Convexity Not Required: Estimation of Smooth Moment Condition Models

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.