Identification and inference for semiparametric single index transformation models

This paper considers a semiparametric single index model in which the dependent variable is subject to a nonparametric transformation. The model has the form $G_0\left(Y\right)=g_0\left(X^{\top }{\theta }_0\right)+e$, where $X$ is a random vector of regressors, $Y$ is the dependent variable and $e$ is the random noise, the monotonic function $G_0$, the smooth function $g_0$ and the index vector ${\theta }_0$ are all unknown. This model is quite general in the sense that it nests many popular regression models as special cases. We first propose identification strategies for the three unknown quantities, based on which estimators are then constructed. The kernel density weighted average derivative estimator of $\delta$ (proportional to ${\theta }_0$) has a $V$-statistic representation and its asymptotical normality is established under the small bandwidth asymptotics. The kernel estimator of the transformation function $G_0$ is a functional of the conditional distribution estimator of $Y$ given $X^{\top }{\theta }_0$ and is shown to be $\sqrt{n}$-consistent and asymptotically normal. The sieve estimator of $g_0$ is shown to enjoy the standard nonparametric asymptotic properties. A specification test for the single index structure and extension to allow for endogeneous regressors are also developed. In addition, data-driven choices of the smoothing parameters are discussed. Simulation results illustrate the nice finite sample performance of the proposed estimators and specification test. An empirical application to studying the impact of family income on child achievement demonstrates the practical merits of the proposed model.