Semiparametric Efficient Tests

Abstract

This paper proposes efficient tests for restrictions on finite-dimensional parameters in regular semiparametric models. Our theory overcomes the main limitation of the existing theory, which requires explicit computation and estimation of certain projections onto infinite-dimensional tangent spaces and a case-by-case analysis. We consider generic semiparametric models defined by an infinite number of moment conditions, including a finite-dimensional parameter of interest and possibly containing moment-specific nonparametric nuisance parameters. We investigate tests based on functionals of the sample analog of the moments, and show that the optimal functional takes the form of a Radon-Nikodym derivative or nonparametric Likelihood Ratio (LR). We first show that the resulting LR test is efficient in our general semiparametric setting. The LR is generally infea- sible, as it assumes knowledge of a certain spectrum. We then propose and justify feasible efficient tests based on a novel nonparametric estimator of the so-called efficient score, without requiring direct computation of projections onto tangent spaces or sample splitting techniques. Thus, the proposed efficient tests are widely applicable, while being straightforward to implement. Finally, to illustrate the benefits of the approach, we apply the new methods to a semiparametric linear quantile regression model with a continuum of quantiles. Optimal inferences in this model were not available because classical efficiency arguments are difficult to apply. In contrast, our methods deliver relatively simple efficient tests in this example.

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