Quasi-Bayes for Plausible Moment Restrictions

Summary

The inference engine of Plausible GMM. A continuous-updating GMM criterion $Q_T(\theta ,\mu )$ is exponentiated and combined with the joint prior $\pi (\theta ,\mu )=\pi (\theta \mid \mu )\pi (\mu )$ to form a quasi-posterior $p_T(\theta ,\mu )$. Because it is built directly from the sample criterion and the prior, it remains well defined even when $(\theta ,\mu )$ is not point-identified. Marginalizing gives $p_T(\theta )$ (the chief object of interest) and $p_T(\mu )$ (posterior beliefs about plausibility); minimizing quasi-posterior expected loss gives optimal decisions. The posterior generally requires MCMC.

Overview

Given the Plausible Moment Restriction Model $m(\theta )=\mu$ with prior $\pi (\mu )$, we want to update beliefs about the structural parameter $\theta$. Because $(\theta ,\mu )$ is partially identified, a likelihood is unavailable/awkward; instead we use a quasi-Bayesian posterior (QBP) — the Laplace-type estimator approach of Chernozhukov & Hong (2003) — which replaces the log-likelihood with a GMM criterion function.

Main Content

Definition: Continuous-updating GMM criterion ( $\S$2.2, Eq. 1)

Let $\hat{m}(\theta ):=\frac{1}{T}{\sum }_{t=1}^{T}g(Z_t,\theta )$ be the empirical moment. The criterion for $(\theta ,\mu )$ is

$$Q_T(\theta ,\mu )=-T(\hat{m}(\theta )-\mu {)}^{\top }{\hat{\Omega }}_T(\theta {)}^{-1}(\hat{m}(\theta )-\mu ),$$

where ${\hat{\Omega }}_T(\theta )$ is a positive-definite estimate of

$$\Omega (\theta )=\underset{T\to \infty }{\lim }\mathrm{Var}\left(\sqrt{T}(\hat{m}(\theta )-m(\theta ))\right).$$

Under i.i.d. $Z_t$ one may use the centered outer-product

$${\hat{\Omega }}_T(\theta )=\frac{1}{T}\sum _{t=1}^T(g(Z_t,\theta )-\hat{m}(\theta ))(g(Z_t,\theta )-\hat{m}(\theta ){)}^{\top }.$$

This is the continuous-updating GMM objective, now penalizing the distance between the empirical moment $\hat{m}(\theta )$ and the plausibility characteristic $\mu$ (rather than from $0$).

Definition: Quasi-posterior ( $\S$2.2, Eq. 2)

With joint prior $\pi (\theta ,\mu )=\pi (\theta \mid \mu )\pi (\mu )$ and joint support $\Xi$, the quasi-posterior is

$$p_T(\theta ,\mu )=\frac{\exp \left(\frac{1}{2}{Q}_T(\theta ,\mu )\right)\pi (\theta ,\mu )}{{\int }_{\Xi }\exp \left(\frac{1}{2}{Q}_T(\theta ,\mu )\right)\pi (\theta ,\mu )d\mu d\theta }.$$

Constructed from the sample criterion and prior alone, it is well defined even when $(\theta ,\mu )$ is not point-identified. A flat prior on $\theta$, $\pi (\theta \mid \mu )\propto 1$, is common; the framework also allows economically motivated informative priors on $\theta$.

Definition: Marginal quasi-posteriors ( $\S$2.2)

Integrate out the other parameter:

$$p_T(\theta )={\int }_{\mathcal{M}}{p}_T(\theta ,\mu )d\mu ,p_T(\mu )={\int }_{\Theta }{p}_T(\theta ,\mu )d\theta .$$

$p_T(\theta )$ captures posterior information about the economically meaningful parameter (the chief object of interest). $p_T(\mu )$ summarizes posterior beliefs about the plausibility term — how much misspecification the data + prior suggest.

Definition: Optimal quasi-Bayes decision ( $\S$2.2, Eq. 3)

For a loss $\ell (\theta ,\mu ,d)$ and decision $d\in \mathcal{D}$, the optimal decision minimizes quasi-posterior expected risk:

$$s_T(p_T)\in \arg \underset{d\in \mathcal{D}}{\min }{\int }_{\Xi }\ell (\theta ,\mu ,d)p_T(\theta ,\mu )d\mu d\theta .$$

In most applications the loss depends only on $\theta$, but $\ell$ is allowed to depend on $\mu$ as well.

Interpretation and computation

• Approximate Bayesian interpretation. The quasi-posterior and its summaries (optimal decisions, credible intervals) admit an approximate Bayesian interpretation. Florens & Simoni (2021) and Andrews & Mikusheva (2022) show that, under the dogmatic prior $\mu \equiv 0$, this object corresponds to a genuine posterior for $\theta$ as the prior becomes diffuse; augmenting the parameter space to include $\mu$ extends this.
• Non-dogmatic prior matters. With a non-degenerate prior over $\mu$, the optimal Bayes decision depends explicitly on both the prior for $\theta$ (as in Andrews–Mikusheva) and the prior for $\mu$.
• Frequentist anchor. Under correct specification ($m({\theta }_{\ast })\equiv 0$) and strong identification, Chernozhukov–Hong (2003) show inference from the quasi-posterior is asymptotically equivalent to efficient GMM; posterior means/credible intervals are usable as point estimators/confidence intervals even when $Q_T$ is hard to optimize directly. With a non-degenerate prior over $\mu$, credible intervals no longer deliver the usual frequentist coverage but retain an approximate Bayes interpretation (Moon & Schorfheide 2012; Gustafson 2015) and an ex ante two-stage coverage (see Plausible GMM - Overview).
• Computation. $p_T(\theta ,\mu )$ is generally not analytic; it is approximated by Markov Chain Monte Carlo (Robert & Casella 2005) or other methods. A tractable Gaussian approximation under a local prior is given in Gaussian Local Prior Approximation.

Examples

See Plausible GMM - Institutions and GDP Application, where this quasi-posterior is simulated under several priors over $\mu$ and compared to the dogmatic ($\mu \equiv 0$) Chernozhukov–Hong posterior.

Connections

• The criterion $Q_T$ is the continuous-updating GMM objective; the QBP is the simulation-based Laplace-Type Estimator (LTE) of Chernozhukov–Hong.
• Replaces a likelihood with a criterion function — same device as in quasi-Bayes asymptotics.
• The shift from ”$\hat{m}(\theta )$ near $0$” to ”$\hat{m}(\theta )$ near $\mu$” is exactly the plausibility relaxation.

See Also

• Plausible Moment Restriction Model — the model $m(\theta )=\mu$ this conditions on
• Gaussian Local Prior Approximation — closed-form Gaussian approximation to $p_T(\theta )$
• Plausible GMM - Overview — concentration, decision, and coverage results that build on this