Nonparametric Estimation of Derivative Functions with Data-Driven Optimally Selected Smoothing Parameters

Abstract

Estimating gradients is of crucial importance across a broad range of applied economic domains. Here we consider data-driven bandwidth selection based on the gradient of an unknown regression function. This is a difficult problem empirically given that direct observation of the value of the gradient is typically not observed.The procedure developed here delivers bandwidths which behave asymptotically as though they were selected knowing the true gradient. This procedure is shown valid for semiparametric single index models. Simulated examples showcase the finite sample attraction of this new mechanism and confirm the theoretical predictions.