University of Washington - Department of Mathematics
A framework will be described in which linear regression, as a way of approximating a random variable by other random variables, can be carried out in a variety of modes which 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 can be developed much further than that.
Axiomatic concepts of error measure, deviation measure and risk measure can be coordinated with certain â€œstatisticsâ€ that likewise say something about a random variable. Special attention can be paid to parametric forms of regression which arise in connection with factor models. It appears that when different aspects of risk enter an optimization problem under uncertainty, different forms of regression ought to be invoked for each of those aspects.