Nonparametric Identification Using Instrumental Variables: Sufficient Conditions For Completeness

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 $z.$