## Tuesday, October 21, 2014

(All rants, including this one, are here.)

Countless times, from me to Chair/Dean xxx at Some Other University:

I am happy to help with your evaluation of Professor zzz. This email will serve as my letter. [email here]...
Countless times, from Chair/Dean xxx to me:
Fantasy response from me to Chair/Dean xxx:
Sure, no problem at all. My time is completely worthless, so I'm happy to oblige, despite the fact that email conveys precisely the same information and is every bit as legally binding (whatever that even means in this context) as a "signed" "letter" on "letterhead." So now I’ll copy my email, try to find some dusty old Word doc letterhead on my hard drive, paste the email into the Word doc, try to beat it into submission depending on how poor the formatting / font / color / blocking looks when first pasted, print from Word to pdf, attach the pdf to a new email, and re-send it to you. How 1990’s.
Actually last week I did send something approximating the fantasy email to a dean at a leading institution. I suspect that he didn't find it amusing. (I never heard back.) But as I also said at the end of that email,
"Please don’t be annoyed. I...know that these sorts of 'requirements' have nothing to do with you per se. Instead I’m just trying to push us both forward in our joint battle with red tape."

## Monday, October 13, 2014

### Lawrence R. Klein Legacy Colloquium

In Memoriam

The Department of Economics of the University of Pennsylvania, with kind support from the School of Arts and Sciences, the Wharton School, PIER and IER, is pleased is pleased to host a colloquium, "The Legacy of Lawrence R. Klein: Macroeconomic Measurement, Theory, Prediction and Policy," on Penn’s campus, Saturday, October 25, 2014. The full program and related information are here. We look forward to honoring Larry’s legacy throughout the day. Please join us if you can.

Featuring:
• Olav Bjerkholt, Professor of Economics, University of Oslo
• Harold L. Cole, Professor of Economics and Editor of International Economic Review, University of Pennsylvania
• Thomas F. Cooley, Paganelli-Bull Professor of Economics, New York University
• Francis X. Diebold, Paul F. Miller, Jr. and E. Warren Shafer Miller Professor of Economics, University of Pennsylvania
• Jesus Fernandez-Villaverde, Professor of Economics, University of Pennsylvania
• Dirk Krueger, Professor and Chair of the Department of Economics, University of Pennsylvania
• Enrique G. Mendoza, Presidential Professor of Economics and Director of Penn Institute for Economic Research, University of Pennsylvania
• Glenn D. Rudebusch, Executive Vice President and Director of Research, Federal Reserve Bank of San Francisco
• Frank Schorfheide, Professor of Economics, University of Pennsylvania
• Christopher A. Sims, John F. Sherrerd ‘52 University Professor of Economics, Princeton University
• Ignazio Visco, Governor of the Bank of Italy

## Monday, October 6, 2014

### Intuition for Prediction Under Bregman Loss

Elements of the Bregman family of loss functions, denoted $$B(y, \hat{y})$$, take the form:
$$B(y, \hat{y}) = \phi(y) - \phi(\hat{y}) - \phi'(\hat{y}) (y-\hat{y})$$ where $$\phi: \mathcal{Y} \rightarrow R$$ is any strictly convex function, and $$\mathcal{Y}$$ is the support of $$Y$$.

Several readers have asked for intuition for equivalence between the predictive optimality of $$E[y|\mathcal{F}]$$ and Bregman loss function $$B(y, \hat{y})$$.  The simplest answers come from the proof itself, which is straightforward.

First consider $$B(y, \hat{y}) \Rightarrow E[y|\mathcal{F}]$$.  The derivative of expected Bregman loss with respect to $$\hat{y}$$ is
$$\frac{\partial}{\partial \hat{y}} E[B(y, \hat{y})] = \frac{\partial}{\partial \hat{y}} \int B(y,\hat{y}) \;f(y|\mathcal{F}) \; dy$$
$$= \int \frac{\partial}{\partial \hat{y}} \left ( \phi(y) - \phi(\hat{y}) - \phi'(\hat{y}) (y-\hat{y}) \right ) \; f(y|\mathcal{F}) \; dy$$
$$= \int (-\phi'(\hat{y}) - \phi''(\hat{y}) (y-\hat{y}) + \phi'(\hat{y})) \; f(y|\mathcal{F}) \; dy$$
$$= -\phi''(\hat{y}) \left( E[y|\mathcal{F}] - \hat{y} \right).$$
Hence the first order condition is
$$-\phi''(\hat{y}) \left(E[y|\mathcal{F}] - \hat{y} \right) = 0,$$
so the optimal forecast is the conditional mean, $$E[y|\mathcal{F}]$$.

Now consider $$E[y|\mathcal{F}] \Rightarrow B(y, \hat{y})$$. It's a simple task of reverse-engineering. We need the f.o.c. to be of the form
$$const \times \left(E[y|\mathcal{F}] - \hat{y} \right) = 0,$$
so that the optimal forecast is the conditional mean, $$E[y|\mathcal{F}]$$. Inspection reveals that $$B(y, \hat{y})$$ (and only $$B(y, \hat{y})$$) does the trick.

One might still want more intuition for the optimality of the conditional mean under Bregman loss, despite its asymmetry.  The answer, I conjecture, is that the Bregman family is not asymmetric! At least not for an appropriate definition of asymmetry in the general $$L(y, \hat{y})$$ case, which is more complicated and subtle than the $$L(e)$$ case.  Asymmetric loss plots like those in Patton (2014), on which I reported last week, are for fixed $$y$$ (in Patton's case, $$y=2$$ ), whereas for a complete treatment we need to look across all $$y$$. More on that soon.

[I would like to thank -- without implicating -- Minchul Shin for helpful discussions.]