Experience, Information Asymmetry, and Rational Forecast Bias

Abstract

We use a Bayesian model of updating forecasts in which the bias in forecast endogenously determines how the forecaster’s own estimates weigh into the posterior beliefs. Our model predicts a concave relationship between accuracy in forecast and posterior weight that is put on the forecaster’s self-assessment. We then use a panel regression to test our analytical findings and find that an analyst’s experience is indeed concavely related to the forecast error.

This study examines whether it is ever rational for analysts to post biased estimates and how information asymmetry and analyst experience factor into the decision. Using a construct where analysts wish to minimize their forecasting error, we model forecasted earnings when analysts combine private information with consensus estimates to determine the optimal forecast bias, i.e., the deviation from the consensus. We show that the analyst’s rational bias increases with information asymmetry, but is concavely related with experience. Novice analysts post estimates similar to the consensus but as they become more experienced and develop private information channels, their estimates become biased as they deviate from the consensus. Highly seasoned analysts, who have superior analytical skills and valuable relationships, need not post biased forecasts.

Appendices

Appendix 1: Proofs

2.1.1 The Objective Function

Given that the analyst’s forecast is a weighted average of the analyst’s unconditional estimate and the consensus, F = wE + (1 – w)E _c , the objective function can be expressed as

$$ \underset{w|b}{ \min}\left[{w}^2{\left({\tau}_0+\tau (b)\right)}^{-2}+{\left(1-w\right)}^2{\tau}_c^{-2}+2\rho w\left(1-w\right){\left({\tau}_0+\tau (b)\right)}^{-1}{\tau}_c^{-1}\right] $$

(2.6)

The first-order condition then is

$$ 2w{\left({\tau}_0+\tau (b)\right)}^{-2}-2\left(1-w\right){\tau}_c^{-2}+2\rho \left(1-w\right){\left({\tau}_0+\tau (b)\right)}^{-1}{\tau}_c^{-1}-2\rho w{\left({\tau}_0+\tau (b)\right)}^{-1}{\tau}_c^{-1}\equiv 0 $$

By collecting terms, we then have

$$ w\left\{{\left({\tau}_0+\tau (b)\right)}^{-2}+{\tau}_c^{-2}-2\rho {\left({\tau}_0+\tau (b)\right)}^{-1}{\tau}_c^{-1}\right\}={\tau}_c^{-2}-\rho {\left({\tau}_0+\tau (b)\right)}^{-1}{\tau}_c^{-1} $$

This means that the optimal weight is

$$ w=\frac{{\left({\tau}_0+\tau (b)\right)}^2-\rho {\tau}_c\left({\tau}_0+\tau (b)\right)}{{\left({\tau}_0+\tau (b)\right)}^2+{\tau}_c^2-2\rho {\tau}_c\left({\tau}_0+\tau (b)\right)} $$

(2.7)

2.1.2 Proof of Proposition 1

By taking the derivative of Eq. 2.7 with respect to τ ₀, we have

$$ \frac{\partial w}{\partial {\tau}_0}=\frac{2{\tau}_c^2\left({\tau}_0+\tau (b)\right)-2\rho {\tau}_c{\left({\tau}_0+\tau (b)\right)}^2-\rho {\tau}_c^3}{{\left[{\left({\tau}_0+\tau (b)\right)}^2+{\tau}_c^2-2\rho {\tau}_c\left({\tau}_0+\tau (b)\right)\right]}^2} $$

(2.8)

Clearly, since the denominator of ∂w/∂τ ₀ is positive, then the sign is only a function of the numerator. This implies that the sign changes when the numerator, 2τ _c (τ ₀ + τ(b)) − 2ρ(τ ₀ + τ(b))² − ρτ ² _c , is at maximum. To find the maximum, we solve for τ ₀ that satisfies the first-order conditions of the numerator. The first-order condition yields τ _c − 2ρ(τ ₀ + τ(b)) ≡ 0. Thus, at optimal weight τ ₀ + τ(b) = 0.5ρ ⁻¹ τ _c .

2.1.3 Proof of Proposition 2

By taking the derivative of Eq. 2.7 with respect to bias, we have

$$ \frac{\partial w}{\partial b}=\frac{\left[2{\tau}_c^2\left({\tau}_0+\tau (b)\right)-2\rho {\tau}_c{\left({\tau}_0+\tau (b)\right)}^2-\rho {\tau}_c^3\right]\frac{\partial \tau }{\partial b}}{{\left[{\left({\tau}_0+\tau (b)\right)}^2+{\tau}_c^2-2\rho {\tau}_c\left({\tau}_0+\tau (b)\right)\right]}^2} $$

(2.9)

Clearly, since the denominator of ∂w/∂b is positive, then the sign is only a function of the numerator. This implies (1) that since ∂τ/∂b is positive, then the optimal weight would be monotonically increasing with ∂τ/∂b or information asymmetry, and (2) that the optimal weight is nonlinearly, concavely related to private information precision. Since the first term in the numerator is a quadratic function of analyst’s own precision, the maximum in the function is the point at which the numerator changes sign. This point, however, is exactly the same point at which ∂w/∂τ ₀ maximizes. For biases at which τ ₀ + τ(b) falls below 0.5ρ ⁻¹ τ _c ., then so long as bias increases so does the optimal weight.

Appendix 2: Alternate Samples

See Table 2.6.

Table 2.6 Alternate samples

Appendix 3: Alternate Proxies for Information Asymmetry for Table 2.5

See Table 2.7.

Table 2.7 Alternate proxies for information asymmetry

Appendix 3 presents the results of regressing forecast error on the regressors specified in each column. Inf. asymmetry is analyst coverage in Panel A and relative firm size in Panel B. Same quarter is a dummy variable equal to one if the estimate is in the same quarter as the actual and zero otherwise. Same quarter is orthogonalized (on NumRevisions) to ensure that multicollinearity is not a problem between these two variables. Controls is a vector of firm-specific variables including broker reputation, broker size, accruals, intang. assets, and return std dev. Brokerage reputation is a ranking of brokerage reputation, where 0 is the worst and 9 is the best. Brokerage size is the natural log of the number of companies per quarter a brokerage house follows. Brokerage reputation is orthogonalized (on brokerage size) to ensure that multicollinearity is not a problem between these two variables. Accruals are the accrued revenue/liabilities utilized for earnings smoothing. Intang. assets are the covered firm’s intangible assets value relative to its total assets. Return standard deviation is the standard deviation of the covered firm’s return. I is a vector of one-digit SIC industry dummies. T is a vector of time dummies.