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).

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...

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!)

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!

FinancialConnectedness.org Site Now Up

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.