Monday, September 29, 2014

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).






































Monday, September 22, 2014

Prelude to a Mind-Blowing Result

A mind-blowing optimal prediction result will come next week. This post sets the stage.

My earlier post, "Musings on Prediction Under Asymmetric Loss," got me thinking and re-thinking about the predictive conditions under which the conditional mean is optimal, in the sense of minimizing expected loss.

To strip things to the simplest case possible, consider a conditionally-Gaussian process.

(1) Under quadratic loss, the conditional mean is of course optimal. But the conditional mean is also optimal under other loss functions, like absolute-error loss (in general the conditional median is optimal under absolute-error loss, but by symmetry of the conditionally-Gaussian process, the conditional median is the conditional mean).

(2) Under asymmetric loss like linex or linlin, the conditional mean is generally not the optimal prediction. One would naturally expect the optimal forecast to be biased, to lower the probability of making errors of the more hated sign. That intuition is generally correct. More precisely, the following result from Christoffersen and Diebold (1997) obtains:
If \(y_{t}\) is a conditionally Gaussian process and \( L(e_{t+h} )\) is any loss function defined on the \(h\)-step-ahead prediction error \(e_{t+h |t}\), then the \(L\)-optimal predictor is of the form \begin{equation} y_{t+h | t} = \mu _{t+h,t} +  \alpha _{t}, \end{equation}where \( \mu _{t+h,t} = E(y_{t+h} | \Omega_t) \), \( \Omega_t = y_t, y_{t-1}, ...\), and \(\alpha _{t}\) depends only on the loss function \(L\) and the conditional prediction-error variance \( var(e _{t+h} | \Omega _{t} )\).
That is, the optimal forecast is a "shifted" version of the conditional mean, where the generally time-varying bias depends only on the loss function (no explanation needed) and on the conditional variance (explanation: when the conditional variance is high, you're more likely to make a large error, including an error of the sign you hate, so under asymmetric loss it's optimal to inject more bias at such times).

(1) and (2) are true. A broad and correct lesson emerging from them is that the conditional mean is the central object for optimal prediction under any loss function. Either it is the optimal prediction, or it's a key ingredient.

But casual readings of (1) and (2) can produce false interpretations. 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? Isn't the folk theorem basically just (2)?

Things get really interesting.

To be continued...

Monday, September 15, 2014

1976 NBER-Census Time Series Conference

What a great blast from the past -- check out the program of the 1976 NBER-Census Time-Series Conference. (Thanks to Bill Wei for forwarding, via Hang Kim.)

The 1976 conference was a pioneer in bridging time-series econometrics and statistics. Econometricians at the table included Zellner, Engle, Granger, Klein, Sims, Howrey, Wallis, Nelson, Sargent, Geweke, and Chow. Statisticians included Tukey, Durbin, Bloomfield, Cleveland, Watts, and Parzen. Wow!

The 1976 conference also clearly provided the model for the subsequent long-running and hugely-successful NBER-NSF Time-Series Conference, the hallmark of which is also bridging the time-series econometrics and statistics communities. An historical listing is here, and the tradition continues with the upcoming 2014 NBER-NSF meeting at the Federal Reserve Bank of St. Louis. (Registration deadline Wednesday!)

Monday, September 8, 2014

Network Econometrics at Dinner

At a seminar dinner at Duke last week, I asked the leading young econometrician at the table for his forecast of the Next Big Thing, now that the partial-identification set-estimation literature has matured. The speed and forcefulness of his answer -- network econometrics -- raised my eyebrows, and I agree with it. (Obviously I've been working on network econometrics, so maybe he was just stroking me, but I don't think so.) Related, the Acemoglu-Jackson 2014 NBER Methods Lectures, "Theory and Application of Network Models," are now online (both videos and slides). Great stuff!

Tuesday, September 2, 2014

FinancialConnectedness.org Site Now Up



kamil_yilmaz

The Financial and Macroeconomic Connectedness site is now up, thanks largely to the hard work of Kamil Yilmaz and Mert Demirer. Check it out at http://financialconnectedness.org. It implements the Diebold-Yilmaz framework for network connecteness measurement in global stock, sovereign bond, FX and CDS markets, both statically and dynamically (in real time). It includes results, data, code, bibliography, etc. Presently it's all financial markets and no macro (e.g., no global business cycle connectedness), but macro is coming soon. Check back in the coming months as the site grows and evolves.

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.

Monday, August 18, 2014

Models Didn't Cause the Crisis

Some of the comments engendered by the Black Swan post remind me of something I've wanted to say for a while: In sharp contrast to much popular perception, the financial crisis wasn't caused by models or modelers.

Rather, the crisis was caused by huge numbers of smart, self-interested people involved with the financial services industry -- buy-side industry, sell-side industry, institutional and retail customers, regulators, everyone -- responding rationally to the distorted incentives created by too-big-to-fail (TBTF), sometimes consciously, often unconsciously. Of course modelers were part of the crowd looking the other way, but that misses the point: TBTF coaxed everyone into looking the other way. So the key to financial crisis management isn't as simple as executing the modelers, who perform invaluable and ongoing tasks. Instead it's credibly committing to end TBTF, but no one has found a way. Ironically, Dodd-Frank steps backward, institutionalizing TBTF, potentially making the financial system riskier now than ever. Need it really be so hard to end TBTF? As Nick Kiefer once wisely said (as the cognoscenti rolled their eyes), "If they're too big to fail, then break them up."

[For more, see my earlier financial regulation posts:  part 1part 2 and part 3.]

Monday, August 11, 2014

You Can Now Browse by Topic

You can now browse No Hesitations by topic.  Check it out -- just look in the right column, scrolling down a bit. I hope it's useful.

On Rude and Risky "Calls for Papers"

You have likely seen calls for papers that include this script, or something similar: 
You will not hear from the organizers unless they decide to use your paper.
It started with one leading group's calls, which go so even farther:
You will not hear from the organizers unless they decide to use your paper.  They are not journal editors or program committee chairmen for a society. 
Now it's spreading.

Bad form, folks.

(1) It's rude. Submissions are not spam to be acted upon by the organizers if interesting, and deleted otherwise. On the contrary, they're solicited, so the least the organizer can do is acknowledge receipt and outcome with costless "thanks for your submission" and "sorry but we couldn't use your paper" emails (which, by the way, are automatically sent in leading software like Conference Maker). As for gratuitous additions like "They are not journal editors or program committee chairmen...," well, I'll hold my tongue.

(2) It's risky. Consider an author whose fine submission somehow fails to reach the organizer, which happens surprisingly often. The lost opportunity hurts everyone -- the author whose career would have been enhanced, the organizer whose reputation would have been enhanced, and the conference participants whose knowledge would have been enhanced, not to mention the general advancement of science -- and no one is the wiser. That doesn't happen when the announced procedure includes acknowledgement of submissions, in which case the above author would simply email the organizer saying, "Hey, where's my acknowledgement? Didn't you receive my submission?"

(Note the interplay between (1) and (2). Social norms like "courtesy" arise in part to promote efficiency.)

Monday, August 4, 2014

The Black Swan Spectrum

Speaking of the newly-updated draft of Econometrics, now for some fun. Here's a question from the Chapter 6 EPC (exercises, problems and complements). Where does your reaction fall on the A-B spectrum below?
Nassim Taleb is a financial markets trader (and Wharton graduate) turned pop author. His book, The Black Swan, deals with many of the issues raised in this chapter. "Black swans"' are seemingly impossible or very low-probability events -- after all, swans are supposed to be white -- that occur with annoying regularity in reality. Read his book. Where does your reaction fall on the A-B spectrum below? 
A. Taleb offers crucial lessons for econometricians, heightening awareness in ways otherwise difficult to achieve. After reading Taleb, it's hard to stop worrying about non-normality, model misspecification, and so on.
B. Taleb belabors the obvious for hundreds of pages, arrogantly "informing"' us that non-normality is prevalent, that all models are misspecified, and so on. Moreover, it takes a model to beat a model, and Taleb offers nothing new.
The book is worth reading, regardless of where your reaction falls on the A-B spectrum.