Kernel Estimation of Average Derivatives and Differences

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Abstract
In this article we consider the problem of estimating average derivatives and differences using kernel estimators. Our analysis focuses on developing new methods that are appropriate in the context of bounded and discrete regressors and do not require higher-order kernels for consistency or asymptotic normality. We derive a new nonparametric estimator that we call the average difference estimator. 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 0. To illustrate the properties of our estimator, we provide some evidence from a Monte Carlo experiment. We also consider an application that focuses on estimating derivatives of earning functions using repeated cross-sectional data from the Current Population Survey (CPS) for a number of narrowly defined occupations. We find that the average difference estimator yields plausible estimates for the average derivative of the earnings functions with respect to hours worked in all subsamples of the CPS considered in this article.