Risk Tuning with Generalized Linear Regression

A framework is set up in which linear regression, as a way of approximating a random variable by other random variables, can be carried out in a variety of ways, which, moreover, can be tuned to the needs of a particular model in finance, or operations research more broadly. Although the idea of adapting the form of regression to the circumstances at hand has already found advocates in promoting quantile regression as an alternative to classical least-squares approaches, it is carried here much farther than that. Axiomatic concepts of error measure, deviation measure, and risk measure are coordinated with certain “statistics” that likewise say something about a random variable. Problems of regression utilizing these concepts are analyzed and the character of their solutions is explored in a range of examples. Special attention is paid to parametric forms of regression which arise in connection with factor models. It is argued that when different aspects of risk enter an optimization problem, different forms of regression ought to be invoked for each of those aspects.