## Monday, August 25, 2014

### Musings on Prediction Under Asymmetric Loss

As has been known for more than a half-century, linear-quadratic-Gaussian (LQG) decision/control problems deliver certainty equivalence (CE). That is, in LQG situations we can first predict/extract (form a conditional expectation) and then simply plug the result into the rest of the problem. Hence the huge literature on prediction under quadratic loss, without specific reference to the eventual decision environment.

But two-step forecast-decision separation (i.e., CE) is very special. Situations of asymmetric loss, for example, immediately diverge from LQG, so certainty equivalence is lost. That is, the two-step CE prescription of “forecast first, and then make a decision conditional on the forecast” no longer works under asymmetric loss.

Yet forecasting under asymmetric loss -- again without reference to the decision environment -- seems to pass the market test. People are interested in it, and a significant literature has arisen. (See, for example, Elliott and Timmermann, "Economic Forecasting," Journal of Economic Literature, 46, 3-56.)

What gives? Perhaps the implicit hope is that CE two-step procedures might be acceptably-close approximations to fully-optimal procedures even in non-CE situations. Maybe they are, sometimes. Or perhaps we haven't thought enough about non-CE environments, and the literature on prediction under asymmetric loss is misguided. Maybe it is, sometimes. Maybe it's a little of both.