A Frisch-Waugh-Lovell theorem for empirical likelihood

Abstract

A Frisch-Waugh-Lovell-type (FWL) theorem for empirical likelihood estimation with instrumental variables is presented, which resembles the standard FWL theorem in ordinary least squares (OLS), but its partitioning procedure employs the empirical likelihood weights at the solution rather than the original sample distribution. This result is leveraged to simplify the computational process through an iterative algorithm, where exogenous variables are partitioned out using weighted least squares, and the weights are updated between iterations. Furthermore, it is demonstrated that iterations converge locally to the original empirical likelihood estimate at a stochastically super-linear rate. A feasible iterative constrained optimization algorithm for calculating empirical-likelihood-based confidence intervals is provided, along with a discussion of its properties. Monte Carlo simulations indicate that the iterative algorithm is robust and produces results within the numerical tolerance of the original empirical likelihood estimator in finite samples, while significantly improves computation in large-scale problems. Additionally, the algorithm performs effectively in an illustrative application using the return to education framework.