Convexity Not Required: Estimation of Smooth Moment Condition Models

TL;DR

This paper demonstrates that for smooth moment condition models, convexity is not necessary for global convergence of certain algorithms, broadening the scope of reliable estimation methods in economic modeling.

Contribution

It introduces conditions involving the Jacobian that ensure global convergence of gradient-based algorithms without requiring convexity, applicable even under moderate misspecification.

Findings

Gradient descent and Gauss-Newton algorithms are globally convergent under the proposed conditions.

The methods are robust to non-convexity and moderate reparameterizations.

Numerical and empirical examples validate the theoretical results.

Abstract

Generalized and Simulated Method of Moments are often used to estimate structural Economic models. Yet, it is commonly reported that optimization is challenging because the corresponding objective function is non-convex. For smooth problems, this paper shows that convexity is not required: under conditions involving the Jacobian of the moments, certain algorithms are globally convergent. These include a gradient-descent and a Gauss-Newton algorithm with appropriate choice of tuning parameters. The results are robust to 1) non-convexity, 2) one-to-one moderately non-linear reparameterizations, and 3) moderate misspecification. The conditions preclude non-global optima. Numerical and empirical examples illustrate the condition, non-convexity, and convergence properties of different optimizers.