Empirical Likelihood for Robust Poverty Comparisons

Robust rankings of poverty are ones that do not rely on a single poverty measure with a single poverty line. Mathematically, such robust rankings of two populations specifies a continuum of unconditional moment inequality constraints. If these constraints could be imposed in estimation then a statistical test can be performed using an empirical likelihood-ratio (ELR) test, which is a nonparametric version of the likelihood-ratio test in parametric inference. While these constraints cannot be imposed exactly, we show that these can be imposed approximately with the approximation disappearing asymptotically. We then propose a bootstrap test procedure that implements the resulting approximate ELR test. The paper derives the asymptotic properties of this test, presents Monte Carlo experiments that show improved power compared to existing tests such as that of Linton et al. (2010), and provides an empirical illustration to Canadian income distribution data. More generally, the bootstrap test procedure provides a uniformly asymptotically valid nonparametric test of a continuum of unconditional moment inequality constraints. The proofs exploit the fact that the constrained optimization problem is a concave semi-infinite programming optimization problem.