Likelihood inference on semiparametric models: Average derivative and treatment effect

In the past few decades, much progress has been made in semiparametric modeling and estimation methods for econometric analysis. This paper is concerned with inference (i.e., confidence intervals and hypothesis testing) in semiparametric models. In contrast to the conventional approach based on t-ratios, we advocate likelihood-based inference. In particular, we study two widely applied semiparametric problems, weighted average derivatives and treatment effects, and propose semiparametric empirical likelihood and jackknife empirical likelihood methods. We derive the limiting behavior of these empirical likelihood statistics and investigate their finite sample performance via Monte Carlo simulation. Furthermore, we extend the (delete-1) jackknife empirical likelihood toward the delete-d version with growing d and establish general asymptotic theory. This extension is crucial to deal with non-smooth objects, such as quantiles and quantile average derivatives or treatment effects, due to the well-known inconsistency phenomena of the jackknife under non-smoothness.