Plausible Moment Restriction Model

Summary

The model that underlies Plausible GMM. A structural model implies $q\ge k$ moment conditions $m(\theta )=\mu$, where the $q$-dimensional plausibility characteristic $\mu$ measures how far the moments are from holding exactly. Classical GMM imposes the dogmatic prior ${\mu }_{\ast }\equiv 0$ (moments hold exactly). Plausible GMM instead places a proper, non-degenerate prior $\pi (\mu )$ over ${\mu }_{\ast }$, concentrated near $0$, whose spread encodes the researcher’s beliefs about possible economic violations. Because $\mu$ is unrestricted up to its prior, ${\theta }_{\ast }$ and ${\mu }_{\ast }$ are not jointly identified.

Overview

We observe i.i.d. data $\{Z_t{\}}_{t=1}^T$ from an unknown distribution ${\mathbb{P}}_{{\mu }_{\ast }}$. A posited structural economic model provides a set of moment restrictions indexed by a $q$-dimensional parameter ${\mu }_{\ast }\in \mathcal{M}$, for a $k$-dimensional structural parameter $\theta \in \Theta$ with $q\ge k$.

The conceptual move is to treat the exactness of the moment conditions as itself an object of belief: rather than asserting the moments hold exactly, the researcher asserts they plausibly hold, and quantifies the plausibility with a prior.

Main Content

Definition: Moment function and target ( $\S$2.1)

The structural model implies $q\ge k$ moment equations for the $k$-dimensional parameter $\theta$:

$$m(\theta )={\mathbb{E}}_{{\mathbb{P}}_{{\mu }_{\ast }}}\left[g(Z_t,\theta )\right],$$

and there exists a target parameter ${\theta }_{\ast }$ and a vector ${\mu }_{\ast }$ — the plausibility characteristic — satisfying

$$m({\theta }_{\ast })={\mu }_{\ast }.$$

${\mu }_{\ast }$ measures the degree to which the structural moment restrictions fail to hold exactly.

Definition: Dogmatic vs. plausible prior ( $\S$2.1)

• Classical / dogmatic GMM: ${\mu }_{\ast }$ is treated as a known fixed vector, WLOG ${\mu }_{\ast }\equiv 0$. This is equivalent to a dogmatic prior that the moment equations hold exactly — i.e. the model is correctly specified.
• Plausible GMM: the researcher departs from the dogmatic belief by using a proper, non-degenerate prior $\pi (\mu )$ over ${\mu }_{\ast }$. Concentrating $\pi (\mu )$ near $0$ captures the belief that the restrictions are likely correct; the spread/shape away from $0$ captures beliefs about economically motivated deviations.

Why a prior is needed. With no restrictions on ${\mu }_{\ast }$, it is impossible to update beliefs about ${\theta }_{\ast }$ or ${\mathbb{P}}_{{\mu }_{\ast }}$ from the structural model: for any posited $\theta$ and ${\mathbb{P}}_{\mu }$ one can always set $\mu ={\mathbb{E}}_{{\mathbb{P}}_{\mu }}[g(Z_t,\theta )]$ so the structural moment equation is satisfied. The structural restriction adds no information if ${\mu }_{\ast }$ is left completely unrestricted. A proper prior over ${\mu }_{\ast }$ is what lets the moments be informative about ${\theta }_{\ast }$ while falling short of imposing the implausible restriction that they hold exactly.

Definition: Roots and the support assumption ( $\S$2.1)

Denote any root of $m(\theta )=\mu$ by $\theta (\mu )$. For the formal results, $\pi (\mu )$ is assumed to place strictly positive mass over a region $\Gamma$ such that a solution $\theta (\mu )$ exists for every $\mu \in \Gamma$.

• Just-identified ($q=k$): essentially trivially satisfied for any prior (Hall & Inoue 2003).
• Over-identified ($q>k$): not guaranteed — care is needed when adding moment conditions about which beliefs are weak, unless the researcher uses very diffuse priors.

Partial identification by over-parameterization

The pair $(\theta ,\mu )$ is over-parameterized: the data identify $m(\theta )$, but splitting it into “structural signal” $\theta$ and “misspecification” $\mu$ requires the prior. Hence ${\theta }_{\ast }$ and ${\mu }_{\ast }$ are not jointly identified, and the impact of the prior is not asymptotically negligible — this is what makes Plausible GMM a genuine partial-identification problem rather than a standard regular estimation problem.

Examples

Example: IV exclusion restriction made plausible ( $\S$2.1)

Setup. Constant-coefficient linear model

$$Y_t=X_t{\theta }_{\ast }+U_t,$$

with $X_t$ endogenous, ${\mathbb{E}}_{{\mathbb{P}}_{{\mu }_{\ast }}}[X_t{U}_t]\ne 0$. We also observe a variable $D_t$ believed (from economic/institutional reasoning) to satisfy the exclusion restriction ${\mathbb{E}}_{{\mathbb{P}}_{{\mu }_{\ast }}}[D_t{U}_t]=0$, giving the moment condition that identifies ${\theta }_{\ast }$.

The worry. Suppose an unobserved confound $M_t$ covaries with both $Y_t$ and $D_t$: $U_t=M_t+V_t$ with $\mathbb{E}[D_t{M}_t]={\mu }_{\ast }\ne 0$ and $\mathbb{E}[D_t{V}_t]=0$. Imposing the (false) exact restriction $\mathbb{E}[D_t(Y_t-\theta X_t)]=0$ and solving yields

$$\theta ={\left({\mathbb{E}}_{{\mathbb{P}}_{{\mu }_{\ast }}}[D_t{X}_t]\right)}^{-1}{\mathbb{E}}_{{\mathbb{P}}_{{\mu }_{\ast }}}[D_t{Y}_t]={\theta }_{\ast }+{\left({\mathbb{E}}_{{\mathbb{P}}_{{\mu }_{\ast }}}[D_t{X}_t]\right)}^{-1}{\mu }_{\ast }\ne {\theta }_{\ast }.$$

The exact-IV estimand is biased by exactly the misspecification term.

The plausible fix. Instead impose ${\mathbb{E}}_{{\mathbb{P}}_{{\mu }_{\ast }}}[D_t(Y_t-\theta X_t)]=\mu$ with, e.g., $\mu \sim \mathcal{N}(0,{\sigma }^2)$. The prior mass concentrated at $0$ encodes the belief the instrument is “close to” valid; $\{\mu =0\}$ (perfect validity) has prior probability zero, reflecting that exact validity is incredibly unlikely. The prior variance ${\sigma }^2$ controls beliefs about the strength of the unobserved confound while keeping ${\mu }_{\ast }$ technically unbounded. The proper prior gives a concrete description of the moment restriction being plausibly — but not certainly — satisfied.

Connections

• Generalizes the classical moment-condition / GMM model, which is the special case ${\mu }_{\ast }\equiv 0$.
• The plausibility characteristic $\mu$ plays the role of the “sensitivity parameter” in frequentist sensitivity analysis — but here it carries a prior rather than being varied over a fixed set.
• Feeds directly into the inference machinery in Quasi-Bayes for Plausible Moment Restrictions and the tractable special case in Gaussian Local Prior Approximation.

See Also

• Plausible GMM - Overview — where this setup fits in the paper’s contribution
• Instrumental Variables — the exclusion-restriction example in its classical (exact) form
• Quasi-Bayes for Plausible Moment Restrictions — turning the model into a posterior