Plausible GMM - Institutions and GDP Application

Summary

An empirical illustration of Plausible GMM revisiting Acemoglu, Johnson & Robinson (2001) — the effect of institutions on GDP, estimated by linear IV using settler mortality as the instrument ($T=64$ ex-colonies). The exclusion restriction is treated as plausible but not exact: an augmented model adds a latent misspecification term $C_t=(1,W_t,D_t^{\top })\pi$, and a proper prior on $\pi$ (hence on $\mu$) encodes the belief that the GDP elasticity w.r.t. settler mortality is, with high probability, no larger than ~10%. The quasi-posterior for the institutions coefficient ${\beta }_X$ stays informative and is relatively robust to the prior over misspecification, growing only modestly more diffuse as the prior widens.

Overview

This is the first of the paper’s two empirical applications (the second — IV quantile regression of 401(k) participation, revisiting Chernozhukov & Hansen 2004 — is in the Supplemental Appendix and not in the main PDF). It shows how to specify priors over both $\theta$ and the plausibility characteristic $\mu$, and how to read prior-sensitivity of the resulting quasi-posterior.

Main Content

Model and data

• Outcome $Y_t$: log PPP-adjusted GDP per capita, 1995, for $t=1,\dots ,64$ ex-European colonies.
• Regressor of interest $X_t$: a ten-point index of protection against expropriation risk — a proxy for institutional quality.
• Control $W_t$: normalized distance from the equator (geography).
• Instruments $D_t$: log settler mortality (the Linear IV(1) baseline, just-identified); Linear IV(2) adds the proportion of the population of European descent in 1900 (over-identified).

Linear IV model and moment condition ( $\S$3.1)

$$Y_t=\alpha +{\beta }_X{X}_t+{\beta }_W{W}_t+U_t,\theta =(\alpha ,{\beta }_X,{\beta }_W{)}^{\top },$$

with moment function

$$g(Z_t,\theta )=(1,W_t,D_t^{\top }{)}^{\top }(Y_t-\alpha -X_t{\beta }_X-W_t{\beta }_W),$$

where $D_t$ is the vector of instruments. See Instrumental Variables for the classical (exact-exclusion) version.

Prior on the structural parameter $\theta$

$$\theta \sim \mathcal{N}\left(0,\mathrm{diag}(100,4,64)\right).$$

The variance for ${\beta }_X$ is set by economic reasoning: $X_t$ ranges over a 10-point scale (empirical 25th/75th percentiles $5.6$ and $7.8$), and a coefficient ${\beta }_X\approx 0.5$ already implies that moving from the 25th to 75th percentile of institutions is associated with ~1 log unit (~170%) higher GDP — economically large. So a magnitude $|{\beta }_X|>4$ is given low prior probability (sd $=2$). The same logic sets the priors on $\alpha$ and ${\beta }_W$.

Prior on misspecification $\mu$ (the plausibility characteristic)

Augmented "plausible" model ( $\S$3.1)

Allow the exclusion/exogeneity restrictions to fail via a latent term linear in the exogenous variables:

$$Y_t=\alpha +{\beta }_X{X}_t+{\beta }_W{W}_t+C_t+U_t,C_t=(1,W_t,D_t^{\top })\pi .$$

For Linear IV(1), the moment equation becomes, for a given $\pi$,

$$\mathbb{E}[g(Z_t,\theta )]=\mathbb{E}\left[(1,W_t,D_t^{\top }{)}^{\top }(1,W_t,D_t^{\top })\right]\pi =\mu .$$

A proper prior over $\pi$ encodes subjective beliefs about misspecification.

Calibrating the prior. Center $\pi$ at $0$ (the arguments for exclusion/exogeneity are compelling enough to center beliefs at “no violation”). For the entry associated with the excluded instrument $D_t$ (log settler mortality, centuries before 1995), one reasonably believes the direct effect is small: with high probability the GDP elasticity w.r.t. settler mortality is no larger than 10% (that entry of $\pi$ no larger than $0.1$). This is encoded as a mean-zero Gaussian with standard deviation $0.05$.

Baseline prior PGMM-g ( $\S$3.1)

$$\mu \sim \mathcal{N}\left(0,{\Sigma }_T{\Omega }_d{\Sigma }_T^{\top }\right),{\Sigma }_T=T^{-1}\sum _{t=1}^T(1,W_t,D_t^{\top }{)}^{\top }(1,W_t,D_t^{\top }),{\Omega }_d=0.05^2{I}_3.$$

For Linear IV(2) (extra instrument = proportion European descent), assuming its direct impact is, w.h.p., no greater than 1% (semi-elasticity), extend with

$${\Omega }_d=\mathrm{diag}\left(0.05^2{I}_3,0.005^2\right),$$

the final diagonal entry corresponding to the new instrument. The $1/T$-scaled ${\Sigma }_T$ realizes the local Gaussian prior $\mathcal{N}({\mu }_0,\Lambda /T)$ in data-scaled form.

Sensitivity priors

Label	Prior over $\mu$	Purpose
CH	dogmatic $\mu \equiv 0$ (Chernozhukov–Hong)	exact moments — benchmark
PGMM-g	$\mathcal{N}(0,{\Sigma }_T{\Omega }_d{\Sigma }_T^{\top })$	baseline plausible prior
PGMM(d)-g	$\mathcal{N}(0,c{\Sigma }_T{\Omega }_d{\Sigma }_T^{\top })$, $c=4$	more diffuse Gaussian
PGMM-u	Uniform over the elliptical region $C=\{({\Sigma }_T{\Omega }_d{\Sigma }_T^{\top }{)}^{1/2}c:c^{\top }c\le {\chi }_{0.68}^2(q)\}$	uniform over the 68% HDR of the Gaussian prior

Results (Figure 1)

Prior-sensitivity of the institutions coefficient ${\beta }_X$ ($\S$3.1, Fig. 1)

Upper panel (marginal quasi-posteriors for ${\beta }_X$, Linear IV(1)).

• In terms of ${\beta }_X$, the quasi-posteriors are relatively robust to the prior over $\mu$.
• They become somewhat more diffuse as prior dispersion increases ($\mu \equiv 0\to$ PGMM-g $\to$ PGMM(d)-g), but the changes are small despite the large increase in prior dispersion for $\mu$.
• The benchmark Gaussian (PGMM-g) and the related uniform (PGMM-u) produce very similar quasi-posteriors — by design of the uniform prior.
• The marginal prior for ${\beta }_X$ (dotted) is far more diffuse than any posterior — the data + moments are informative.

Lower panel (95% HPD interval for ${\beta }_X$ vs. prior scale $c$).

• The posterior midpoint is stable around $\approx 2.1$ across $c\in \{0,1,4,4.2,4.5,5.1\}$, declining slightly to $\approx 1.8$ at the largest scale.
• Interval bounds widen modestly as $c$ grows, and the intervals exclude $0$ throughout the displayed range — institutions retain a positive, economically meaningful association with GDP even under substantial relaxation of the exclusion restriction.

Linear IV(2) (over-identified) shows similar patterns (Supplemental Appendix Fig. SA.1); posteriors for elements of $\mu$ roughly align with their priors (Fig. SA.2).

Takeaway. Allowing for plausible (not exact) instrument validity makes inference less precise but more honest, and here the qualitative conclusion of Acemoglu–Johnson–Robinson — institutions matter for GDP — survives. The approach thus enhances the credibility of the empirical result. This is the “no free lunch” principle in action.

Connections

• A concrete instance of the plausible IV exclusion-restriction example.
• Uses the local Gaussian prior (data-scaled ${\Sigma }_T$) and the quasi-posterior (simulated, plus the Gaussian approximation).
• Classical exact-IV baseline: Instrumental Variables; the dogmatic posterior benchmark is Chernozhukov–Hong (2003).

See Also

• Plausible GMM - Overview — the framework and its guarantees
• Gaussian Local Prior Approximation — why posteriors widen with prior dispersion
• Sensitivity Analysis in Observational Studies — the frequentist analogue of varying $\mu$