Estimating derivatives of function-valued parameters in a class of moment condition models

We develop a general approach to estimating the derivative of a function-valued parameter θo(u) that is identified for every value of u as the solution to a moment condition. This setup in particular covers interesting models for conditional distributions, such as quantile regression or distribution regression. Exploiting that θo(u) solves a moment condition, we obtain an explicit expression for its derivative from the Implicit Function Theorem, and then estimate the components of this expression by suitable sample analogues. The last step generally involves (local linear) smoothing of the empirical moment condition. Our estimators can then be used for a variety of purposes, including the estimation of conditional density functions, quantile partial effects, and the distribution of bidders’ valuations in structural auction models.