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Corrected Forecast Combinations

Chu-An Liu, Andrey L. Vasnev

TL;DR

The paper addresses the issue that combined forecasts frequently exhibit strongly autocorrelated errors, which can limit the gains from forecast combination. It introduces corrected forecast combinations, starting with a simple mean-forecast correction $f^{CEW} = f^{EW} + γ\, e^{EW}_{T|T-1}$ and extending to corrections of the time-series optimal weights via GLS-based joint estimation. A key theoretical result is that the optimal correction is $b_T^{EW} = E(e^{EW}_{T+h|T} | I_T)$, and empirical SPF evidence suggests a practical, robust starting point of $γ \approx 0.5$ across multiple macro indicators and periods, excluding COVID outliers. The work demonstrates that correcting for serial dependence can yield substantial improvements, sometimes surpassing the gains from the original forecast combination, and provides a practical framework (GLS/COW2024) for implementing these corrections in real forecasting settings.

Abstract

This paper proposes corrected forecast combinations when the original combined forecast errors are serially dependent. Motivated by the classic Bates and Granger (1969) example, we show that combined forecast errors can be strongly autocorrelated and that a simple correction--adding a fraction of the previous combined error to the next-period combined forecast--can deliver sizable improvements in forecast accuracy, often exceeding the original gains from combining. We formalize the approach within the conditional risk framework of Gibbs and Vasnev (2024), in which the combined error decomposes into a predictable component (measurable at the forecast origin) and an innovation. We then link this correction to efficient estimation of combination weights under time-series dependence via GLS, allowing joint estimation of weights and an error-covariance structure. Using the U.S. Survey of Professional Forecasters for major macroeconomic indices across various subsamples (including pre and post-2000, GFC, and COVID), we find that a parsimonious correction of the mean forecast with a coefficient around 0.5 is a robust starting point and often yields material improvements in forecast accuracy. For optimal-weight forecasts, the correction substantially mitigates the forecast combination puzzle by turning poorly performing out-of-sample optimal-weight combinations into competitive forecasts.

Corrected Forecast Combinations

TL;DR

The paper addresses the issue that combined forecasts frequently exhibit strongly autocorrelated errors, which can limit the gains from forecast combination. It introduces corrected forecast combinations, starting with a simple mean-forecast correction and extending to corrections of the time-series optimal weights via GLS-based joint estimation. A key theoretical result is that the optimal correction is , and empirical SPF evidence suggests a practical, robust starting point of across multiple macro indicators and periods, excluding COVID outliers. The work demonstrates that correcting for serial dependence can yield substantial improvements, sometimes surpassing the gains from the original forecast combination, and provides a practical framework (GLS/COW2024) for implementing these corrections in real forecasting settings.

Abstract

This paper proposes corrected forecast combinations when the original combined forecast errors are serially dependent. Motivated by the classic Bates and Granger (1969) example, we show that combined forecast errors can be strongly autocorrelated and that a simple correction--adding a fraction of the previous combined error to the next-period combined forecast--can deliver sizable improvements in forecast accuracy, often exceeding the original gains from combining. We formalize the approach within the conditional risk framework of Gibbs and Vasnev (2024), in which the combined error decomposes into a predictable component (measurable at the forecast origin) and an innovation. We then link this correction to efficient estimation of combination weights under time-series dependence via GLS, allowing joint estimation of weights and an error-covariance structure. Using the U.S. Survey of Professional Forecasters for major macroeconomic indices across various subsamples (including pre and post-2000, GFC, and COVID), we find that a parsimonious correction of the mean forecast with a coefficient around 0.5 is a robust starting point and often yields material improvements in forecast accuracy. For optimal-weight forecasts, the correction substantially mitigates the forecast combination puzzle by turning poorly performing out-of-sample optimal-weight combinations into competitive forecasts.
Paper Structure (10 sections, 4 theorems, 17 equations, 4 figures, 4 tables)

This paper contains 10 sections, 4 theorems, 17 equations, 4 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Theory
  3. Correction of the mean forecast
  4. Correction of the (unconditionally) optimal forecast
  5. Conditional Framework of COW2024
  6. Generalization
  7. Empirical illustration: The US Survey of Professional Forecasters
  8. Corrected Mean Forecast
  9. Corrected Optimal Forecast
  10. Conclusion

Key Result

Theorem 1

Given the additional available information $I_T$, the corrected forecast $f_{T+h | T}^{\text{CEW}}$ using $b_{T}^{\text{EW}} = \E(e_{T+h | T}^{\text{EW}} | I_T)$ achieves lower conditional and unconditional MSE, i.e.,

Figures (4)

  • Figure 1: Improvements from combination (EW) and correction (CEW) in the motivating example from bates1969combination. Points ES and BJ correspond to the original Brown's exponential smoothing and Box-Jenkins forecasts.
  • Figure 2: Four major macroeconomic indicators considered in the empirical study. The line represents the actual realizations of each macroeconomic variable, while the light crosses represent the individual expert forecasts from the US SPF.
  • Figure 3: Historically optimal correction factor for UNEMP computed using the past available data and used for the mean forecast correction.
  • Figure 4: Autocorrelation of the forecast errors.

Theorems & Definitions (4)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4