Smoothing bias in the measurement of marginal effects

This paper examines the tendency of kernel regression estimators to underestimate marginal effects, or derivatives. By developing the connection between smoothing and ‘errors-in-variables’ structure, we explain how smoothing itself leads to attenuation bias in derivatives, or bias toward zero. When the true model is linear and the regressors normal, the attenuation bias is uniform, and can be accurately explained by standard ‘errors-in-variables’ bias formulae. These formulations also indicate the seriousness of the attenuation bias with different numbers of regressors and different sample sizes. For the situation where the true model is nonlinear, we examine the related errors-in-variables structure and indicate what features affect the size of attenuation bias in derivatives. Finally, we explain the connections between our analysis and the standard bias expansion for kernel regression estimators, including brief consideration of technical devices such as higher-order kernels.