Semiparametric inference for non-LAN models

This thesis consists of three essays in theory of econometrics and statistics, focusing on the issue of semiparametric efficiency in non-LAN (Locally Asymptotically Normality) models. The first essay starts with a univariate case of the unit root testing problem, of which the limit experiment is of the LABF (Locally Asymptotically Brownian Functional) model. A novel approach is designed for developing the semiparametric power envelope and a family of rank-based tests that are semiparametrically efficient is proposed. The second essay generalizes the approach to all LAQ (Locally Asymptotically Quadratic) models. Moreover, it expands the rank statistics in a unique way from the univariate case to the multivariate case. Using these results, in the third essay, the semiparametric power envelop of all invariant tests for stock return predictability is developed. And subsequently, a new family of tests that are more efficient than the existing ones is proposed.