Estimation of Nonparametric Models With Simultaneity
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存在联立关系的非参数模型估计
- 研究问题
- 在存在联立关系的非参数、非加性结构模型中，如何利用可观测变量的分布特征估计结构导数，并刻画观测等价？
- 方法与思路
- 作者将结构导数与可观测密度的导数相联系，提出基于矩阵运算及在特定工具变量取值处计算密度导数的两种方法。
- 材料陈述的贡献
- 摘要称论文给出两类模型的观测等价新刻画，并据此构造易于计算的结构导数估计量，两种逐点估计均一致且渐近正态。
- 阅读关注点
- 核对每个方程的排他性回归变量与单工具变量设定、密度导数的估计要求，以及矩阵可逆性和两步取值搜索的条件。
- 阅读边界
- 摘要未说明密度估计的带宽、收敛速度或有限样本效果；逐点渐近结果也不能直接推及整个结构函数的统一推断。
原文摘要
We introduce methods for estimating nonparametric, nonadditive models with simultaneity. The methods are developed by directly connecting the elements of the structural system to be estimated with features of the density of the observable variables, such as ratios of derivatives or averages of products of derivatives of this density. The estimators are therefore easily computed functionals of a nonparametric estimator of the density of the observable variables. We consider in detail a model where to each structural equation there corresponds an exclusive regressor and a model with one equation of interest and one instrument that is included in a second equation. For both models, we provide new characterizations of observational equivalence on a set, in terms of the density of the observable variables and derivatives of the structural functions. Based on those characterizations, we develop two estimation methods. In the first method, the estimators of the structural derivatives are calculated by a simple matrix inversion and matrix multiplication, analogous to a standard least squares estimator, but with the elements of the matrices being averages of products of derivatives of nonparametric density estimators. In the second method, the estimators of the structural derivatives are calculated in two steps. In a first step, values of the instrument are found at which the density of the observable variables satisfies some properties. In the second step, the estimators are calculated directly from the values of derivatives of the density of the observable variables evaluated at the found values of the instrument. We show that both pointwise estimators are consistent and asymptotically normal.