Monday, November 24, 2014

More on Big Data

An earlier post, "Big Data the Big Hassle," waxed negative. So let me now give credit where credit is due.

What's true in time-series econometrics is that it's very hard to list the third-most-important, or even second-most-important, contribution of Big Data. Which makes all the more remarkable the mind-boggling -- I mean completely off-the-charts -- success of the first-most-important contribution: volatility estimation from high-frequency trading data. Yacine Ait-Sahalia and Jean Jacod give a masterful overview in their new book, High-Frequency Financial Econometrics.

What do financial econometricians learn from high-frequency data? Although largely uninformative for some purposes (e.g., trend estimation), high-frequency data are highly informative for others (volatility estimation), an insight traces at least to Merton's early work. Roughly put: as we sample returns arbitrarily finely, we can infer underlying volatility arbitrarily well. Accurate volatility estimation and forecasting, in turn, are crucial for financial risk management, asset pricing, and portfolio allocation. And it's all facilitated by by the trade-by-trade data captured in modern electronic markets.

In stressing "high frequency" financial data, I have thus far implicitly stressed only the massive time-series dimension, with its now nearly-continuous record. But of course we're ultimately concerned with covariance matrices, not just scalar variances, for tens of thousands of assets, so the cross-section dimension is huge as well. (A new term: "Big Big Data"? No, please, no.) Indeed multivariate now defines significant parts of both the theoretical and applied research frontiers; see Andersen et al. (2013).

Monday, November 17, 2014

Quantitative Tools for Macro Policy Analysis

Penn's First Annual PIER Workshop on Quantitative Tools for Macroeconomic Policy Analysis will take place in May 2015.  The poster appears below (and here if the one below is a bit too small), and the website is here. We are interested in contacting anyone who might benefit from attending. Research staff at central banks and related organizations are an obvious focal point, but all are welcome. Please help spread the word, and of course, please consider attending. We hope to see you there!

Monday, November 10, 2014

Penn Econometrics Reading Group Materials Online

Locals who come to the Friday research/reading group will obviously be interested in this post, but others may also be interested in following and influencing the group's path.

The schedule has been online here for a while. Starting now, it will contain not only paper titles but also links to papers when available. (Five are there now.) We'll leave the titles and papers up, instead of deleting them as was the earlier custom. We'll also try to post presenters' slides as we move forward.

Don't hesitate to suggest new papers that would be good for the Group.

Sunday, November 2, 2014

A Tribute to Lawrence R. Klein

(Remarks given at the Klein Legacy Dinner, October 24, 2014, Lower Egyptian Gallery, University of Pennsylvania Museum of Archaeology and Anthropology.)

I owe an immense debt of gratitude to Larry Klein, who helped guide, support and inspire my career for more than three decades. Let me offer just a few vignettes.

Circa 1979 I was an undergraduate studying finance and economics at Penn's Wharton School, where I had my first economics job. I was as a research assistant at Larry's firm, Wharton Econometric Forecasting Associates (WEFA). I didn't know Larry at the time; I got the job via a professor whose course I had taken, who was a friend of a friend of Larry's. I worked for a year or so, perhaps ten or fifteen hours per week, on regional electricity demand modeling and forecasting. Down the hall were the U.S. quarterly and annual modeling groups, where I eventually moved and spent another year. Lots of fascinating people roamed the maze of cubicles, from eccentric genius-at-large Mike McCarthy, to Larry and Sonia Klein themselves, widely revered within WEFA as god and goddess. During fall of 1980 I took Larry's Wharton graduate macro-econometrics course and got to know him. He won the Nobel Prize that semester, on a class day, resulting in a classroom filled with television cameras. What a heady mix!

I stayed at Penn for graduate studies, moving in 1981 from Wharton to Arts and Sciences, home of the Department of Economics and Larry Klein. I have no doubt that my decision to stay at Penn, and to move to the Economics Department, was heavily dependent on Larry's presence there. During the summer following my first year of the Ph.D. program, I worked on a variety of country models for Project LINK, under the supervision Larry and another leading modeler in the Klein tradition, Peter Pauly.  It turned out that the LINK summer job pushed me over the annual salary cap for a graduate student -- \$6000 or so 1982 dollars, if I remember correctly -- so Larry and Peter paid me the balance in kind, taking me to the Project LINK annual meeting in Wiesbaden, Germany. More excitement, and also my first trip abroad.

Both Larry and Peter helped supervise my 1986 Penn Ph.D. dissertation, on ARCH modeling of asset return volatility. I couldn't imagine a better trio of advisors: Marc Nerlove as main advisor, with committee members Larry and Peter (who introduced me to ARCH). I took a job at the Federal Reserve Board, with the Special Studies Section led by Peter Tinsley, a pioneer in optimal control of macro-econometric models. Circa 1986 Larry had more Ph.D. students at the Board than anyone else, by a wide margin. Surely that helped me land the Special Studies job. Another Klein student, Glenn Rudebusch, also went from Penn to the Board that year, and we wound up co-authoring a dozen articles and two books over nearly thirty years. My work and lasting friendship with Glenn trace in significant part to our melding in the Klein crucible.

I returned to Penn in 1989 as an assistant professor. Although I have no behind-the-scenes knowledge, it's hard to imagine that Larry's input didn't contribute to my invitation to return. Those early years were memorable for many things, including econometric socializing. During the 1990's my wife Susan and I had lots of parties at our home for faculty and students. The Kleins were often part of the group, as were Bob and Anita Summers, Herb and Helene Levine, Bobby and Julie Mariano, Jere Behrman and Barbara Ventresco, Jerry Adams, and many more. I recall a big party on one of Penn's annual Economics Days, which that year celebrated The Keynesian Revolution, Larry's landmark 1947 monograph.

The story continues, but I'll mention just one more thing. I was honored and humbled to deliver the Lawrence R. Klein Lecture at the 2005 Project LINK annual meeting in Mexico City, some 25 years after Larry invited a green 22-year-old to observe the 1982 meeting in Wiesbaden.

I have stressed guidance and support, but in closing let me not forget inspiration, which Larry also provided for three decades, in spades. He was the penultimate scholar, focused and steady, and the penultimate gentleman, remarkably gracious under pressure.

A key point, of course, is that it's not about what Larry provided me, whether guidance, support or inspiration -- I'm just one member of this large group. Larry generously provided for all of us, and for thousands of others who couldn't be here tonight, enriching all our lives. Thanks Larry. We look forward to working daily to honor and advance your legacy.

---

(For more, see the materials here.)

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.]

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 $$y_{t+h | t} = \mu _{t+h,t} + \alpha _{t},$$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!)