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Calculating Effective Degrees of Freedom for Forecast Combinations and Ensemble Models

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  • 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
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    References listed on IDEAS

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    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

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