Gaussian Local Prior Approximation

Summary

The tractable special case of the quasi-posterior: when the prior over the plausibility term is local Gaussian, $\mu \sim \mathcal{N}({\mu }_0,\Lambda /T)$, and identification is strong, the marginal quasi-posterior for $\theta$ is approximately $\mathcal{N}(\hat{\theta },V/T)$. The mode $\hat{\theta }$ is a GMM estimator using a plausibility-adjusted weighting matrix $A_{\theta }\ne \Omega (\theta {)}^{-1}$ that places the most weight on moments the researcher is most confident in. The quasi-posterior variance is strictly larger than the efficient-GMM variance — the formal “no free lunch”: allowing for misspecification costs precision. Efficient GMM is recovered as $\Lambda \to 0$ (no uncertainty about the moments).

Overview

The full quasi-posterior of Quasi-Bayes for Plausible Moment Restrictions requires MCMC. This section gives a closed-form Gaussian approximation valid when the prior on the plausibility characteristic $\mu$ is normal with small (order $1/T$) variance and the model is strongly identified. It makes transparent the forces shaping the quasi-posterior — most importantly, how it endogenously trades off moment precision against the researcher’s uncertainty about each moment’s validity.

Main Content

Definition: Local Gaussian prior ( $\S$2.3, Eq. 4)

$$\mu \sim \mathcal{N}\left({\mu }_0,\frac{\Lambda }{T}\right),$$

for a fixed $q$-vector ${\mu }_0$ and fixed full-rank $q\times q$ matrix $\Lambda$. A simple choice is diagonal $\Lambda$ with entries ${\lambda }_k$: a small ${\lambda }_k$ means little uncertainty about the plausibility of the $k$-th moment; a large ${\lambda }_k$ means high uncertainty.

“Local.” Variance of order $1/T$ means prior misspecification uncertainty shrinks at the same rate as sampling uncertainty in the moments, so neither dominates the large-$T$ asymptotics (cf. Conley, Hansen & Rossi 2012; Armstrong & Kolesár 2021).

Regularity assumptions. Set ${\mu }_0=0$ and assume a flat prior on $\theta$. Assume strong identification: $m(\theta (\mu ))=\mu$ has a unique solution $\theta (\mu )$, with the linearization around ${\theta }_0\equiv \theta ({\mu }_0)$

$$m(\theta (\mu ))=G(\theta (\mu )-{\theta }_0)+o\left(\Vert \theta (\mu )-{\theta }_0\Vert \right),G=\frac{\partial \mathbb{E}[\hat{m}(\theta )]}{\partial \theta }{|}_{\theta ={\theta }_0},$$

where $G^{\top }G$ has minimal eigenvalue bounded away from zero.

Definition: Plausibility-adjusted weighting matrix ( $\S$2.3)

$${\hat{A}}_{T,\theta }={\hat{\Omega }}_T(\theta {)}^{-1}-{\hat{\Omega }}_T(\theta {)}^{-1}[{\Lambda }^{-1}+{\hat{\Omega }}_T(\theta {)}^{-1}{]}^{-1}{\hat{\Omega }}_T(\theta {)}^{-1},$$

with population counterpart

$$A_{\theta }=\Omega (\theta {)}^{-1}-\Omega (\theta {)}^{-1}[{\Lambda }^{-1}+\Omega (\theta {)}^{-1}{]}^{-1}\Omega (\theta {)}^{-1}.$$

Unlike the efficient GMM weight $\Omega (\theta {)}^{-1}$, $A_{\theta }$ down-weights moments with large plausibility-uncertainty ${\lambda }_k$, reflecting the extra uncertainty from not knowing $\mu$ exactly. As $\Lambda \to 0$ (i.e. ${\Lambda }^{-1}\to \infty$), $A_{\theta }\to \Omega (\theta {)}^{-1}$ and efficient GMM is recovered.

Result: Gaussian quasi-posterior approximation ( $\S$2.3, Eq. 5)

Under the local Gaussian prior and strong identification, the marginal quasi-posterior is approximately proportional to

$$\exp \left(-T\Vert \hat{m}(\hat{\theta })+G(\theta -\hat{\theta }){\Vert }_{{\hat{A}}_{T,\theta }}^2/2\right),$$

where the mode $\hat{\theta }$ is the GMM estimator using weighting matrix ${\hat{A}}_{T,\theta }$. Equivalently,

$$\theta \approx \mathcal{N}\left(\hat{\theta },\frac{V}{T}\right),V={\left(G^{\top }{\hat{A}}_{T,\theta }G\right)}^{-1}.$$

Three noteworthy features

1. Center. $\hat{\theta }$ is the classical GMM estimator with weight $A_{{\theta }_0}$ rather than the efficient $\Omega ({\theta }_0{)}^{-1}$. Efficient weighting reflects only sampling variation in the moments; $A_{{\theta }_0}$ incorporates both sampling and plausibility uncertainty, placing most weight where combined uncertainty is lowest.

2. Quasi-posterior variance is inflated.

   $$V={\left(G^{\top }{A}_{{\theta }_0}G\right)}^{-1}\ge {\left(G^{\top }\Omega ({\theta }_0{)}^{-1}G\right)}^{-1},$$

   the right side being the usual efficient-GMM asymptotic variance. The gap is the extra uncertainty from lack of certainty about moment validity.

3. Sampling distribution of the center.

   $$\sqrt{T}(\hat{\theta }-{\theta }_0){\to }_d\mathcal{N}(0,\bar{V}),\bar{V}={\left(G^{\top }{A}_{{\theta }_0}G\right)}^{-1}{G}^{\top }{A}_{{\theta }_0}\Omega ({\theta }_0)A_{{\theta }_0}G{\left(G^{\top }{A}_{{\theta }_0}G\right)}^{-1},$$

   with $\bar{V}\le V$ (since $A_{{\theta }_0}\Omega ({\theta }_0)A_{{\theta }_0}\le A_{{\theta }_0}$). The quasi-posterior variance $V$ exceeds the actual sampling variance $\bar{V}$ of $\hat{\theta }$ because the latter is computed under the dogmatic belief $\mu \equiv 0$ and only reflects misspecification through reweighting, not through the prior.

Key takeaway: "No free lunch" ( $\S$2.3)

Incorporating a non-dogmatic prior over moment-condition violations necessarily produces less informative inference than the dogmatic case: the quasi-posterior variance $V$ is larger than efficient GMM. This is desirable — inference then more accurately reflects what can actually be learned when model uncertainty exists. Efficient GMM is the limiting, over-confident special case $\Lambda \to 0$.

Connections

• A computable stand-in for the general quasi-posterior (Eq. 2), valid under local Gaussian priors.
• The weighting matrix $A_{\theta }$ realizes, from a quasi-Bayesian angle, the precision-vs-misspecification trade-off that Armstrong & Kolesár (2021) obtain via a minimax criterion — see Plausible GMM - Overview.
• Recovers efficient GMM / Chernozhukov–Hong asymptotics as $\Lambda \to 0$.
• The local prior $\mathcal{N}({\mu }_0,\Lambda /T)$ is the device used in the institutions–GDP application (with data-scaled ${\Sigma }_T$).

See Also

• Quasi-Bayes for Plausible Moment Restrictions — the exact quasi-posterior this approximates
• Plausible GMM - Institutions and GDP Application — the approximation and full simulation in practice
• Plausible GMM - Overview — Bernstein–von Mises concentration results (full versions in the Supplemental Appendix)