SLIM: Stochastic Learning and Inference in Overidentified Models

TL;DR

SLIM introduces a scalable stochastic framework for nonlinear GMM that achieves fast convergence, efficiency, and reliable inference without requiring initial estimators or convexity, demonstrated on large-scale demand models.

Contribution

It develops a novel stochastic approximation method for overidentified nonlinear GMM that is scalable, efficient, and does not depend on initial estimators or convexity assumptions.

Findings

SLIM solves large nonlinear GMM problems significantly faster than traditional methods.

The method achieves full-sample GMM efficiency with optional second-order refinement.

Finite-sample inference is reliable using the proposed plug-in and debiased J-test procedures.

Abstract

We propose SLIM (Stochastic Learning and Inference in overidentified Models), a scalable stochastic approximation framework for nonlinear GMM. SLIM forms iterative updates from independent mini-batches of moments and their derivatives, producing unbiased directions that ensure almost-sure convergence. It requires neither a consistent initial estimator nor global convexity and accommodates both fixed-sample and random-sampling asymptotics. We further develop an optional second-order refinement achieving full-sample GMM efficiency and inference procedures based on random scaling and plug-in methods, including plug-in, debiased plug-in, and online versions of the Sargan--Hansen $J$-test tailored to stochastic learning. In Monte Carlo experiments based on a nonlinear demand system with 576 moment conditions, 380 parameters, and $n = 10^5$, SLIM solves the model in under 1.4 hours, whereas full-sample GMM in Stata on a powerful laptop converges only after 18 hours. The debiased plug-in $J$-test delivers satisfactory finite-sample inference, and SLIM scales smoothly to $n = 10^6$.