Fast Implementation of Density-Weighted Average Derivative Estimation

Given random variables X ∈ IR and Y such that E[Y |X = x] = m(x), the average derivative δ0 is defined as δ0 = E[∇m(X)], i.e., as the expected value of the gradient of the regression function. Average derivative estimation has several applications in econometric theory (Stoker, 1992) and thus it is crucial to have a fast implementation of this estimator for practical purposes. We present such an implementation for a variation known as density-weighted average derivative estimation. This algorithm is based on the ideas of binning or Weighted Averaging of Rounded Points (WARPing). The basic idea of this method is to discretize the original data into a d-variate histogram and to replace in the nonparametric smoothing steps the actual observations by the appropriate bincenters. The non-parametric smoothing steps become thus a (multi-dimensional) convolution between the (discretized) data and the (discretized) smoothing kernel. A Monte-Carlo study demonstrates that with this binned implementation substantial reduction in computing time can be achieved. But it will also become clear that in higher dimension the choice of how to bin is crucial.