Index: Plausible GMM

Plausible GMM

Routing Summary

Quasi-Bayesian inference for structural / moment-condition models when the moment restrictions are plausible but not exact (Chernozhukov, Hansen, Kong & Wang 2026). A proper prior $\pi (\mu )$ is placed on the degree of misspecification ${\mu }_{\ast }$, turning misspecification into a partial-identification problem. Contains 5 notes.

• New here / want the big picture? → Plausible GMM - Overview
• The model $m(\theta )=\mu$ and the IV motivation? → Plausible Moment Restriction Model
• The criterion function & quasi-posterior (Eqs. 1–3)? → Quasi-Bayes for Plausible Moment Restrictions
• Closed-form Gaussian approximation & “no free lunch” (Eq. 5)? → Gaussian Local Prior Approximation
• Worked empirical example (institutions → GDP)? → Plausible GMM - Institutions and GDP Application

Concept Map

Concept	Note	Type	Depends On	Key Result
Plausibility characteristic ${\mu }_{\ast }$; dogmatic vs. plausible prior	Plausible Moment Restriction Model	definition	Instrumental Variables	$m({\theta }_{\ast })={\mu }_{\ast }$; classical GMM is the dogmatic ${\mu }_{\ast }\equiv 0$ case
Plausible IV exclusion restriction	Plausible Moment Restriction Model	example	—	$\mathbb{E}[D_t(Y_t-\theta X_t)]=\mu \sim \mathcal{N}(0,{\sigma }^2)$
CU-GMM criterion $Q_T$; quasi-posterior $p_T$	Quasi-Bayes for Plausible Moment Restrictions	concept	Method of Simulated Moments	$p_T\propto \exp (\frac{1}{2}{Q}_T)\pi$; well defined without point ID
Local Gaussian prior $\mu \sim \mathcal{N}({\mu }_0,\Lambda /T)$	Gaussian Local Prior Approximation	theorem	Quasi-Bayes for Plausible Moment Restrictions	$\theta \approx \mathcal{N}(\hat{\theta },V/T)$, $V=(G^{\top }{\hat{A}}_{T,\theta }G{)}^{-1}$
Plausibility-adjusted weighting / no free lunch	Gaussian Local Prior Approximation	theorem	—	$A_{\theta }\ne {\Omega }^{-1}$; $V\ge$ efficient-GMM var; $\to$ efficient GMM as $\Lambda \to 0$
Institutions → GDP plausible IV	Plausible GMM - Institutions and GDP Application	example	Instrumental Variables	Posterior for ${\beta }_X$ robust to prior over $\mu$; excludes 0

Notes

• Plausible GMM - Overview — CONTAINS: research question, 5 contributions (PGMM framework, BvM concentration, optimal decisions, ex-ante frequentist coverage, endogenous robust weighting), literature map, gaps.
• Plausible Moment Restriction Model — CONTAINS: Def. moment function & target $m({\theta }_{\ast })={\mu }_{\ast }$; Def. dogmatic vs. plausible prior; Def. roots & support assumption; Example: plausible IV exclusion restriction.
• Quasi-Bayes for Plausible Moment Restrictions — CONTAINS: Def. CU-GMM criterion $Q_T$ (Eq. 1); Def. quasi-posterior $p_T$ (Eq. 2); Def. marginals; Def. optimal quasi-Bayes decision (Eq. 3); MCMC note.
• Gaussian Local Prior Approximation — CONTAINS: Def. local Gaussian prior (Eq. 4); Def. plausibility-adjusted weighting ${\hat{A}}_{T,\theta }$; Result: Gaussian quasi-posterior approximation (Eq. 5); three features (center, inflated variance, sampling dist.); “no free lunch”.
• Plausible GMM - Institutions and GDP Application — CONTAINS: linear IV model & moments; prior on $\theta$; augmented misspecification model $C_t=(1,W_t,D_t^{\top })\pi$; PGMM-g / PGMM(d)-g / PGMM-u / CH priors; Figure 1 prior-sensitivity of ${\beta }_X$.

Sources

• Plausible GMM - A Quasi-Bayesian Approach — Chernozhukov, Hansen, Kong & Wang (2026), arXiv:2507.00555 (econ.EM). Main body §1–§3.1 only; §4 theorems + 401(k) application are in the unincluded Supplemental Appendix.

See Also

• Instrumental Variables — classical exact-exclusion IV (the relaxed case here)
• Method of Simulated Moments / Simulation-Based Estimation - Overview — GMM family
• Asymptotics and Frequentist Connections — Bernstein–von Mises / quasi-Bayes background
• Sensitivity Analysis in Observational Studies — frequentist analogue of a prior over $\mu$