Extremum estimation and numerical derivatives
Extremum estimation and numerical derivatives
复制标题
极值估计和数值导数
DOI：
10.1016/j.jeconom.2014.05.019
复制
发表时间：
2015
影响因子：
6.3
通讯作者：
Denis Nekipelov
中科院分区：
文献类型：
--
作者：
H. Hong;A. Mahajan;Denis Nekipelov
有限差分近似广泛用于实证工作中来评估估计函数的导数。例如，许多标准优化例程依赖于有限差分公式来进行梯度计算和估计标准误差。然而，这种近似对所得估计量的统计特性的影响仅在少数特殊情况下进行了研究。本文研究了常用的有限差分方法对所得估计量的大样本特性的影响。我们发现，首先，需要根据样本大小调整步长。其次，高阶有限差分公式减少了类似于高阶核的渐近偏差。第三，我们为梯度和方向导数的有限差分近似的一致一致性提供了弱充分条件。第四，我们分析基于数值梯度的极值估计量，发现所得估计量的渐近分布可能取决于步长的序列。我们陈述了基于数值导数的极值估计量一致且渐近正态的条件。第五，我们将我们的结果推广到半参数估计问题。最后，我们证明我们的结果适用于一系列非标准估计程序。
Finite-difference approximations are widely used in empirical work to evaluate derivatives of estimated functions. For instance, many standard optimization routines rely on finite-difference formulas for gradient calculations and estimating standard errors. However, the effect of such approximations on the statistical properties of the resulting estimators has only been studied in a few special cases. This paper investigates the impact of commonly used finite-difference methods on the large sample properties of the resulting estimators. We find that first, one needs to adjust the step size as a function of the sample size. Second, higher-order finite difference formulas reduce the asymptotic bias analogous to higher order kernels. Third, we provide weak sufficient conditions for uniform consistency of the finite-difference approximations for gradients and directional derivatives. Fourth, we analyze numerical gradient-based extremum estimators and find that the asymptotic distribution of the resulting estimators may depend on the sequence of step sizes. We state conditions under which the numerical derivative based extremum estimator is consistent and asymptotically normal. Fifth, we generalize our results to semiparametric estimation problems. Finally, we demonstrate that our results apply to a range of nonstandard estimation procedures.