–1033.CrossRefGoogle Scholar
  48. Moral-Benito, E. (2015). Model averaging in economics: An overview. Journal of Economic Surveys, 29(1), 46–75. Scholar
  49. Newbold, P., & Granger, C. J. W. (1974). Experience with forecasting univariate time series and the combination of forecasts. Journal of the Royal Statistical Society A, 137, 131–165.CrossRefGoogle Scholar
  50. Pauwels, L. L., Radchenko, P., & Vasnev, A. (2018). Higher moment constraints for predictive density combinations. SSRN Electronic Journal.
  51. Pauwels, L. L., & Vasnev, A. L. (2016). A note on the estimation of optimal weights for density forecast combinations. International Journal of Forecasting, 32, 391–397.CrossRefGoogle Scholar
  52. Pesaran, M. H., & Timmermann, A. (2007). Selection of estimation window in the presence of breaks. Journal of Econometrics, 137, 134–161.CrossRefGoogle Scholar
  53. Pinar M., Stengos, T., & Yazgan, M. E. (2012). Is there an optimal forecast combination? a stochastic dominance approach to forecast combination. Waterloo, On: The Rimini Centre for Economic Analysis.Google Scholar
  54. Smith, J., & Wallis, K. F. (2009). A simple explanation of the forecast combination puzzle. Oxford Bulletin of Economics and Statistics, 71, 331–355.CrossRefGoogle Scholar
  55. Stigler, S. M. (1973). Laplace, fisher and the discovery of the concept of sufficiency. Biometrika, 60(3), 439–445.Google Scholar
  56. Stock, J. H., & Watson, M. W. (1998). A comparison of linear and nonlinear univariate models for forecasting macroeconomic time series. Cambridge: National Bureau of Economic Research (NBER).CrossRefGoogle Scholar
  57. Stock, J. H., & Watson, M. W. (2003). How did leading indicator forecasts perform during the 2001 recession? Quarterly Economic Review – Federal Reserve Bank of Richmond, 89, 71–90.Google Scholar
  58. Stock, J. H., & Watson, M. W. (2004). Combination forecasts of output growth in a seven-country data set. Journal of Forecasting, 23, 405–430.CrossRefGoogle Scholar
  59. Tay, A. S., & Wallis, K. F. (2000). Density forecasting: A survey. Journal of Forecasting, 19, 235–254.CrossRefGoogle Scholar
  60. Tian, J., & Anderson, H. M. (2014). Forecast combination under structural break uncertainty. International Journal of Forecasting, 30, 161–175.CrossRefGoogle Scholar
  61. Timmermann, A. (2006). Forecast combinations. In Handbook of economic forecasting (Chap. 4, Vol. 1, pp. 135–196). Scholar
  62. Wallis, K. F. (2005). Combining density and interval forecasts: A modest proposal. Oxford Bulletin of Economics and Statistics, 67, 983–994.CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of Economics, Finance and PropertyCurtin UniversityPerthAustralia
  2. 2.Discipline of Business AnalyticsUniversity of SydneySydneyAustralia

Personalised recommendations