工具变量的一个性质

A Property of Instrumental Variable

当线性回归模型中的解释变量是随机变量且与随机干扰项同期线性相关时,本文证明了,随机解释变量与工具变量的线性相关系数的平方必然小于或等于1减去随机解释变量与随机干扰项的线性相关系数的平方.这就提醒我们,当随机解释变量与随机干扰项高度相关时,随机解释变量与工具变量的相关系数必然接近0,如果样本容量再不够大,工具变量法估计值将不稳定.本文所作的数值模拟结果也验证了这一推断.

This paper proves that,when the explanatory variable in a linear regression model is a random variable and is linearly correlated with the random interference term contemporaneously,the square of the linear correlation coefficient of the random explanatory variable and the instrumental variable must be less than or equal to 1 minus the square of the linear correlation coefficient of the random explanatory variable and the random interference term.This imply that,when the random explanatory variable is highly correlated with the random interference term,the correlation efficient of the random explanatory and the instrumental variable must be near to 0,in conditions with the sample size not large enough,the instrumental variable estimated value will be unstable,the simulation results in this paper also verifies this inference.