NONPARAMETRIC IDENTIFICATION USING INSTRUMENTAL VARIABLES: SUFFICIENT CONDITIONS FOR COMPLETENESS
研究了内生回归变量与工具变量模型中回归函数的非参数识别，给出了条件分布完备性的充分条件，对从事非参数计量经济学研究的学者有参考价值。
This paper provides sufficient conditions for the nonparametric identification of the regression function $m\left( \cdot \right)$ in a regression model with an endogenous regressor x and an instrumental variable z . It has been shown that the identification of the regression function from the conditional expectation of the dependent variable on the instrument relies on the completeness of the distribution of the endogenous regressor conditional on the instrument, i.e., $f\left( {x|z} \right)$ . We show that (1) if the deviation of the conditional density $f\left( {x|{z_k}} \right)$ from a known complete sequence of functions is less than a sequence of values determined by the complete sequence in some distinct sequence $\left\{ {{z_k}:k = 1,2,3, \ldots } \right\}$ converging to ${z_0}$ , then $f\left( {x|z} \right)$ itself is complete, and (2) if the conditional density $f\left( {x|z} \right)$ can form a linearly independent sequence $\{ f( \cdot |{z_k}):k = 1,2, \ldots \}$ in some distinct sequence $\left\{ {{z_k}:k = 1,2,3, \ldots } \right\}$ converging to ${z_0}$ and its relative deviation from a known complete sequence of functions under some norm is finite then $f\left( {x|z} \right)$ itself is complete. We use these general results to provide specific sufficient conditions for completeness in three different specifications of the relationship between the endogenous regressor x and the instrumental variable $z.$