Literature

Published work behind the methods, and what each source supports

The methods are not reproductions of any single paper. Each ingredient still has a long published history. The sources below explain why the formulas take the form they do. They do not validate the shipped parameters, which come from their own training samples.

Forecast combination

Both methods combine a model forecast with consensus. That is the forecast combination problem.

• Precision weighting: Weights inversely proportional to error variance originate with Bates and Granger (1969). The independent Bayesian weights have this form.
• Regression weights: Granger and Ramanathan (1984) estimate combination weights by least squares, with an intercept and without forcing the weights to sum to one. The anchored estimate’s regression on the gap and revision follows this route.
• Surveys: Clemen (1989) and Timmermann (2006) review the field, including when estimated weights beat simple ones. Wang et al. (2023) extend the review to time-varying, nonlinear, correlation-aware and probabilistic combinations.
• Unstable covariance weights: Smith and Wallis (2009) show that estimation error in optimal weights can make them worse than simpler weights. This is the argument for the independent method as the default over the correlation adjusted one.
• Textbook treatment: Elliott and Timmermann (2016) cover combination weights with and without covariance, the case for simple weights, and forecast evaluation.

Dependent sources

• Conditioning on correlated readings: Winkler (1981) combines normal distributions from dependent sources through their joint covariance. The correlation adjusted method is this conditional expectation.
• Value of a correlated source: Clemen and Winkler (1985) show how error correlation limits what an additional source can contribute. This explains the gap between the regression’s gap weight and the independent precision weight.

Bayesian updating

• Normal-normal conjugacy: The independent posterior is the standard conjugate update of a normal prior with normal observations, as in
• Heavy tails: Financial changes are fatter-tailed than a Gaussian, first documented by Mandelbrot (1963). This motivates the Student-t interval.
• Sequential Bayesian forecasting: West and Harrison (1997) develop normal updating over time and the combination of forecasts within one Bayesian framework.

Consensus bias filter

• Kalman filter: The recursive update is due to Kalman (1960). Hamilton (1994) gives the standard econometric derivation.
• Local level model: The filter tracks a random-walk bias observed with noise. Harvey (1990) and Durbin and Koopman (2012) treat this model and its initialization.
• Exponential smoothing: In steady state the local-level filter is simple exponential smoothing (Hyndman et al. 2008), which gives the process and measurement noise settings a familiar reading.

Attenuation of a noisy signal

• Reliability ratio: Regressing an outcome on a noisy reading of the truth shrinks the slope by the share of the reading’s variance that is signal (Fuller 1987). This is why the anchored gap weight lies well below one.

Analyst behavior behind consensus

• Walk-down and customary beats: Forecasts drift down to beatable levels before a report (Richardson et al. 2004; Matsumoto 2002), and meeting or beating expectations is rewarded (Bartov et al. 2002). The anchored estimate’s intercept captures the resulting customary beat.
• Under-reaction to news: Analysts revise in the right direction but not far enough (Abarbanell and Bernard 1992; Gleason and Lee 2003). Revisions therefore predict later forecast errors (Coibion and Gorodnichenko 2015; Bouchaud et al. 2019), which is why the revision coefficient exceeds one.
• Herding: Career concerns pull forecasts together (Hong et al. 2000), so tight dispersion is weak evidence of predictability.
• Dispersion and analyst count: Barron et al. (1998) relate forecast dispersion and the number of analysts to the uncertainty in the mean forecast. The dispersion and count weighting options rest on the same relationship.
• Survey: Kothari et al. (2016) review analyst forecasts and their biases.

Alternative data and surprises

• Real-time sales: Froot et al. (2017) show that real-time corporate sales measures predict revenue and earnings surprises. The anchored gap is this kind of signal, measured against consensus.

Evaluating an adjustment

• Comparing accuracy: Diebold and Mariano (1995) give a test for whether one forecast’s errors are significantly smaller than another’s. The paired comparisons in the forecast improvement tutorial are the descriptive step before such a test.

References

Abarbanell, Jeffery S., and Victor L. Bernard. 1992. “Tests of Analysts’ Overreaction/Underreaction to Earnings Information as an Explanation for Anomalous Stock Price Behavior.” The Journal of Finance 47 (3): 1181–207. https://doi.org/10.1111/j.1540-6261.1992.tb04010.x.

Barron, Orie E., Oliver Kim, Steve C. Lim, and Douglas E. Stevens. 1998. “Using Analysts’ Forecasts to Measure Properties of Analysts’ Information Environment.” The Accounting Review 73 (4): 421–33.

Bartov, Eli, Dan Givoly, and Carla Hayn. 2002. “The Rewards to Meeting or Beating Earnings Expectations.” Journal of Accounting and Economics 33 (2): 173–204. https://doi.org/10.1016/S0165-4101(02)00045-9.

Bates, J. M., and C. W. J. Granger. 1969. “The Combination of Forecasts.” Operational Research Quarterly 20 (4): 451–68. https://doi.org/10.2307/3008764.

Bouchaud, Jean-Philippe, Philipp Krüger, Augustin Landier, and David Thesmar. 2019. “Sticky Expectations and the Profitability Anomaly.” The Journal of Finance 74 (2): 639–74. https://doi.org/10.1111/jofi.12734.

Clemen, Robert T. 1989. “Combining Forecasts: A Review and Annotated Bibliography.” International Journal of Forecasting 5 (4): 559–83. https://doi.org/10.1016/0169-2070(89)90012-5.

Clemen, Robert T., and Robert L. Winkler. 1985. “Limits for the Precision and Value of Information from Dependent Sources.” Operations Research 33 (2): 427–42. https://doi.org/10.1287/opre.33.2.427.

Coibion, Olivier, and Yuriy Gorodnichenko. 2015. “Information Rigidity and the Expectations Formation Process: A Simple Framework and New Facts.” American Economic Review 105 (8): 2644–78. https://doi.org/10.1257/aer.20110306.

Diebold, Francis X., and Roberto S. Mariano. 1995. “Comparing Predictive Accuracy.” Journal of Business & Economic Statistics 13 (3): 253–63. https://doi.org/10.1080/07350015.1995.10524599.

Durbin, James, and Siem Jan Koopman. 2012. Time Series Analysis by State Space Methods. 2nd ed. Oxford University Press. https://doi.org/10.1093/acprof:oso/9780199641178.001.0001.

Elliott, Graham, and Allan Timmermann. 2016. Economic Forecasting. Princeton University Press.

Froot, Kenneth, Namho Kang, Gideon Ozik, and Ronnie Sadka. 2017. “What Do Measures of Real-Time Corporate Sales Say about Earnings Surprises and Post-Announcement Returns?” Journal of Financial Economics 125 (1): 143–62. https://doi.org/10.1016/j.jfineco.2017.04.008.

Fuller, Wayne A. 1987. Measurement Error Models. Wiley. https://doi.org/10.1002/9780470316665.

Gelman, Andrew, John B. Carlin, Hal S. Stern, David B. Dunson, Aki Vehtari, and Donald B. Rubin. 2013. Bayesian Data Analysis. 3rd ed. Chapman; Hall/CRC. https://doi.org/10.1201/b16018.

Gleason, Cristi A., and Charles M. C. Lee. 2003. “Analyst Forecast Revisions and Market Price Discovery.” The Accounting Review 78 (1): 193–225. https://doi.org/10.2308/accr.2003.78.1.193.

Granger, Clive W. J., and Ramu Ramanathan. 1984. “Improved Methods of Combining Forecasts.” Journal of Forecasting 3 (2): 197–204. https://doi.org/10.1002/for.3980030207.

Hamilton, James D. 1994. Time Series Analysis. Princeton University Press. https://doi.org/10.1515/9780691218632.

Harvey, Andrew C. 1990. Forecasting, Structural Time Series Models and the Kalman Filter. Cambridge University Press. https://doi.org/10.1017/CBO9781107049994.

Hong, Harrison, Jeffrey D. Kubik, and Amit Solomon. 2000. “Security Analysts’ Career Concerns and Herding of Earnings Forecasts.” The RAND Journal of Economics 31 (1): 121–44. https://doi.org/10.2307/2601032.

Hyndman, Rob J., Anne B. Koehler, J. Keith Ord, and Ralph D. Snyder. 2008. Forecasting with Exponential Smoothing: The State Space Approach. Springer. https://doi.org/10.1007/978-3-540-71918-2.

Kalman, R. E. 1960. “A New Approach to Linear Filtering and Prediction Problems.” Journal of Basic Engineering 82 (1): 35–45. https://doi.org/10.1115/1.3662552.

Kothari, S. P., Eric So, and Rodrigo Verdi. 2016. “Analysts’ Forecasts and Asset Pricing: A Survey.” Annual Review of Financial Economics 8: 197–219. https://doi.org/10.1146/annurev-financial-121415-032930.

Mandelbrot, Benoit. 1963. “The Variation of Certain Speculative Prices.” The Journal of Business 36 (4): 394–419. https://doi.org/10.1086/294632.

Matsumoto, Dawn A. 2002. “Management’s Incentives to Avoid Negative Earnings Surprises.” The Accounting Review 77 (3): 483–514. https://doi.org/10.2308/accr.2002.77.3.483.

Richardson, Scott, Siew Hong Teoh, and Peter D. Wysocki. 2004. “The Walk-down to Beatable Analyst Forecasts: The Role of Equity Issuance and Insider Trading Incentives.” Contemporary Accounting Research 21 (4): 885–924. https://doi.org/10.1506/KHNW-PJYL-ADUB-0RP6.

Smith, Jeremy, and Kenneth F. Wallis. 2009. “A Simple Explanation of the Forecast Combination Puzzle.” Oxford Bulletin of Economics and Statistics 71 (3): 331–55. https://doi.org/10.1111/j.1468-0084.2008.00541.x.

Timmermann, Allan. 2006. “Forecast Combinations.” In Handbook of Economic Forecasting, edited by Graham Elliott, Clive W. J. Granger, and Allan Timmermann, vol. 1. Elsevier. https://doi.org/10.1016/S1574-0706(05)01004-9.

Wang, Xiaoqian, Rob J. Hyndman, Feng Li, and Yanfei Kang. 2023. “Forecast Combinations: An over 50-Year Review.” International Journal of Forecasting 39 (4): 1518–47. https://doi.org/10.1016/j.ijforecast.2022.11.005.

West, Mike, and Jeff Harrison. 1997. Bayesian Forecasting and Dynamic Models. 2nd ed. Springer. https://doi.org/10.1007/b98971.

Winkler, Robert L. 1981. “Combining Probability Distributions from Dependent Information Sources.” Management Science 27 (4): 479–88. https://doi.org/10.1287/mnsc.27.4.479.