An efficient stochastic generalized method of moments estimator for nonlinear moment models
Abstract
We study the asymptotic properties of the nonlinear stochastic GMM algorithm proposed by Chen, Lee, Liao, Seo, Shin and Song for recursively estimating parameters defined by possibly nonlinear and overidentified moment restrictions E[g(X,β*)]=0. Each observation updates estimates of the moment Jacobian, inverse covariance matrix and Gauss--Newton preconditioner at a cost independent of sample size. For linear and nonlinear instrumental-variable moments, we derive an exact rank-two Sherman--Morrison--Woodbury update. The algorithm combines preconditioned stochastic approximation with iterate averaging. Under a preconditioned monotonicity condition, satisfied globally in linear models and locally around a consistent pilot estimate in general, we establish almost sure convergence and rates, as well as a functional central limit theorem with the efficient GMM variance. These results require four moments of the moment function and its Jacobian when the step exponent exceeds 2/3. The inverse Gauss--Newton matrix consistently estimates this variance, and a projected linearised statistic enables online testing of overidentifying restrictions. We extend the results to L-dependent data with long-run variance weights under stronger moment assumptions. Simulations show that the averaged estimator approaches offline two-step GMM in accuracy and coverage at 106 observations, while requiring a fraction of its computing time and fixed memory at up to 107 observations. The online test achieves correct size and comparable power. In an application estimating the effect of a third child on weeks worked using 254,654 women from the 1980 US Census, single-pass estimates agree with offline GMM within a fraction of a standard error on average across data orderings, and both methods yield the same overidentification-test conclusion in most orderings.
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BibTeX
@misc{nguyen2026efficient,
title = {{An efficient stochastic generalized method of moments estimator for nonlinear moment models}},
author = {Duy Nguyen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.12873},
eprint = {2610.12873},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.12873},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An efficient stochastic generalized method of moments estimator for nonlinear moment models
AU - Duy Nguyen
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.12873
UR - https://arxiv.org/abs/2610.12873
AB - We study the asymptotic properties of the nonlinear stochastic GMM algorithm proposed by Chen, Lee, Liao, Seo, Shin and Song for recursively estimating parameters defined by possibly nonlinear and overidentified moment restrictions E[g(X,β^*)]=0. Each observation updates estimates of the moment Jacobian, inverse covariance matrix and Gauss--Newton preconditioner at a cost independent of sample size. For linear and nonlinear instrumental-variable moments, we derive an exact rank-two Sherman--Morrison--Woodbury update. The algorithm combines preconditioned stochastic approximation with iterate averaging. Under a preconditioned monotonicity condition, satisfied globally in linear models and locally around a consistent pilot estimate in general, we establish almost sure convergence and rates, as well as a functional central limit theorem with the efficient GMM variance. These results require four moments of the moment function and its Jacobian when the step exponent exceeds 2/3. The inverse Gauss--Newton matrix consistently estimates this variance, and a projected linearised statistic enables online testing of overidentifying restrictions. We extend the results to L-dependent data with long-run variance weights under stronger moment assumptions. Simulations show that the averaged estimator approaches offline two-step GMM in accuracy and coverage at 10^6 observations, while requiring a fraction of its computing time and fixed memory at up to 10^7 observations. The online test achieves correct size and comparable power. In an application estimating the effect of a third child on weeks worked using 254,654 women from the 1980 US Census, single-pass estimates agree with offline GMM within a fraction of a standard error on average across data orderings, and both methods yield the same overidentification-test conclusion in most orderings.
ER -