Decision Theory Applied to an Instrumental Variables Model
Decision Theory Applied to an Instrumental Variables Model
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决策理论应用于工具变量模型
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发表时间：
2007
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通讯作者：
G. Chamberlain
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作者：
G. Chamberlain
本文将决策理论中的一些一般概念应用于一个简单的工具变量模型。有两个内生变量由一个结构方程联系在一起; k个外生变量被排除在这个结构方程之外，并提供了工具变量（IV）。内生变量的简化形式分布条件下的外生变量对应于独立提请从一个二元正态分布与线性回归函数和一个已知的协方差矩阵。模型的规范形式具有参数向量（rho，phi，omega），其中phi是感兴趣的参数，并且被归一化为单位圆上的点。工具变量的简化形式系数被分成标量参数rho和参数向量omega，后者被归一化为（k - 1）维单位球面上的一个点; rho度量内生变量和工具变量之间的关联强度，omega是方向的度量。在IV模型中引入了先验分布。参数phi、rho和omega被视为独立的随机变量。φ的分布在单位圆上是均匀的; Ω的分布在维度为k-1的单位球面上是均匀的。这些选择来自于极大极小问题的解决方案。罗的副院长是左将军。事实证明，给定ρ的任何正值，phi的贝叶斯估计量不依赖于ρ;它等于最大似然估计量。这个贝叶斯估计量具有常数风险;因为它相对于适当的先验最小化平均风险，所以它是极小极大的。经济计量学会2007年版权所有。
This paper applies some general concepts in decision theory to a simple instrumental variables model. There are two endogenous variables linked by a single structural equation; k of the exogenous variables are excluded from this structural equation and provide the instrumental variables (IV). The reduced-form distribution of the endogenous variables conditional on the exogenous variables corresponds to independent draws from a bivariate normal distribution with linear regression functions and a known covariance matrix. A canonical form of the model has parameter vector (rho, phi, omega), where phi is the parameter of interest and is normalized to be a point on the unit circle. The reduced-form coefficients on the instrumental variables are split into a scalar parameter rho and a parameter vector omega, which is normalized to be a point on the (k - 1)-dimensional unit sphere; rho measures the strength of the association between the endogenous variables and the instrumental variables, and omega is a measure of direction. A prior distribution is introduced for the IV model. The parameters phi, rho, and omega are treated as independent random variables. The distribution for phi is uniform on the unit circle; the distribution for omega is uniform on the unit sphere with dimension k-1. These choices arise from the solution of a minimax problem. The prior for rho is left general. It turns out that given any positive value for rho, the Bayes estimator of phi does not depend on rho; it equals the maximum-likelihood estimator. This Bayes estimator has constant risk; because it minimizes average risk with respect to a proper prior, it is minimax. Copyright The Econometric Society 2007.