Specification and inference in linear models

Abstract

In the framework of I.I.D. sampling, a general class of linear models is analyzed. Incidental parameters are shown to naturally arise in this class of models. More fundamentally, special attention is paid to the high dimensionality of the parameter space. The objective of the paper is to offer a strategy for progressively specifying a model within that class of linear models. By so doing, we aim at displaying the precise role of each assumption, at offering alternatives to unnecessarily restrictive specifications, and, thereby, at improving the robustness of the inference procedures we discuss. Decompositions of the inference process are obtained through a systematic use of (Bayesian) cuts. Maximum Likelihood Estimation and Bayesian Inference are discussed. An objective of the progressive specification is to preserve the computational tractability and the interpretability of the procedures we develop by relying on known properties of the usual multivariate regression model.