A Disciplined Cross-Section: Bayesian Model Averaging over the Factor Zoo Under the Joint-Hypothesis Constraint

Fama's joint-hypothesis problem is formalized as a Bayesian posterior decomposition, delivering a stopping rule for factor inclusion

The paper constructs a Bayesian model-averaging framework over the full factor zoo (K ≥ 200) that treats the joint-hypothesis constraint as an explicit posterior object, not a rhetorical device.

• Likelihood decomposed into pricing-model adequacy and market-efficiency components, both estimable
• Posterior-odds stopping rule derived from the joint-hypothesis structure, asymptotically equivalent to FDR control
• Nests FF3, FF5, HLZ t=3, and KMZ complexity as special-case prior configurations

Over 400 candidate factors exist, with 65% replication failure, creating a triple crisis

Harvey, Liu, and Zhu (2016) catalogued 316 factors; Hou, Xue, and Zhang (2020) found 65% of 452 anomalies fail replication. The field faces replication, multiple-testing, and model-selection crises simultaneously.

• Chen-Zimmermann (2022) open-source library provides 207 replicable factors
• Fama's own progression from 3 to 5 factors implies a need for a stopping rule
• Existing answers (HLZ t=3, FDR, dense models) are not cast in Fama's language

The joint-hypothesis constraint is written as a Bayesian likelihood factorization

Fama (1970) stated the constraint verbally; this paper writes it as p(DM_j,E) = L_pricing · L_efficient, making both components independently estimable.

• Posterior on efficiency is model-averaged: p(ED) = Σ p(M_j, E=1D)
• Previous Bayesian work (Barillas-Shanken 2018) limited to 6 models, not 2^K
• Spike-and-slab priors enable stochastic search over 10⁶⁰ model space

Two Fama-consistent hyperparameters encode parsimony and theory motivation

π encodes preference for compact models; π boosts prior inclusion probability for theory-motivated factors (MKT, SMB, HML, RMW, CMA, momentum).

• Parsimony prior: p(f_k ∈ M_j) = π · g(M_j; π ), decreasing in model size
• Theory prior: p(f_k ∈ M_j) ∝ exp(π ) for theory factors, 1 otherwise
• No assumption hard-coded; every assumption is a tunable prior

Residual alphas are decomposed into omitted-risk vs. genuine-mispricing components

L_efficient compares probability that observed alphas arise from omitted systematic risk versus true mispricing, using persistence, cross-sectional structure, and international covariation.

• Persistence: omitted-risk alphas persistent; mispricing alphas transient (McLean-Pontiff 2016)
• Cross-sectional structure: omitted-risk alphas load on firm characteristics
• International covariation: systematic risk appears across markets (Fama-French 2012, 2017)

Gibbs sampler with spike-and-slab priors identifies 7–9 factors in U.S. data

500,000 MCMC draws (100,000 burn-in) over 207 factors from Chen-Zimmermann library, using CRSP/Compustat 1963–2025. Baseline hyperparameters π =1.0, π =1.0.

• 8 factors exceed posterior inclusion threshold τ*=0.72 at 5% Bayesian FDR
• FF5 factors all survive (MKT 0.998, SMB 0.94, HML 0.91, RMW 0.89, CMA 0.85)
• Momentum (0.82), short-term reversal (0.78), accruals (0.74) also included

Posterior-odds stopping rule controls Bayesian FDR and is asymptotically equivalent to Benjamini-Hochberg

1: Include factor f_k if p(f_kD) > τ*, where τ* is derived from priors and data via the joint-hypothesis decomposition — not chosen by convention.

• Bayesian FDR control follows Newton et al. (2004) direct-posterior-probability approach
• Posterior consistency: p(f_kD) → 1 for true factors, 0 for false (Narisetty-He 2014)
• τ* = f(π , π , L_efficient) — emerges from framework, not arbitrary

Framework nests FF3, FF5, HLZ t=3, and KMZ complexity as special prior cases

Under strong parsimony and 1963–1991 data, posterior mode recovers FF3. Weakened parsimony with 2015 data recovers FF5. Flat priors reproduce HLZ t=3. Zero parsimony reproduces KMZ dense-model result.

• FF3: π →∞, π >>0, data 1963–1991 → MKT, SMB, HML
• FF5: π moderate, data to 2015 → MKT, SMB, HML, RMW, CMA
• HLZ: flat priors, τ → t≈3 threshold; KMZ: π →0 → ridge regression

7–9 factor posterior holds across European, Japanese, and Asia-Pacific markets

Using Fama-French international datasets, the posterior identifies similar factor sets, confirming robustness. Out-of-sample cross-sectional R² competitive with dense ML approaches.

• International replication follows Fama-French (2012, 2017) methodology
• Posterior mode at 8 factors, 90% credible interval [6, 11]
• Models with >15 factors have <0.02 posterior probability

Factor model misspecification propagates welfare costs through pension, insurance, and sovereign debt

The Private Pareto Theorem (Postnieks 2026a) shows bilateral optimality does not guarantee system-level welfare. Getting factor pricing right is necessary for efficient capital allocation.

• Factor models determine cost of capital, pension expected returns, insurance risk assessments
• Mispricing propagates through credit markets, pension funding, insurance, sovereign debt
• BMA framework is a methodological contribution with welfare implications across 61 domain papers

The framework resolves a sixty-year gap: Fama's implied stopping rule is now explicit

The posterior-odds rule answers 'when do you stop adding factors?' within Fama's own methodological frame. The answer is a procedure, not a number.

• Rational-equilibrium null maintained throughout; no behavioral content
• Five falsification bounties (F1–F5) specify exact conditions for each core claim to fail
• Open-source replication: seed=20260419, 500,000 draws, full repository documented