Optimal smoothing parameter selection in single-index model derivative estimation

Abstract.

Single-index model is one of the most popular semiparametric models in applied econometrics. Estimation of the derivative function is often of crucial importance, as studying “marginal effects” serves as a cornerstone of microeconomics. A prerequisite for the successful application of nonparametric/semiparametric kernel estimation methods is to select smoothing parameters properly to balance the estimation squared bias and variance. Henderson et al. (2015) propose a novel method for selecting the smoothing parameters optimally for derivative function estimation. However, their method suffers from the “curse of diemnsionality” problem in a multivariate nonparametric regression model. In this article, we extend the work of Henderson et al. (2015) to estimation of the derivative function of a single-index model. Specifically, we propose a data-driven method to select smoothing parameters optimally for single-index model derivative function estimation. We also derive the asymptotic distribution of the resulting local linear estimator of the derivative function. Both simulations and empirical applications show that the proposed method works well in practice.

Keywords:

• Derivative estimation
• optimal smoothing parameter selection
• single-index model

JEL Classification::

• C14

Notes

1. For here O_e(⋅) denotes an exact probability order. For example, A = O_e(1) means that A = O_p(1) but A≠o_p(1).

2. The pth order local polynomial estimators’ (p≥3) leading biases are negligible compared with the bias term of the local-linear estimator. Hence, higher-order (p≥4) polynomials work for our theoretical purpose. However, the pth order polynomial estimation requires one to estimate (p+1) estimators of the unknown function and its derivative functions up to the pth order. Considering the extra computational burden, higher order (p≥4) polynomials would not offer better performance.

Additional information

Funding

Liu’s research is supported by the National Natural Science Foundation of China (grant numbers 72273112, 72173107).