Nonparametric estimation of coefficient function derivatives in varying coefficient models

In varying-coefficient models, accurately estimating the derivatives of coefficient functions is crucial, particularly for optimal bandwidth selection and confidence interval construction. Despite its importance, methods have largely ignored derivative estimation. In this paper, we addresse this gap by proposing a novel weighted difference quotient approach to estimate both first and second order derivatives of coefficient functions under the differences in the smoothness of the coefficient functions. We derived the asymptotic properties of our estimator and introduced a data-driven tuning parameter selection method. Simulations and real-data analyses confirmed the superior performance of our approach compared to existing techniques. Our method provides the most accurate estimates, effectively estimating both the first and second derivatives.