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 a global rank condition involving the Jacobian of the sample 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 non-linear reparameterizations, and 3) moderate misspecification. In contrast, Newton-Raphson and quasi-Newton methods can fail to converge for the same estimation because of non-convexity. A simple example illustrates a non-convex GMM estimation problem that satisfies the aforementioned rank condition. Empirical applications to random coefficient demand estimation and impulse response matching further illustrate the results.

翻译：广义矩方法和模拟矩方法常用于估计结构性经济模型。然而，优化过程因对应的目标函数非凸而常被报告为具有挑战性。对于光滑问题，本文表明凸性并非必需：在涉及样本矩雅可比矩阵的全局秩条件下，某些算法能够全局收敛。这包括梯度下降法和高斯-牛顿算法，只要适当选择调优参数。该结果对以下情况具有鲁棒性：1）非凸性；2）一对一非线性重新参数化；3）适度误设定。相比之下，牛顿-拉夫森方法和拟牛顿方法因非凸性可能无法收敛于同一估计。一个简单示例展示了满足上述秩条件的非凸GMM估计问题。对随机系数需求估计和脉冲响应匹配的实证应用进一步说明了这些结果。