Decomposition and reproducing property of local polynomial equivalent kernels in varying coefficient models

We consider local polynomial estimation for varying coefficient models and derive corresponding equivalent kernels that provide insights into the role of smoothing on the data and fill a gap in the literature. We show that the asymptotic equivalent kernels have an explicit decomposition with three parts: the inverse of the conditional moment matrix of covariates given the smoothing variable, the covariate vector, and the equivalent kernels of univariable local polynomials. We discuss finite-sample reproducing property which leads to zero bias in linear models with interactions between covariates and polynomials of the smoothing variable. By expressing the model in a centered form, equivalent kernels of estimating the intercept function are asymptotically identical to those of univariable local polynomials and estimators of slope functions are local analogues of slope estimators in linear models with weights assigned by equivalent kernels. Two examples are given to illustrate the weighting schemes and reproducing property.