Kernel Estimation of Average Derivatives and Differences

In this paper, we derive nonparametric average difference estimators. We show that this estimator is consistent and root-$N$ asymptotically normally distributed. Furthermore, the average difference estimator converges to the well-known average derivative estimator as the increment used to compute the difference converges to zero. We apply this estimator to test for differences between average and marginal compensation of workers. We estimate different versions of the model using repeated cross-sectional data from the CPS for a number of narrowly defined occupations. The average difference estimator yields plausible estimates for the average marginal compensation in all subsamples of the CPS considered in this paper. Our results highlight the importance of choosing bandwidth parameters in nonparametric estimation. If important covariates are measured discretely, standard approaches for choosing optimal bandwidth parameters do not necessarily apply. Our main empirical findings suggest that, at least for the preferred range of bandwidth parameters, marginal compensation exceeds average compensation, which suggests that average compensation is at best a noisy measure for the unobserved productivity of workers.