Encompassing: concepts and implementation

The encompassing principle has been carefully and precisely defined in various contexts, since its first appearance in the 1980s literature in numerous papers by Hendry, Mizon and Richard. Since then, several distinct notions of encompassing have been proposed and still coexist in the literature. We describe, illustrate and connect these notions in this paper. We start with the intuitive properties of exact encompassing between estimated models and compare it with its testable counterpart, approximate encompassing. We examine these notions and their main properties within static and dynamic, parametric and non-parametric, classical and Bayesian models and estimators. Encompassing or the lack of encompassing, is also studied via the concepts of parsimonious and partial encompassing. Pseudo-true values, which are central elements in measuring and testing approximate encompassing, are defined in line with the concept of specificity between models. We also examine the role played by the data generating process in the different approaches in the literature.