Estimation of Conditional Ranks and Tests of Exogeneity in Nonparametric Nonseparable Models
针对非参数不可分回归模型，提出基于条件分布V=F_{Y|Z}(Y|Z)与工具变量W独立性的外生性检验，无需估计结构函数，并给出渐近性质与自助法临界值。
Consider a nonparametric nonseparable regression model Y = ϕ(Z, U), where ϕ(Z, U) is strictly increasing in U and U ∼ U[0, 1]. We suppose that there exists an instrument W that is independent of U. The observable random variables are Y, Z, and W, all one-dimensional. We construct test statistics for the hypothesis that Z is exogenous, that is, that U is independent of Z. The test statistics are based on the observation that Z is exogenous if and only if V = FY|Z(Y|Z) is independent of W, and hence they do not require the estimation of the function ϕ. The asymptotic properties of the proposed tests are proved, and a bootstrap approximation of the critical values of the tests is shown to be consistent and to work for finite samples via simulations. An empirical example using the U.K. Family Expenditure Survey is also given. As a byproduct of our results we obtain the asymptotic properties of a kernel estimator of the distribution of V, which equals U when Z is exogenous. We show that this estimator converges to the uniform distribution at faster rate than the parametric n− 1/2-rate.