Quasi-Bayesian Hierarchical Models

We develop the Quasi-Bayesian Hierarchical Model (QBHM) for grouped GMM settings.

The framework combines Bayesian hierarchical modelling with Laplace-type estimation: it preserves each group-specific objective function, while introducing a pooling term for economically comparable parameters.

When the number of studies is fixed, the QBHM estimator-the quasi-posterior mean-has the same asymptotic distribution as GMM when estimating strongly identified study parameters.

For weakly identified studies, we analyze the asymptotic properties of the method via a weak-GMM limit experiment: an asymptotic approximation in which the sample-moment criterion remains a random function over the weak parameter space, and the upper-level pooling relation induces a family of priors over weak values.

In this experiment, the weak-limit QBHM rule is a Bayes rule under squared loss for the hierarchy-induced weak-limit prior, which provides a decision-theoretic justification for our procedure.

We also extend our results to mixed within-study blocks, allowing a single study to contain both strongly and weakly identified parameters.

Pooling can also reduce the pointwise asymptotic mean squared error (MSE) relative to unpooled estimation when the bias--variance tradeoff is favorable.

Gaussian likelihood, nonlinear weak-GMM, and weak-IV calculations show when this happens, while simulations and a microenterprise application illustrate the method.