Nonparametric Derivative Estimation for Related-Effect Panel Data

In a "fixed-effect" panel data model with a nonparametric regression function \rho(x_{it}), the usual first-differencing yields a nonparametric regression function \mu(x_{it},x_{i,t+1}) with the restriction \mu(x_{it},x_{i,t+1}) = \rho(x_{i,t+1}) - \rho(x_{it}). Although \mu(x_{it},x_{i,t+1}) can be easily estimated nonparametrically with a kernel method, it is not clear that how to identify and estimate \partial\rho(x_{it})/\partial x_{it} (and \rho(x_{it})) using a kernel method, and this task becomes more difficult when a time-invariant variable c_i enters \rho(x_{it}). In this paper, we propose a kernel estimator that is a linear combination of partial derivative estimators for \partial\mu(x_{it},x_{i,t+1},c_i)/\partial x_{i,t+1} and \partial\mu(x_{it},x_{i,t+1},c_i)/\partial x_{i,t}, prove its consistency for \partial\rho(x_{it})/\partial x_{it} and derive the asymptotic distribution. An extensive Monte Carlo study is presented. Also multiple periods longer than two and mixed continuous/discrete regressor cases are considered to enhance the applicability.