A Mind-Blowing Optimal Prediction Result

I concluded my previous post with:

Consider, for example, the following folk theorem: "Under asymmetric loss, the optimal prediction is conditionally biased." The folk theorem is false. But how can that be?

What's true is this: The conditional mean is the L-optimal forecast if and only if the loss function L is in the Bregman family, given by
$$L(y, \hat{y}) = \phi (y) - \phi (\hat{y}) - \phi ' ( \hat{y}) (y - \hat{y}).$$ Quadratic loss is in the Bregman family, so the optimal prediction is the conditional mean.  But the Bregman family has many asymmetric members, for which the conditional mean remains optimal despite the loss asymmetry. It just happens that the most heavily-studied asymmetric loss functions are not in the Bregman family (e.g., linex, linlin), so the optimal prediction is not the conditional mean.

So the Bregman result (basically unseen in econometrics until Patton's fine new 2014 paper) is not only (1) a beautiful and perfectly-precise (necessary and sufficient) characterization of optimality of the conditional mean, but also (2) a clear statement that the conditional mean can be optimal even under highly-asymmetric loss.

Truly mind-blowing! Indeed it sounds bizarre, if not impossible. You'd think that such asymmetric Bregman families must must be somehow pathological or contrived. Nope. Consider for example, Kneiting's (2011) "homogeneous" Bregman family obtained by taking \( \phi (x; k) = |x|^k \) for \( k>1 \), and Patton's (2014) "exponential" Bregman family, obtained by taking \( \phi (x; a) =  2 a^{-2} exp(ax) \) for \(a \ne 0  \). Patton (2014) plots them (see Figure 1 from his paper, reproduced below with his kind permission). The Kneiting homogeneous Bregman family has a few funky plateaus on the left, but certainly nothing bizarre, and the Patton exponential Bregman family has nothing funky whatsoever. Look, for example, at the upper right element of Patton's figure. Perfectly natural looking -- and highly asymmetric.

For your reading pleasure, see: Bregman (1967), Savage (1971), Christoffersen and Diebold (1997), Gneiting (2011), Patton (2014).