Forecasting with Feedback

Layer 1: Overview

This paper develops a strategic model of point forecast production in environments where the forecast itself influences the outcome being predicted — what the authors call “forecasting with feedback.” The canonical example is Federal Reserve staff (Greenbook) inflation forecasts: these forecasts guide FOMC interest rate decisions, and those rate decisions in turn affect realized inflation. The central theoretical claim, proved formally, is that even a forecaster with purely quadratic (mean-squared-error) loss will optimally produce biased forecasts in such environments, provided there is some uncertainty about how strongly the decision maker (DM) will react to the forecast. This finding offers a third interpretation of observed forecast biases — beyond the two dominant explanations in the prior literature, namely forecaster irrationality and asymmetric loss functions.

The model has three components. First, an outcome equation: y_{t+1} = theta_t + a_t + epsilon_{t+1}, where theta_t is a private signal (the state of the economy) observed only by the forecaster, a_t is the DM’s action, and epsilon_{t+1} is unforecastable noise. Second, a DM reaction function: a_t = x_t * [y_T - E(theta_t | f_t)], analogous to a Taylor rule, where y_T is a known target, and x_t is a strength-of-reaction multiplier drawn from a distribution with mean mu and variance tau^2; x_t is the DM’s private information. Third, the forecaster minimizes expected squared error, anticipating the DM’s endogenous response. The model is linear and closed-form solutions are derived.

The key mechanism is a bias-variance tradeoff. Because the DM’s action responds to the forecast, the variance of the realized outcome itself becomes a function of the forecast. When the DM’s reaction strength x_t is uncertain (tau^2 > 0), this variance-of-outcome term is not trivially minimized by an unbiased forecast. The forecaster reduces outcome volatility by attenuating the sensitivity of the forecast to the state — shrinking the forecast slope toward zero relative to what an unbiased forecast would require — at the cost of introducing systematic bias. When tau^2 = 0 (no uncertainty about the DM’s reaction), the forecaster can perfectly anticipate and correct for the DM’s response, and the optimal forecast is unbiased. Feedback alone, without uncertainty, does not produce bias.

The paper derives equilibrium forecasts in a Perfect Bayesian Equilibrium where the DM holds correct (rational) beliefs about the forecasting rule. Key analytical results include: (i) the equilibrium exists when tau^2 <= 1/4; (ii) the equilibrium conditional bias equals [(1 - sqrt(1 - 4*tau^2))/2] * (theta_t - y_T), which changes sign depending on whether the state is above or below the target — the forecaster gravitates toward the target; (iii) the Mincer-Zarnowitz (MZ) regression slope (the slope from regressing realized outcomes on forecasts) can be large and positive, close to zero, or even negative, depending on mu and tau^2; (iv) when mu = 1 (the DM on average fully closes the gap to the target), the equilibrium MZ slope is exactly zero for any tau^2 value.

The paper motivates these results with two documented empirical patterns in Greenbook 4-quarter-ahead inflation forecasts from 1980q1 to 2019q4. First, using 40-quarter rolling windows, bias in Greenbook forecasts is persistent but sign-changing over time — a pattern consistent with the model’s prediction that the sign of bias tracks whether the state theta_t is above or below the inflation target y_T. Second, the MZ slope (from 40-quarter rolling-window regressions) hovers near unity in the mid-1980s through early 1990s, returns to unity by the late 1990s, then drops sharply to significantly negative territory by the mid-2000s, before becoming indistinguishable from zero in the final portion of the sample — a pattern consistent with the model’s prediction that the MZ slope shifts radically with changes in mu and tau^2. Both facts are computed using the last revision of the GDP deflator.

The policy and methodological implications are significant. Standard forecast rationality tests (Mincer-Zarnowitz regressions, bias tests) are designed to detect irrationality or asymmetric loss, but in feedback environments these same test statistics can indicate “failure” even when the forecaster is fully rational under quadratic loss. Studies conducting rationality tests or estimating loss functions must either explicitly assume away feedback (and justify that assumption) or account for the feedback mechanism.

Layer 2: Deep Dive

What is the identification strategy, and what are the main threats to identification?

The paper is primarily theoretical: it derives closed-form equilibrium forecasting rules and forecast statistics from first principles within a stylized game-theoretic model. There is no econometric identification exercise. The Greenbook evidence is descriptive and motivational — rolling-window bias estimates and MZ slope estimates are presented as stylized facts consistent with the theory, not as causal identification. The main caveat the authors themselves make is that the model is not claimed to be an exclusive or exhaustive explanation of the documented GB forecast patterns. Inflation forecasting is complex, and many other factors (learning, structural breaks, regime changes in monetary policy, data revisions) could contribute to the observed patterns. The authors explicitly disclaim any claim to exclusivity.

What is the core mathematical mechanism, and how does uncertainty play a necessary role?

The forecaster’s MSE decomposes into a conditional variance term and a squared-bias term: MSE = Var[a*(f_t) | theta_t] + bias^2(f_t | theta_t) + sigma^2. The critical insight is that when x_t (the reaction-strength multiplier) is uncertain, the variance of the DM’s action — and hence of the outcome — depends on the level of the forecast itself. Specifically, Var[a*(f_t) | theta_t] = tau^2 * (y_T - f_t/c + b/c)^2. So choosing a larger or smaller forecast changes not just the bias term but also the variance term. The optimal resolution of this tradeoff requires an attenuated (biased) forecast slope. When tau^2 = 0 (no uncertainty), the variance term vanishes entirely and the forecaster can correct for feedback in full by solving a fixed-point problem, producing an unbiased forecast. The paper explicitly proves (taking limits as tau^2 to 0 in the bias and MZ slope formulas) that both return to zero and one respectively, confirming that uncertainty is a necessary condition for bias.

What is the equilibrium concept and what are its properties?

The equilibrium is a linear Perfect Bayesian Equilibrium (PBE). The DM conjectures that the forecast is a linear function f_t = b + c*theta_t, uses that conjecture to form expectations E(theta_t | f_t) = (f_t - b)/c, and chooses her action optimally. Equilibrium requires that the DM’s conjectured intercept and slope (b, c) coincide with those actually used by the forecaster. The paper shows (Corollary 1) that such a linear PBE exists when tau^2 <= 1/4, and that the equilibrium is fully revealing — the DM can learn the true state theta_t from the forecast because the forecast is a one-to-one function of the state. Two linear equilibria exist: the paper focuses on the Pareto-preferred one (lower forecaster loss, lower absolute bias), which is also the one whose limit as tau^2 approaches 0 corresponds to the natural optimal forecast.

What sign and magnitude patterns does the equilibrium bias exhibit?

From Corollary 2(a), the conditional equilibrium bias is: E(y_{t+1} - f_t^dagger | theta_t) = [(1 - sqrt(1 - 4tau^2)) / 2] * (theta_t - y_T). The multiplier (1 - sqrt(1 - 4tau^2))/2 is always positive (for tau^2 in (0, 1/4]), so the sign of the bias is determined entirely by the sign of (theta_t - y_T). When theta_t > y_T (state above target), bias is positive — the forecaster underpredicts, shrinking the forecast toward the target. When theta_t < y_T, bias is negative — the forecaster overpredicts, again gravitating toward the target. This sign-change mechanism, driven by changing economic conditions relative to a fixed target, is cited as consistent with the persistent but sign-changing bias observed in Greenbook inflation forecasts from 1980 to 2019.

What does the model predict about the Mincer-Zarnowitz slope, and how variable can it be?

From Corollary 2(b), the MZ slope in equilibrium is a highly nonlinear function of mu and tau^2. Figure 3 in the paper (discussed in the text) shows that the slope can be large and positive, positive but close to zero, negative, or even very steeply negative, for different combinations of mu and tau^2. A key special case: when mu = 1 (DM fully closes the gap to target on average), E(y_{t+1} | f_t^dagger) = y_T for all values of the forecast, giving an MZ slope of exactly zero and intercept equal to y_T. The authors note that when mu is close to 1 and tau^2 is small, even small deviations of mu from unity can produce large positive or negative MZ slopes. The model can thus account for the dramatic shift in the GB MZ slope documented in the paper — from around unity in the 1980s-1990s, to significantly negative territory in the mid-2000s, to approximately zero thereafter.

What is the relationship between the DM’s reaction function and the Taylor rule, and how is it microfounded?

The DM’s reaction function is a_t* = x_t * [y_T - E(theta_t | f_t)], directly analogous in spirit to a Taylor rule (Taylor, 1993). Online Appendix A provides a formal microfoundation: if the DM minimizes a quadratic loss in (y_{t+1} - y_T)^2 plus a quadratic adjustment cost w_t * a_t^2 — where w_t is a private, randomly drawn adjustment cost parameter — then the optimal action is precisely a_t* = x_t * [y_T - E(theta_t | f_t)] with x_t = 1/(1 + w_t). This microfoundation connects the model to the literature on central bank optimal control and provides a rational justification for the reaction function structure used throughout the paper.

How does this paper relate to and differ from the Crawford-Sobel (1982) cheap talk model?

The paper borrows the sender-receiver communication game structure from Crawford and Sobel (1982), with the forecaster as sender and the DM as receiver. However, it departs in two important ways. First, in Crawford-Sobel, the sender’s payoff depends only on the state and the action, not directly on the message (the forecast). In this paper, the forecast enters the forecaster’s loss function directly through the outcome equation (y = theta + a + epsilon, and the forecast determines a which determines y which enters the loss), making it a model of ‘costly talk’ in the sense of Kartik, Ottaviani, and Squintani (2007). Second, in standard communication games the realized outcome is exogenous — the DM’s action affects only her own payoff but not the variable being forecast. Here, the DM’s action causally determines the realized outcome that the forecaster was trying to predict. This feedback causality is absent in the standard setup and is the source of the paper’s novel results.

How does this paper relate to Bernanke and Woodford (1997)?

Bernanke and Woodford (1997) also study professional inflation forecasts and monetary policy in a rational expectations equilibrium framework, and raise the question of whether an informative equilibrium exists — concluding it may not. This paper differs in three respects: it assumes the forecaster has private information (state theta_t) that the DM cannot directly observe; it works in an environment with uncertainty about the DM’s reaction (x_t is random); and rather than focusing on equilibrium existence, it derives the statistical properties of equilibrium forecasts — the bias formula, MZ regression coefficients — which Bernanke and Woodford do not. The authors describe their work as providing ’the first formal treatment of the statistical properties of forecasts’ in feedback environments.

What heterogeneity and parameter sensitivity is documented?

The paper documents sensitivity of forecast properties to mu (mean policy reaction strength) and tau^2 (variance of policy reaction strength). The DM’s average aggressiveness mu affects both the sign and magnitude of the MZ slope: for cautious DMs (mu near 0.1), the equilibrium MZ slope is relatively close to unity; for aggressive DMs (mu near 1), the slope can flatten toward zero; for moderate but increasing mu (with tau^2 above a threshold of approximately 0.05), the slope flattens monotonically. A higher tau^2 at given mu generally attenuates the slope toward zero, but the relationship is nonlinear. When mu is precisely one, the MZ slope is exactly zero regardless of tau^2. The equilibrium bias magnitude scales with [(1 - sqrt(1 - 4*tau^2))/2], which increases in tau^2. The sign of bias is determined by the direction of (theta_t - y_T). The paper does not present cross-sectional or time-series panel heterogeneity — the parametric sensitivity analysis in Figure 3 constitutes the heterogeneity exercise.

What robustness checks are run for the Greenbook empirical patterns?

The authors state (in a footnote) that the documented patterns — persistent but sign-changing bias in 4-quarter-ahead GB inflation forecasts from 1980q1 to 2019q4 — are robust to using the second release of the GDP deflator rather than the last release. The main results use the last release. The choice of 40-quarter (10-year) rolling window is applied uniformly for both the bias plot and the MZ slope plot. No additional robustness checks (alternative window lengths, alternative forecast horizons, formal structural break tests) are explicitly documented in the paper, though the authors cite Rossi and Sekhposyan (2016), who use formal rationality tests and confirm that GB forecast rationality breaks down around 2005 — consistent with the pattern the authors document via the rolling MZ slope.

What does the model say about the forecaster’s inability to commit, and could commitment help?

In the baseline model, the forecaster cannot commit to a fixed forecasting rule ex ante because the state theta_t is not directly observable by the DM. The authors note in Section 3.3 that modeling forecasters with commitment is a straightforward extension, and that commitment can actually increase forecaster welfare in equilibrium. However, this extension is not formally developed in the paper. The intuition is that if the forecaster could credibly commit to a more informative forecast rule, the DM could react more precisely, reducing the variance of outcomes; but without commitment, the strategic equilibrium involves an attenuated (biased) forecast.

What are the implications for forecast rationality tests and loss function estimation?

The paper’s central methodological warning is that standard forecast rationality tests (MZ regression tests for zero intercept and unit slope; bias tests) and loss function estimation exercises are contaminated in environments with policy feedback. If feedback is present and x_t is uncertain, a fully rational forecaster with quadratic loss will produce forecasts that fail standard rationality tests — showing nonzero bias, non-unit MZ slopes (potentially even negative), and forecast errors correlated with the forecaster’s own information. Researchers conducting such tests must either: (a) explicitly assume no feedback applies (and justify this assumption in their specific application), or (b) carefully model the feedback mechanism and account for it. Studies that interpret GB forecast irrationality (e.g., Rossi and Sekhposyan 2016) or asymmetric loss (e.g., Capistran 2008) as the explanation for observed GB forecast properties may be confounded by the feedback mechanism identified in this paper.

What are the conditions under which a linear equilibrium does or does not exist?

From Corollary 1 and Remark 3 following it: a linear PBE exists if and only if tau^2 <= 1/4. When tau^2 > 1/4, the forecaster always wants to attenuate the slope more than the DM expects, so no fixed-point equilibrium in linear strategies exists. The paper also notes a sufficient condition for equilibrium existence: if the support of x_t is contained in [0, 1] (the DM never overreacts and never underreacts by more than half), then tau^2 <= 1/4 is automatically satisfied and an equilibrium always exists. Two linear equilibria exist when tau^2 <= 1/4, but the paper focuses on the Pareto-preferred one, which has lower forecaster loss, lower absolute bias, and a natural limiting behavior as tau^2 approaches 0.

What scope conditions limit the applicability of the results?

Several scope conditions are made explicit: (1) The outcome equation is linear; nonlinear outcome determination would change quantitative results but the feedback mechanism would persist qualitatively. (2) The model is a single-period (point-in-time) game, not a multi-period learning model — it does not analyze how beliefs about mu and tau^2 evolve over time. (3) The independence assumption between x_t and theta_t is a benchmark; if policy aggressiveness varies with economic conditions, additional effects arise. (4) The focus on linear equilibria rules out non-linear forecasting strategies. (5) The results apply to unconditional forecasts (where the forecaster anticipates the DM’s response); conditional forecasts (conditioned on a pre-specified action) behave differently. (6) The empirical Greenbook evidence is illustrative, not a formal test of the model — the authors explicitly state they do not claim their model provides an exclusive explanation of GB forecast properties.

Key Concepts

Forecasting with feedback: A forecasting environment in which the DM’s action — taken in response to the forecast — causally affects the realized value of the variable being forecast, so that the forecast influences its own target outcome. Distinguished from no-feedback environments (e.g., weather forecasting) where decisions made on the basis of the forecast do not affect the outcome.

Unconditional forecast: A forecast that anticipates and factors in the expected response of the decision maker to the forecast itself, rather than being conditioned on a pre-specified (potentially counterfactual) action. The paper’s model produces unconditional forecasts; conditional forecasts (conditioned on a given policy path) are a distinct and narrower concept.

Bias-variance tradeoff (in feedback forecasting): The tradeoff that arises when the DM’s reaction to the forecast is uncertain: a less informative (attenuated) forecast reduces the variance of the outcome (by inducing a less volatile policy action) but introduces systematic bias. The optimal forecast under quadratic loss resolves this tradeoff by attenuating the forecast slope below what an unbiased forecast would require, producing an optimally biased forecast.

Reaction function (DM’s): The rule by which the decision maker translates a forecast into a policy action: a_t* = x_t * [y_T - E(theta_t | f_t)], analogous to a Taylor rule. The multiplier x_t captures the strength of the policy response and is drawn from a distribution with mean mu and variance tau^2; it is the DM’s private information and a key source of the forecaster’s uncertainty.

Mincer-Zarnowitz (MZ) regression: The linear regression of the realized outcome on the forecast: y_{t+1} = alpha + beta * f_t + error. Under the canonical null of rational forecasting with quadratic loss and no feedback, the intercept alpha should be zero and the slope beta should be one. The paper shows that under optimal forecasting with feedback, alpha and beta can take a wide range of values, including negative beta, even when the forecaster is rational.

Equilibrium forecast slope (c-dagger): The slope of the linear forecasting rule in Perfect Bayesian Equilibrium, given by c^dagger = (1/2) - mu + sqrt(1 - 4*tau^2)/2. This slope is less than one and can be negative depending on mu and tau^2, reflecting the attenuation of the forecast toward the policy target that arises from the bias-variance tradeoff under uncertain DM reactions.

Greenbook (GB) inflation forecasts: Inflation forecasts produced by Federal Reserve staff (now called Tealbook forecasts), used as empirical motivation in the paper. The paper documents two stylized facts for 4-quarter-ahead GB forecasts from 1980q1 to 2019q4: (i) persistent but sign-changing bias in rolling 40-quarter windows, and (ii) a dramatic shift in the rolling MZ slope from approximately unity in the 1980s-1990s to significantly negative in the mid-2000s and approximately zero in the final part of the sample.

Policy feedback (as a confound for rationality tests): The paper’s use of this term to describe the mechanism by which the presence of feedback invalidates the standard interpretation of forecast rationality test outcomes: a forecaster who is fully rational (quadratic loss, no private agenda) and operating in a feedback environment will systematically produce forecasts that fail standard MZ-based rationality tests, not because of irrationality or asymmetric loss, but because of the optimal bias-variance tradeoff induced by uncertain policy reactions.