IDEAS home Printed from https://ideas.repec.org/p/bca/bocadp/22-19.html
   My bibliography  Save this paper

Calculating Effective Degrees of Freedom for Forecast Combinations and Ensemble Models

Author

Listed:
  • James Younker

Abstract

Forecast combinations, also known as ensemble models, routinely require practitioners to select a model from a massive number of potential candidates. Ten explanatory variables can be grouped into 2^1078 forecast combinations, and the number of possibilities increases further to 2^(1078+2^1078) if we allow for forecast combinations of forecast combinations. This paper derives a calculation for the effective degrees of freedom of a forecast combination under a set of general conditions for linear models. It also supports this calculation with simulations. The result allows users to perform several other computations, including the F-test and various information criteria. These computations are particularly useful when there are too many candidate models to evaluate out of sample. Furthermore, computing effective degrees of freedom shows that the complexity cost of a forecast combination is driven by the parameters in the weighting scheme and the weighted average of parameters in the auxiliary models as opposed to the number of auxiliary models. This identification of complexity cost contributions can help practitioners make informed choices about forecast combination design.

Suggested Citation

  • James Younker, 2022. "Calculating Effective Degrees of Freedom for Forecast Combinations and Ensemble Models," Discussion Papers 2022-19, Bank of Canada.
    • Handle: RePEc:bca:bocadp:22-19
    as

    Download full text from publisher

    File URL: https://www.bankofcanada.ca/2022/09/staff-discussion-paper-2022-19/
    File Function: Abstract
    Download Restriction: no
    File URL: https://www.bankofcanada.ca/wp-content/uploads/2022/09/sdp2022-19.pdf
    File Function: Full text
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Hsiao, Cheng & Wan, Shui Ki, 2014. "Is there an optimal forecast combination?," Journal of Econometrics, Elsevier, vol. 178(P2), pages 294-309.
    2. A. Mukherjee & K. Chen & N. Wang & J. Zhu, 2015. "On the degrees of freedom of reduced-rank estimators in multivariate regression," Biometrika, Biometrika Trust, vol. 102(2), pages 457-477.
    3. Claeskens, Gerda & Magnus, Jan R. & Vasnev, Andrey L. & Wang, Wendun, 2016. "The forecast combination puzzle: A simple theoretical explanation," International Journal of Forecasting, Elsevier, vol. 32(3), pages 754-762.
    4. Bradley Efron, 2004. "The Estimation of Prediction Error: Covariance Penalties and Cross-Validation," Journal of the American Statistical Association, American Statistical Association, vol. 99, pages 619-632, January.
    5. Samuels, Jon D. & Sekkel, Rodrigo M., 2017. "Model Confidence Sets and forecast combination," International Journal of Forecasting, Elsevier, vol. 33(1), pages 48-60.
    6. Kato, Kengo, 2009. "On the degrees of freedom in shrinkage estimation," Journal of Multivariate Analysis, Elsevier, vol. 100(7), pages 1338-1352, August.
    7. Diebold, Francis X. & Shin, Minchul, 2019. "Machine learning for regularized survey forecast combination: Partially-egalitarian LASSO and its derivatives," International Journal of Forecasting, Elsevier, vol. 35(4), pages 1679-1691.
    8. Magnus, Jan R., 2010. "On the concept of matrix derivative," Journal of Multivariate Analysis, Elsevier, vol. 101(9), pages 2200-2206, October.
    9. Mark W. Watson & James H. Stock, 2004. "Combination forecasts of output growth in a seven-country data set," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 23(6), pages 405-430.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Wang, Xiaoqian & Hyndman, Rob J. & Li, Feng & Kang, Yanfei, 2023. "Forecast combinations: An over 50-year review," International Journal of Forecasting, Elsevier, vol. 39(4), pages 1518-1547.
    2. Zhentao Shi & Liangjun Su & Tian Xie, 2020. "L2-Relaxation: With Applications to Forecast Combination and Portfolio Analysis," Papers 2010.09477, arXiv.org, revised Aug 2022.
    3. Antoine Mandel & Amir Sani, 2017. "A Machine Learning Approach to the Forecast Combination Puzzle," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-01317974, HAL.
    4. Bravo, Jorge M. & Ayuso, Mercedes & Holzmann, Robert & Palmer, Edward, 2021. "Addressing the life expectancy gap in pension policy," Insurance: Mathematics and Economics, Elsevier, vol. 99(C), pages 200-221.
    5. Roccazzella, Francesco & Gambetti, Paolo & Vrins, Frédéric, 2022. "Optimal and robust combination of forecasts via constrained optimization and shrinkage," International Journal of Forecasting, Elsevier, vol. 38(1), pages 97-116.
    6. Daniel Borup & Jonas N. Eriksen & Mads M. Kjær & Martin Thyrsgaard, 2024. "Predicting Bond Return Predictability," Management Science, INFORMS, vol. 70(2), pages 931-951, February.
    7. Qiu, Yue & Zheng, Yuchen, 2023. "Improving box office projections through sentiment analysis: Insights from regularization-based forecast combinations," Economic Modelling, Elsevier, vol. 125(C).
    8. Wei Qian & Craig A. Rolling & Gang Cheng & Yuhong Yang, 2019. "On the Forecast Combination Puzzle," Econometrics, MDPI, vol. 7(3), pages 1-26, September.
    9. Qian, Wei & Rolling, Craig A. & Cheng, Gang & Yang, Yuhong, 2022. "Combining forecasts for universally optimal performance," International Journal of Forecasting, Elsevier, vol. 38(1), pages 193-208.
    10. Qian, Yilin & Thompson, Ryan & Vasnev, Andrey L, 2022. "Global combinations of expert forecasts," Working Papers BAWP-2022-02, University of Sydney Business School, Discipline of Business Analytics.
    11. Guillaume Allaire Pouliot & Zhen Xie, 2022. "Degrees of Freedom and Information Criteria for the Synthetic Control Method," Papers 2207.02943, arXiv.org.
    12. Carlos Henrique Dias Cordeiro de Castro & Fernando Antonio Lucena Aiube, 2023. "Forecasting inflation time series using score‐driven dynamic models and combination methods: The case of Brazil," Journal of Forecasting, John Wiley & Sons, Ltd., vol. 42(2), pages 369-401, March.
    13. Thompson, Ryan & Qian, Yilin & Vasnev, Andrey L., 2024. "Flexible global forecast combinations," Omega, Elsevier, vol. 126(C).
    14. Sun, Yuying & Hong, Yongmiao & Wang, Shouyang & Zhang, Xinyu, 2023. "Penalized time-varying model averaging," Journal of Econometrics, Elsevier, vol. 235(2), pages 1355-1377.
    15. Chan, Felix & Pauwels, Laurent L., 2018. "Some theoretical results on forecast combinations," International Journal of Forecasting, Elsevier, vol. 34(1), pages 64-74.
    16. Antoine Mandel & Amir Sani, 2016. "Learning Time-Varying Forecast Combinations," Documents de travail du Centre d'Economie de la Sorbonne 16036, Université Panthéon-Sorbonne (Paris 1), Centre d'Economie de la Sorbonne.
    17. Uniejewski, Bartosz & Maciejowska, Katarzyna, 2023. "LASSO principal component averaging: A fully automated approach for point forecast pooling," International Journal of Forecasting, Elsevier, vol. 39(4), pages 1839-1852.
    18. Wang, Yudong & Hao, Xianfeng, 2022. "Forecasting the real prices of crude oil: A robust weighted least squares approach," Energy Economics, Elsevier, vol. 116(C).
    19. Chuanhua Wei & Chenping Du & Nana Zheng, 2020. "A Changing Weights Spatial Forecast Combination Approach with an Application to Housing Price Prediction," International Journal of Economics and Finance, Canadian Center of Science and Education, vol. 12(4), pages 1-11, April.
    20. Jing Zeng, 2015. "Combining Country-Specific Forecasts when Forecasting Euro Area Macroeconomic Aggregates," Working Paper Series of the Department of Economics, University of Konstanz 2015-11, Department of Economics, University of Konstanz.

    More about this item

    Keywords

    Econometric and statistical methods;

    JEL classification:

    • C01 - Mathematical and Quantitative Methods - - General - - - Econometrics
    • C02 - Mathematical and Quantitative Methods - - General - - - Mathematical Economics
    • C1 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General
    • C13 - Mathematical and Quantitative Methods - - Econometric and Statistical Methods and Methodology: General - - - Estimation: General
    • C5 - Mathematical and Quantitative Methods - - Econometric Modeling
    • C50 - Mathematical and Quantitative Methods - - Econometric Modeling - - - General
    • C51 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Construction and Estimation
    • C52 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Model Evaluation, Validation, and Selection
    • C53 - Mathematical and Quantitative Methods - - Econometric Modeling - - - Forecasting and Prediction Models; Simulation Methods

    NEP fields

    This paper has been announced in the following NEP Reports:

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:bca:bocadp:22-19. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: the person in charge (email available below). General contact details of provider: https://edirc.repec.org/data/bocgvca.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.