📈 economics

Predictive Synthesis under Sporadic Participation: Evidence from Inflation Density Surveys

This paper addresses the distortion of inflation density forecasts caused by irregular participation in professional surveys by developing coherent Bayesian updating rules that maintain a latent predictive state for each forecaster, thereby improving forecast accuracy and calibration while isolating genuine performance from mechanical panel composition effects.

Original authors: Matthew C. Johnson, Matteo Luciani, Minzhengxiong Zhang, Kenichiro McAlinn

Published 2026-02-06

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CC BY 4.0

Original authors: Matthew C. Johnson, Matteo Luciani, Minzhengxiong Zhang, Kenichiro McAlinn

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to predict the weather for next year. Instead of asking one meteorologist, you ask a team of 16 experts. Usually, you'd just take their average guess to get your final forecast.

But here's the catch: these experts are unreliable. Sometimes they take a vacation, sometimes they quit, and sometimes they come back after a long break. In the real world (specifically, the European Central Bank's survey of economists), this happens all the time.

The Problem: The "Empty Chair" Effect
The paper argues that when you try to average these experts' guesses, standard methods make a big mistake.

Imagine you are averaging the heights of people in a room to guess the average height of a city.

The Standard Way: If a very tall person leaves the room, you just remove their number and re-calculate the average. If a short person walks in, you add them and re-calculate.
The Paper's Critique: The problem is that the average might jump up or down just because the group changed, not because the actual city got taller or shorter. In the world of inflation forecasting, if a "wildcard" economist who always predicts high inflation quits, the standard average might suddenly drop. This makes it look like the economy is calming down, when really, it's just that the noisy voice is gone.

The paper calls this "artificial jumps." It confuses the real economic news with the administrative noise of who showed up to work that day.

The Solution: The "Ghost in the Machine"
The authors propose a smarter way to handle this, using a method called Bayesian Predictive Synthesis.

Instead of treating an absent expert as "gone," their method treats them as a "Ghost in the Machine."

The Latent State: Imagine every expert has a hidden "forecasting brain" (a latent state) that keeps working even when they aren't submitting a report.
When they leave (Exit): When an expert stops submitting, the system doesn't delete them. Instead, it says, "Okay, we can't see their brain right now, but we know how their brain usually relates to the others." It mathematically "marginalizes" them—meaning it averages out their hidden influence based on what the remaining experts are saying. This prevents the final forecast from jumping wildly just because someone left.
When they return (Entry): When a new expert joins, the system doesn't just slap their number into the average. It carefully "projects" them into the existing group, adjusting the math so that the new person fits smoothly into the team's history without causing a shock to the system.

The Analogy: The Orchestra
Think of the forecast as a symphony orchestra playing a piece of music (the inflation forecast).

Standard Method: If a violinist stops playing, the conductor just turns up the volume on the remaining violins to fill the gap. The music sounds different, maybe too loud or too quiet, just because the group changed.
This Paper's Method: The conductor knows that the missing violinist's part is still "in the air." Even though they aren't playing, the conductor adjusts the other instruments to account for the absence of that specific sound, keeping the overall melody smooth and true, regardless of who is currently on stage.

What They Found
The authors tested this "Ghost" method against standard ways of averaging (like just taking the average of whoever is there, or guessing what the missing people would have said).

Better Accuracy: The "Ghost" method predicted inflation more accurately, especially during chaotic times (like the pandemic or inflation surges) when experts were most likely to drop in and out.
Better Calmness: The standard methods created "jumps" in the forecast that looked like big economic changes but were actually just administrative glitches. The new method smoothed these out.
Uncertainty Matters: The biggest win wasn't just getting the number right, but getting the uncertainty right. The new method knew when to say, "We are less sure about this," when an expert left, rather than pretending everything was normal.

The Bottom Line
The paper concludes that when you are combining predictions from a group of people who don't always show up, you can't just treat them like a static list. You have to respect the fact that their "influence" exists even when they are silent. By using this mathematically coherent approach, central banks and policymakers can get a clearer, less "jumpy" picture of the economy, separating real economic shifts from the simple fact that someone forgot to fill out their survey form.

Technical Summary: Predictive Synthesis under Sporadic Participation

Problem Formulation
Central banks and institutional forecasters rely on professional surveys, such as the European Central Bank's Survey of Professional Forecasters (SPF), to assess inflation risks and uncertainty. A critical operational challenge in these settings is sporadic participation: forecasters frequently enter and exit the panel, skip rounds, and reappear after long gaps. This results in a time-varying set of active forecasters ( $A_t$ ) and missing density forecasts.

Standard aggregation methods typically handle this by either:

Dropping missing forecasters and renormalizing weights: This changes the conditioning set of the pooled object, causing artificial jumps in combined predictive means and variances driven by panel composition rather than economic information.
Ad hoc imputation: Techniques like last-observation-carried-forward (LOCF) or mean imputation introduce synthetic information, distorting the joint predictive structure and tail-risk assessments.

These mechanical artifacts confound real-time interpretation, making it difficult to distinguish genuine shifts in economic uncertainty from administrative variations in reporting. The paper formalizes the need for an aggregation rule that maintains a well-defined latent predictive state for every forecaster, even when their forecast is unobserved, ensuring the combined distribution evolves smoothly.

Methodology: Coherent Bayesian Predictive Synthesis (BPS)
The authors propose a coherent Bayesian predictive synthesis framework that treats missing forecasts not as deleted contributors, but as unobserved latent predictive states. The methodology extends the dynamic BPS model of McAlinn and West (2019) to handle time-varying expert sets through two specific operators derived from the coherence (consistency) condition:

The Coherence Condition: The decision maker's (DM) prior predictive distribution for the target variable $y_t$ must be consistent with the joint prior over latent states $x_t$ and the synthesis function $\alpha_t$ . This condition must hold even when only a subset of densities $H_{A_t,t}$ is observed.
Exit as Marginalization: When a forecaster exits (leaves $A_t$ $A_{t}$ ), their latent state $x_{j,t}$ $x_{j, t}$ is not removed from the model. Instead, the synthesis function is updated by marginalizing over the unobserved state of the exiting forecaster using the DM's conditional prior.
- Mechanism: The influence of the exiting forecaster is redistributed to remaining forecasters based on prior cross-covariances. If an exiting forecaster is highly correlated with a continuing one, their weight transfers; if idiosyncratic, their weight is absorbed into the intercept. This prevents artificial discontinuities in the predictive mean and variance.
Entry as Conditional Projection: When a new forecaster enters, the model expands to include their latent state and coefficients. To maintain coherence, the new parameters are constructed via conditional projection such that marginalizing out the new entrant recovers the pre-entry predictive distribution.
- Mechanism: New coefficients are drawn from a prior, and the intercept, existing coefficients, and residual variance are adjusted linearly to ensure the enlarged model is a coherent extension of the previous state.

The approach utilizes a Gaussian dynamic model where latent states evolve over time. The entry and exit operators ensure the posterior for the synthesis coefficients and latent states retains a fixed dimension, allowing for smooth evolution of the predictive distribution despite panel turnover.

Key Contributions
The paper makes three primary contributions:

Formalization of the Problem: It formally defines forecast combination under turnover and missingness for predictive distributions, demonstrating empirically that common ad hoc fixes (renormalization, imputation) materially distort combined uncertainty and confound evaluation in major institutional surveys.
Theoretical Development: It develops coherence-based entry and exit operators within dynamic BPS. These provide a principled, operational aggregation rule that preserves the conditional structure of the panel without requiring a structural model for participation or ad hoc imputation.
Empirical Assessment: It provides a real-time application using nearly 25 years of ECB SPF inflation density forecasts, showing that coherence-based adjustments improve predictive accuracy and the economic interpretability of uncertainty dynamics.

Empirical Results
The authors evaluate the coherent BPS approach against several benchmarks using year-ahead euro-area inflation forecasts:

Benchmarks: Equal-weight (EW) pooling, EW with imputation (LOCF, agent-specific mean imputation), and inverse-MSE weighting.
Performance Metrics: Root Mean Squared Error (RMSE) for point accuracy and Log Predictive Density Ratio (LPDR) for density calibration.
Findings:
- Density Accuracy: Coherent BPS delivers substantially better-calibrated density forecasts, evidenced by large positive cumulative LPDR gains relative to EW. This is particularly pronounced during periods of high volatility and elevated turnover (e.g., the inflation surge and COVID-19 period).
- Point Accuracy: While point forecast improvements (RMSE) are modest during tranquil periods, coherent BPS achieves lower cumulative RMSE during high-volatility episodes.
- Mechanism of Gain: The primary gain comes from correct variance accounting. Naive methods either lose variance (by dropping forecasters) or introduce stale variance (by imputing). Coherent BPS explicitly inflates predictive variance when a source of information is lost (via marginalization) and deflates it when new information arrives, ensuring uncertainty quantification reflects the actual information set.
- Stability: The synthesis weights and intercepts evolve smoothly, avoiding the artificial jumps seen in standard pooling when the panel composition changes.

Significance and Claims
The paper claims that handling forecaster turnover is not merely a secondary data issue but an integral part of the statistical problem in forecast combination.

Real-Time Interpretability: By isolating genuine performance differences from mechanical participation effects, the method ensures that movements in forecast dispersion and tail risk reflect true shifts in economic uncertainty rather than administrative artifacts.
Operational Feasibility: The approach is computationally simple, implementable within standard filtering pipelines, and does not require modeling the causes of missingness, only the handling of it via Bayesian coherence.
Policy Relevance: For central banks, the improved calibration implies that uncertainty measures are more reliable for risk assessment and communication, especially during macroeconomic stress when panel instability is highest.

The authors conclude that while simple pooling may suffice when panels are stable, coherent updating is essential when entry, exit, and intermittent participation are common, providing a stable and interpretable alternative to ad hoc renormalization.

Technical Summary: Predictive Synthesis under Sporadic Participation

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