Forecasting with Feedback

The Big Idea: The "Self-Fulfilling" Crystal Ball

Imagine you are forecasting the number of cases of an infectious disease for next week. You publish your prediction: cases will be high. Because of your forecast, people take precautions — they stay home, mask up, postpone gatherings. As a result, the actual number of cases next week comes in lower than your forecast said. Your prediction didn't just describe the future; it changed the future, because the realization of the variable you were forecasting (cases) depended on the actions people took in response to your forecast.

This is what economists call feedback: the realization of the forecasted variable depends on actions taken in response to the prediction. A pure weather forecast doesn't have this property — whether it rains tomorrow doesn't depend on whether people carry umbrellas. But many economically interesting variables DO have this property: inflation responds to the central bank's policy, which is itself shaped by inflation forecasts; epidemic case counts respond to public-health behavior; financial-market prices respond to consensus expectations.

This paper asks a tricky question: If a forecaster knows their prediction will change the outcome, should they still try to be 100% accurate?

The authors' surprising answer is: No. Sometimes, the smartest thing to do is to be slightly wrong on purpose.

The Cast of Characters

To understand the paper, let's meet our two main characters:

The Forecaster (The Messenger): A smart economist who looks at data and tries to guess what will happen (e.g., "Inflation will be 5%").
The Decision Maker (The Boss): A powerful person (like a Central Bank Chair) who hears the forecast and takes action (e.g., "Okay, I'll raise interest rates to cool things down").

The Problem: The "Jittery Boss"

In the real world, the Forecaster doesn't know exactly how the Boss will react.

If the Boss is predictable, the Forecaster can say, "I know if I say 5%, the Boss will raise rates by exactly 0.5%." The Forecaster can then adjust their guess to cancel out the Boss's reaction and give a perfect prediction.
But in reality, the Boss is uncertain. Maybe the Boss is in a bad mood, maybe the committee is arguing, or maybe the Boss is just unsure. The Boss might raise rates by 0.2%, or 0.8%, or nothing at all.

This uncertainty creates a Risk of Volatility.

The Analogy: The Tightrope Walker

Imagine the Forecaster is a tightrope walker trying to balance a pole (the forecast) while the wind (the Boss's reaction) is blowing unpredictably.

The Standard View: The walker should just look straight ahead and aim for the center of the rope. If they miss, they are "irrational" or "bad at math."
The Paper's View: The walker realizes that if they lean too hard in one direction (a very precise, aggressive forecast), the wind might blow them off the rope completely because the Boss might overreact.

The Strategy: To stay safe, the walker decides to lean slightly away from the edge, even if it means they aren't pointing exactly at the center. They accept a small, predictable error (bias) to avoid a huge, chaotic mistake (variance).

The "Bias-Variance Tradeoff" (The Core Mechanism)

The paper explains that when the Boss's reaction is uncertain, the Forecaster faces a tradeoff:

Bias (Being Systematically Wrong): If the Forecaster predicts a number that is slightly too low, they might calm the Boss down, preventing the Boss from panicking and overreacting.
Variance (Being Unpredictably Wrong): If the Forecaster predicts the "true" number, the Boss might overreact wildly because the Boss is unsure how to interpret the news. This makes the final outcome swing wildly.

The Conclusion: It is mathematically better to be consistently a little bit wrong (biased) to keep the outcome stable, rather than being perfectly right and causing a chaotic swing in the economy.

Why This Matters for Real Life (The Greenbook Example)

The authors looked at real data from the Federal Reserve (the "Greenbook" forecasts). They found two weird things that traditional economics couldn't explain:

The Bias Flips: Sometimes the Fed underestimates inflation, and sometimes they overestimate it. It changes over time.
The Slope Breaks: Usually, if you plot the forecast against the actual result, the line should go up at a 45-degree angle (perfect prediction). But in the Fed's data, this line sometimes goes flat, or even goes down (negative slope).

The Paper's Explanation:
These aren't signs that the economists are "stupid" or "irrational." These are signs that they are strategically playing a game.

When the economy is hot, they might slightly under-predict inflation to stop the Boss from raising rates too aggressively (which could crash the economy).
When the economy is cold, they might over-predict to stop the Boss from panicking.

They are "sandbagging" the forecast to manage the Boss's reaction.

The "Game of Telephone" with a Twist

Think of this like a game of "Telephone," but with a twist:

Normal Game: Person A whispers a message to Person B. Person B acts on it. The message doesn't change the truth.
This Paper's Game: Person A whispers a message to Person B. Person B acts on it, and that action changes the truth.
The Twist: Person A knows Person B is a bit "jittery" (uncertain). So, Person A whispers a slightly distorted message, not to lie, but to keep Person B from freaking out and breaking the system.

The Takeaway

For decades, economists have looked at biased forecasts and thought, "These forecasters are irrational or have weird preferences."

This paper says: Stop blaming the forecasters.
If the forecast influences the decision, and the decision influences the future, then being perfectly accurate is actually a bad strategy. The smartest forecasters are the ones who know how to "game" the system slightly to keep the outcome stable.

In short: In a world where your prediction changes the result, the best prediction isn't the truest one—it's the one that keeps everyone calm.

1. Problem Statement

The paper addresses a fundamental puzzle in forecasting literature: systematically biased forecasts.

Traditional View: Biased forecasts are typically interpreted as evidence of either (i) forecasters' irrationality or (ii) asymmetric loss functions (where the cost of over-prediction differs from under-prediction).
The Gap: Standard rationality tests (like the Mincer-Zarnowitz regression) assume the forecast target is exogenous. However, in many economic contexts (e.g., central bank inflation forecasts), the forecast itself influences the policy decision, which in turn alters the realization of the target variable. This creates a feedback loop.
Core Question: Can optimal forecasts be biased even when the forecaster has a standard quadratic loss function, solely due to the presence of policy feedback and uncertainty about the policymaker's reaction?

2. Methodology

The authors develop a strategic game-theoretic model involving a Forecaster and a Decision Maker (DM).

A. The Environment

Outcome Equation: The realized outcome $y_{t+1}$ is determined by:
$y_{t+1} = \theta_t + a_t + \epsilon_{t+1}$
Where $\theta_t$ is the state of the economy (forecaster's private signal), $a_t$ is the DM's action, and $\epsilon_{t+1}$ is an unforecastable error.
Forecaster's Objective: Minimize Expected Squared Error (Quadratic Loss): $E[(y_{t+1} - f_t)^2 | \theta_t]$ .
Feedback Mechanism: The DM chooses action $a_t$ based on the forecast $f_t$ . The DM uses a "Taylor-rule" type reaction function:
$a_t = x_t [y^T - E(\theta_t | f_t)]$
Where $y^T$ is a target value, and $x_t$ is a random multiplier representing the strength of the policy response.

B. Key Assumptions

Uncertainty: The forecaster does not know the exact value of $x_t$ (the policy multiplier) at the time of forecasting; they only know its distribution (mean $\mu$ , variance $\tau^2$ ). The DM observes $x_t$ .
Strategic Interaction: The DM forms expectations about the relationship between the state $\theta_t$ and the forecast $f_t$ . The paper focuses on Linear Equilibria where the DM conjectures a linear forecast rule ( $f_t = b + c\theta_t$ ), and the forecaster optimizes given this conjecture.
No Commitment: The forecaster cannot commit to a fixed rule ex-ante; they must react to the state $\theta_t$ optimally given the DM's reaction.

C. Solution Concept

The authors solve for the Perfect Bayesian Equilibrium (PBE):

The DM conjectures a linear forecast function.
The forecaster solves the minimization problem (MSE) given the DM's reaction function and the uncertainty in $x_t$ .
The equilibrium is reached when the DM's conjecture about the forecast's slope and intercept matches the forecaster's actual optimal strategy.

3. Key Contributions

Alternative Explanation for Bias: The paper demonstrates that optimal forecasts can be biased under quadratic loss if feedback exists and there is uncertainty about the policy reaction. This challenges the standard interpretation of forecast bias as a sign of irrationality or asymmetric loss.
Bias-Variance Tradeoff in Forecasting: The authors identify a novel mechanism: the forecaster intentionally introduces bias to reduce the variance of the outcome.
- If the forecast is too sensitive to private information, and the policy reaction is uncertain, the outcome becomes highly volatile.
- To minimize Mean Squared Error (MSE), the forecaster "attenuates" the forecast (makes it less sensitive to $\theta_t$ ), effectively shrinking the forecast toward the target. This reduces outcome volatility at the cost of introducing systematic bias.
Breakdown of Canonical Rationality Tests: The paper shows that in feedback environments, the Mincer-Zarnowitz (MZ) regression coefficients (intercept and slope) deviate from the standard "unbiased" benchmarks (0 and 1) even when forecasts are rational.
Equilibrium Analysis: The paper derives explicit formulas for the equilibrium bias and MZ coefficients as functions of the mean ( $\mu$ ) and variance ( $\tau^2$ ) of the policy reaction.

4. Key Results

A. The Optimal Forecast (Proposition 1)

The optimal forecast is linear: $f^*_t = d^* + e^* \theta_t$ .

Attenuation: The slope $e^*$ is strictly less than the slope of the unbiased forecast. The forecaster dampens the response to private information.
Bias: The forecast is biased. The bias depends on the distance between the state and the target, scaled by the uncertainty in the policy reaction.

B. Statistical Properties (Proposition 2 & Corollary 2)

MZ Regression Slope: The slope of the regression of realized outcomes on forecasts ( $E[y_{t+1}|f_t]$ $E [y_{t + 1} ∣ f_{t}]$ ) is not necessarily 1.
- It can be negative, close to zero, or greater than 1, depending on the parameters $\mu$ and $\tau^2$ .
- Specifically, if the DM is aggressive ( $\mu \approx 1$ ) and uncertainty is moderate, the slope can flatten or turn negative.
MZ Regression Intercept: The intercept is generally non-zero.
Sign-Changing Bias: The model generates bias that changes sign over time. If the state $\theta_t$ is above the target $y^T$ , the forecaster under-predicts (bias > 0). If $\theta_t$ is below $y^T$ , the forecaster over-predicts (bias < 0). This explains the "sign-shifting" bias observed in Federal Reserve Greenbook data.

C. The Role of Uncertainty

No Uncertainty ( $\tau^2 = 0$ ): If the forecaster knows the DM's reaction perfectly, the forecast is unbiased. The forecaster can perfectly anticipate and correct for the feedback.
With Uncertainty ( $\tau^2 > 0$ ): The bias-variance tradeoff kicks in, leading to biased forecasts and non-standard MZ coefficients.

D. Empirical Motivation

The authors apply these theoretical insights to Federal Reserve Greenbook (GB) inflation forecasts:

GB forecasts show persistent, sign-changing bias.
The MZ slope for GB forecasts has shifted dramatically over decades (from ~1.0 in the 80s/90s to negative/zero in the 2000s).
The paper argues these patterns are consistent with a model where the Fed's reaction to inflation forecasts is uncertain and variable, rather than evidence of irrationality.

5. Significance and Implications

Redefining Rationality: The paper forces a re-evaluation of "forecast rationality." A failure of standard rationality tests (MZ regression) does not automatically imply irrationality or asymmetric loss; it may simply indicate the presence of feedback and policy uncertainty.
Policy Evaluation: For policymakers and forecasters, the results suggest that in interactive environments, the "optimal" forecast is not the one that predicts the state most accurately, but the one that minimizes the total error (bias + variance) considering the policy reaction.
Methodological Caution: Researchers testing for forecast rationality or estimating loss functions in macroeconomic settings must explicitly account for feedback mechanisms. Ignoring feedback can lead to incorrect conclusions about forecaster behavior.
Theoretical Extension: The model bridges the gap between communication games (Crawford-Sobel) and economic forecasting, introducing "costly talk" where the message (forecast) directly enters the loss function via the outcome.

In summary, Lieli and Nieto-Barthaburu provide a rigorous theoretical framework showing that strategic interaction under uncertainty is sufficient to generate biased, yet optimal, forecasts, fundamentally altering how we interpret forecast errors in policy-relevant environments.