Robustness of the dispersion-coverage identification of analyst herding: dynamic, heterogeneous, and sequential extensions of the finite-player forecasting game

Kim, Kim and Shim (SSRN 2019; published in Global Economic Review 2023) test whether financial analysts have strategic incentives to herd by exploiting an equilibrium property of a finite-player forecasting game: forecast dispersion decreases as the number of forecasters increases if and only if analysts' preferences exhibit strategic complementarity, i.e., a herding motive. Using I/B/E/S earnings forecasts they find strong evidence of herding, but the test rests on a static, symmetric, simultaneous-move model, and the paper itself leaves the dynamic extension (AR(1) fundamentals with per-period signals) and the heterogeneous-analyst extension undeveloped, assumes without testing that the herding parameter does not vary with coverage or forecast horizon, and leaves unexplained the weakening of the herding coefficient as the interval between the first and last forecast in a month lengthens and its insignificance in high-coverage and announcement-day subsamples. The problem posed here is whether this identification of strategic herding is robust: characterize equilibrium and identification in dynamic and heterogeneous versions of the finite-player forecasting game and confront the derived implications with analyst forecast data. A later, independent dynamic herding framework by Keppo and Satopää addresses neighboring questions but not this identification problem, which remains open.

Background

Financial analysts are professional forecasters who publish earnings forecasts for publicly traded firms; the Institutional Brokers Estimate System (I/B/E/S) is the standard database recording these forecasts, from which a consensus (average) forecast is computed. Analysts are said to herd when they deliberately shade their forecasts toward the consensus rather than toward their own private estimate, and to anti-herd (or make 'bold' forecasts) when they deliberately deviate from it. Which regime prevails matters for market efficiency: herding slows the incorporation of dispersed private information into prices, while anti-herding exaggerates dispersion. The empirical literature is conflicting. Trueman (1994) shows theoretically that analysts whose forecasts are disclosed sequentially can herd for non-informational reputational reasons; Hong, Kubik and Solomon (2000) document herding related to career concerns among less experienced analysts; Bernhardt, Campello and Kutsoati (2006) develop a test based on whether the analyst closest to the realized outcome is the consensus and conclude that analysts anti-herd; Ottaviani and Sorensen (2006) show that winner-take-all forecasting contests induce differentiation; and Clements (2018) surveys evidence of both herding and anti-herding among macroeconomic forecasters.

Kim and Shim (2018) provide the theoretical foundation for a new test. In their finite-player forecasting game, each of $n$ analysts observes a public signal $p = θ + α_p^{-1/2}ε$ and a private signal $x_i = θ + α_x^{-1/2}ε_i$ about a firm's earnings $θ$, and chooses a forecast $a_i$ to maximize the quadratic payoff $u_i = -(1/2)(a_i - (1-r)θ - rA_n)^2$, where $A_n = (1/n)\sum_i a_i$ is the average forecast and $r \in (-1,1)$ is the herding preference parameter: $r > 0$ means the analyst wants to be close to the average forecast (strategic complementarity, herding), $r < 0$ means she wants to differentiate (anti-herding), and $r = 0$ means she only cares about accuracy. Because $n$ is finite, each analyst recognizes that her own forecast moves the average — a market-power motive that is absent in continuum-agent coordination models. The unique linear equilibrium puts weight $λ_n = α_x/(α_x + α_p/(1-γ))$ on the private signal, with $γ = r(n-1)/(n-r)$, and equilibrium forecast dispersion equals $(λ_n/\sqrt{α_x})^2$. The key identification result is that $λ_n$ decreases in $n$ if and only if $r > 0$: herding makes forecast dispersion fall as coverage grows, and anti-herding makes it rise.

Kim, Kim and Shim (2019, published 2023) implement this on I/B/E/S data for 1990-2015, regressing forecast dispersion (the standard deviation of analyst forecasts scaled by the mean of current-fiscal-year estimates) on the logarithm of one plus the number of analysts in Fama-MacBeth regressions (year-by-year cross-sectional regressions whose coefficients are averaged over time). The average coefficient is about -0.09 with a t-statistic near -18, indicating herding; the result is robust across forecast horizons from one quarter to long-term growth, is more pronounced for firms with low initial coverage (in a natural experiment based on brokerage house mergers that exogenously change coverage), and weakens monotonically in coverage groups (approximately -0.149 for $n \le 5$, -0.061 for $5 < n \le 10$, -0.041 for $n > 10$). The paper itself, however, flags that the model is static (a dynamic version with an AR(1) fundamental — an autoregressive process of order one, in which the current value of earnings is a linear function of its previous value plus noise — and per-period noisy signals is noted as preserved only 'under certain conditions' but never developed), that it assumes a single herding parameter $r$ common to all analysts and independent of the number of analysts and of the forecast horizon without testing this, and that two empirical patterns are left unexplained: the absolute magnitude of the herding coefficient declines as the time interval between the first and last forecast in a month lengthens (roughly -0.160, -0.134, -0.118 across short, medium, and long interval groups), which is counter-intuitive because more sequential release should if anything strengthen strategic herding; and the coefficient becomes statistically indistinguishable from zero in some specifications, namely the high-coverage group ($n > 10$) with all control variables (about -0.014, t about -1.1) and forecasts issued at or immediately after earnings announcements with all controls. These acknowledged gaps jointly raise a single question: is the dispersion-coverage identification of strategic herding an artifact of the static, symmetric, simultaneous-move assumptions, or does it survive in the dynamic, heterogeneous, and sequential environments that better describe actual forecasting?

Problem Statement

Determine whether the identification of strategic herding used by Kim, Kim and Shim — the equilibrium property of the finite-player forecasting game that forecast dispersion decreases as the number of forecasters increases if and only if the herding parameter $r$ is positive — is robust to relaxing the static, symmetric, simultaneous-move assumptions of the model, and whether it can account for the empirical patterns that the static test leaves unexplained. Concretely, the problem comprises two linked deliverables serving this single objective.

(1) Theoretical extension and identification. Formulate and solve the finite-player forecasting game (public and private noisy signals about earnings, payoff $u_i = -(1/2)(a_i - (1-r)θ - rA_n)^2$ with $A_n$ the average forecast) in environments where (a) the fundamental follows an AR(1) process $θ_t$ and each analyst receives noisy signals in each period, with forecasts released simultaneously or sequentially over time, and (b) analysts are heterogeneous in their herding preference $r_i$ and in the precision of their private information. Treat every sub-case as a required variant and solve each separately: dynamic with simultaneous release, dynamic with sequential release, and heterogeneous with respect to both $r_i$ and private-signal precision. For each variant, characterize the equilibrium (existence, uniqueness, and form), and derive the identification result: what the behavior of forecast dispersion as the number of analysts varies reveals about the herding parameters — in particular, whether the sign test 'dispersion decreasing in $n$ if and only if herding' survives, and if not, what richer observable implications (for example, dynamic or cross-sectional dispersion moments) identify the herding motive or its distribution across analysts. The source paper asserts without proof that under certain conditions the static analysis is preserved in a dynamic version; making this precise, or establishing where it fails, is part of the task.

(2) Empirical confrontation. Implement the derived test on analyst earnings-forecast panel data (I/B/E/S-type data with coverage, horizons, and forecast timing), and answer the questions the source leaves open: whether the herding parameter varies systematically with the number of analysts and with the forecast horizon; what explains the reported decline in the absolute magnitude of the herding coefficient as the interval between the first and last forecast in a month lengthens, and its statistical insignificance in the high-coverage ($n > 10$, with controls) and earnings-announcement-day (with controls) subsamples; and the economic magnitude of the estimated herding effects beyond statistical significance. Where analyst-level panel data of this kind are not available to the solver, confront the derived implications with the source's reported estimates — the coefficient patterns summarized in the Background — and state explicitly what data or observable moments a full empirical test would require.

The two deliverables are linked: the theory supplies the testable implications whose validation on data decides whether analysts' strategic herding, as identified through dispersion and coverage, is a robust structural phenomenon rather than an artifact of the static symmetric simultaneous-move model. No numerical thresholds beyond those reported in the source are imposed; 'robust' means that the identification result, or an explicitly derived replacement for it — observable implications that identify the herding motive or its distribution across analysts — remains valid under the stated relaxations.

References

1. Jin Yeub Kim, Yongjun Kim, Myungkyu Shim, 'Do Financial Analysts Herd?', SSRN working paper (2019), DOI 10.2139/ssrn.3372445, https://doi.org/10.2139/ssrn.3372445
2. Jin Yeub Kim, Yongjun Kim, Myungkyu Shim, 'Do Financial Analysts Herd?', Global Economic Review 52(3):202-219 (2023), DOI 10.1080/1226508X.2023.2230984, https://doi.org/10.1080/1226508X.2023.2230984
3. Jin Yeub Kim, Myungkyu Shim, 'Forecast Dispersion in Finite-Player Forecasting Games', The B.E. Journal of Theoretical Economics 19(1), article 20170023 (2018), DOI 10.1515/bejte-2017-0023, https://doi.org/10.1515/bejte-2017-0023
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8. Brett Trueman, 'Analyst Forecasts and Herding Behavior', Review of Financial Studies 7(1):97-124 (1994), DOI 10.1093/rfs/7.1.97, https://doi.org/10.1093/rfs/7.1.97
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10. Dan Bernhardt, Murillo Campello, Edward Kutsoati, 'Who herds?', Journal of Financial Economics 80(3):657-675 (2006), DOI 10.1016/j.jfineco.2005.07.006, https://doi.org/10.1016/j.jfineco.2005.07.006
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