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.