The Work Behind the Prize

This afternoon (Monday November 4) a panel of four will try to explain the research that Gene Fama and Lars Hansen did to win the Nobel Prize for the University of Chicago community.

This is classic University of Chicago, community of scholars stuff: Yes, we’ve congratulated you.  Now, let’s talk seriously about the ideas and the research.

My job: Explain efficiency, long run returns and volatility in 10 minutes flat. Wish me luck. John Heaton and Jim Heckman will describe Lars Hansen’s work, and Toby Moskowitz will join me on the Fama panel.  Gary Becker will moderate

The announcement is here; RSVP if you want to attend as seating is limited. The event will be web-cast here

Lars Hansen's Nobel

Lars has done so much  deep and pathbreaking research, that I can’t begin to even list it, to say nothing of explain the small part of it that I understand.  I wrote whole chapters of my textbook “Asset Pricing” devoted to just one Hansen paper. Lars writes for the ages, and it often takes 10 years or more for the rest of us to understand what he has done and how important it is.

So I will just try to explain GMM and the consumption estimates, the work most prominently featured in the Nobel citation. Like all of Lars’ work, it looks complex at the outset, but once you see what he did, it is actually brilliant in its simplicity.

The GMM approach basically says, anything you want to do in statistical analysis or econometrics can be written as taking an average.

For example, consider the canonical consumption-based asset pricing model, which is where he and Ken Singleton took GMM out for its first big spin. The model says, we make sense of out of asset returns – we should understand the large expected-return premium for holding stocks, and why that premium varies over time (we’ll talk about that more in the upcoming Shiller post) – by the statement that the expected excess return, discounted by marginal utility growth, should be zero

where Et means conditional expectation, beta and gamma capture investor’s impatience and risk aversion, c is consumption and R is a stock or bond return and Rf is a bond return. E(R-Rf) is the premium – how much you expect to earn on a risky asset over a riskfree one, as compensation for risk. (Non-economists, just ignore the equations. You’ll get the idea). Expected returns vary over time and across assets in puzzling ways, but the expected discounted excess return should always be zero.

How do we take this to data? How do we find parameters beta and gamma that best fit the data? How do we check this over many different times and returns, to see if those two parameters can explain lots of facts? What do we do about that conditional expectation Et, conditional on information in people’s heads? How do we bring in all the variables that seem to forecast returns over time (D/P) and across assets (value, size, etc.)? How do we handle the fact that return variance changes over time, and consumption growth may be autocorrelated?

When Hansen wrote, this was a big headache. No, suggested Lars. Just multiply by any variable z that you think forecasts returns or consumption, and take the unconditional average of this conditional average, and the model predicts  that the unconditional average obeys

So, just take this average in the data. Now, you can do this for lots of different assets R and lots of different “instruments” z, so this represents a lot of averages. Pick beta and gamma that make some of the averages as close to zero as possible. Then look at the other averages and see how close they are to zero.

Lars worked out the statistics of this procedure – how close should the other averages be to zero, and what’s a good measure of the sample uncertainty in beta and gamma estimates – taking in to account a wide variety of statistical problems you could encounter. The latter part and the proofs make the paper hard to read. When Lars says “general” Lars means General!

But using the procedure is actually quite simple and intuitive. All of econometrics comes down to a generalized version of the formula sigma/root T for standard errors of the mean. (I recommend my book “Asset Pricing” which explains how to use GMM in detail.)

Very cool.

The results were not that favorable to the consumption model. If you look hard, you can see the equity premium puzzle – Lars and Ken needed huge gamma to fit the difference between stocks and bonds, but then couldn’t fit the level of interest rates.  But that led to an ongoing search – do we have the right utility function? Are we measuring consumption correctly? And that is now bearing fruit.

GMM is really famous because of how it got used. We get to tests parts of the model without writing down the whole model. Economic models are quantiative parables, and we get to examine and test the important parts of the parable without getting lost in irrelevant details.

What do these words mean? Let me show you an example. The classic permanent income model is a special case of the above, with quadratic utility. If we model income y as an AR(1) with coefficient rho, then the permanent income model says consumption should follow a random walk with innovations equal to the change in the present value of future income:

This is the simplest version of a “complete” model that I can write down. There are fundamental shocks, the epsilon; there is a production technology which says you can put income in the ground and earn a rate of return r, and there is an interesting prediction – consumption smooths over the income shocks.

Now, here is the problem we faced before GMM. First, computing the solutions of this sort of thing for real models is hard, and most of the time we can’t do it and have to go numerical. But just to understand whether we have some first-order way to digest the Fama-Shiller debate, we have to solve big hairy numerical models? Most of which is beside the point? The first equations I showed you were just about investors, and the debate is whether investors are being rational or not. To solve that, I have to worry about production technology and equilibrium?

Second, and far worse, suppose we want to estimate and test this model. If we follow the 1970s formal approach, we immediately have a problem. This model says that the change in consumption is perfectly correlated with income minus rho times last year’s income. Notice the same error epsilon in both equations. I don’t mean sort of equal, correlated, expected to be equal, I mean exactly and precisely equal, ex-post, data point for data point.

If you hand that model to any formal econometric method (maximum likelihood), it sends you home before you start. There is no perfect correlation in the data, for any parameter values. This model is rejected. Full stop.

Wait a minute, you want to say. I didn’t mean this model is a complete perfect description of reality. I meant it is a good first approximation that captures important features of the data. And this correlation between income shocks and consumption shocks is certainly not an important prediction.  I don’t think income is really an AR(1), and most of all I think agents know more about their income than my simple AR(1). But I can’t write that down, because I don’t see all their information. Can’t we just look at the consumption piece of this and worry about production technology some other day?

In this case, yes. Just look whether consumption follows a random walk. Run the change in consumption on a bunch of variables and see if they predict consumption. This is what Bob Hall did in his famous test, the first test of a part of a model that does not specify the whole model, and the first test that allows us to “condition down” and respect the fact that people have more information than we do. (Lars too walks on the shoulders of giants.) Taking the average of my first equation is the same idea, much generalized.

So the GMM approach allows you to look at a piece of a model – the intertemporal consumption part, here – without specifying the whole rest of the model – production technology, shocks, information sets. It allows you to focus on the robust part of the quantitative parable – consumption should not take big predictable movements – and gloss over the parts that are unimportant approximations – the perfect correlation between consumption and income changes.  GMM is a tool for matching quantitative parables to data in a disciplined way.

This use of GMM is part of a large and, I think, very healthy trend in empirical macroeconomics and finance. Roughly at the same time, Kydland and Prescott started “calibrating” models rather than estimating them formally, in part for the same reasons. They wanted to focus on the “interesting” moments and not get distracted by the models’ admitted abstractions and perfect correlations.

Formal statistics asks “can you prove that this model is not a 100% perfect representation of reality” The answer is often “yes,” but on a silly basis. Formal statistics does not allow you to say “does this model captures some really important pieces of the picture?” Is the glass 90% full, even if we can prove it’s missing the last 10%?

But we don’t want to give up on statistics, which much of the calibration literature did. We want to pick parameters in an objective way that gives models their best shot. We want to measure how much uncertainty there is in those parameters. We want to know how precise our predictions for the “testing” moments are. GMM lets you do all these things. If you want to “calibrate” on the means (pick parameters by observations such as the mean consumption/GDP ratio, hours worked, etc.), then “test” on the variances (relative volatility of consumption and output, autocorrelation of output, etc.), GMM will let you do that. And it will tell you how much you really know about parameters (risk aversion, substitution elasticities, etc.) from those “means”, how accurate your predictions about “variances” are, including the degrees of freedom chewed up in estimation!

In asset pricing, similar pathologies can happen. Formal testing will lead you to focus on strange portfolios, thousands of percent long some assets and thousands of percent short others. Well, those aren’t “economically interesting.” There are bid/ask spread, price pressure, short constraints and so on. So, let’s force the model to pick parameters based on interesting, robust moments, and let’s evaluate the model’s performance on the actual assets we care about, not some wild massive long-short (“minimum variance”) portfolio.

Fama long ran OLS regressions when econometricians said to run GLS, because OLS is more robust.  GMM allows you to do just that sort of thing for any kind of model – but then correct the standard errors!

In sum, GMM is a tool, a very flexible tool. It has let us learn what the data have to say, refine models, understand where they work and where they don’t, emphasize the economic intuition, and break out of the straightjacket of “reject” or “don’t reject,” to a much more fruitful empirical style.

Of course, it’s just a tool. There is no formal definition of an “economically interesting” moment, or a “robust” prediction. Well, you have to think, and read critically.

Looking hard but achieving  a remarkable simplicity when you understand it is a key trait of Lars’ work. GMM really is just applying sigma/Root T (generalized) to all the hard problems of econometrics. Once you make the brilliant step of recognizing they can be mapped to a sample mean. His “conditioning information” paper with Scott Richard took me years to digest. But once you understand L2, the central theorem of asset pricing is “to every plane there is an orthogonal line.” Operators in continuous time, and his new work on robust control and recursive preference shares the same elegance.

The trouble with the Nobel is that it leads people to focus on the cited work. Yes, GMM is a classic. I got here in 1985 and everyone already knew it would win a Nobel some day. But don’t let that fool you, the rest of the Lars portfolio is worth studying too. We will be learning from it for years to come. Maybe this will inspire me to write up a few more of his papers. If only he would stop writing them faster than I can digest them.

Source: Becker-Friedman Institute

I won’t even pretend this is unbiased. Lars is a close friend as well as one of my best colleagues at Chicago. I learned most of what I know about finance by shuttling back and forth between Lars’ office and Gene Fama’s, both of whom patiently explained so many things to me. But they did so in totally different terms, and understanding what each was saying in the other’s language led me to whatever synthesis I have been able to achieve. If you like the book “Asset Pricing,” you are seeing the result. He is also a great teacher and devoted mentor to generations of PhD students.

(This is a day late, because I thought I’d have to wait a few more years, so I didn’t have a Hansen essay ready to go. Likewise Shiller, it will take a day or two. Thanks to Anonymous and Greg for reporting a typo in the equations.)

Update: I’m shutting down most comments on these posts. This week, let’s congratulate the winners, and discuss issues again next week.

Fama, Hansen, and Shiller Nobel

Gene Fama, Lars Hansen and Bob Shiller win the Nobel Prize. Congratulations! (Minor complaint: Nobel committee, haven’t you heard of Google? There are lots of nice Gene Fama photographs lying around. What’s with the bad cartoon?)

I’ll write more about each in the coming days. I’ve spent most of my professional life following in their footsteps, so at least I think I understand what they did more than for the typical prize.

As a start, here is an an introduction I wrote for  Gene Fama’s Talk, “The History of the Theory and Evidence on the Efficient Markets Hypothesis” given for the AFA history project. There is a link to this document on my webpage here. The video version is here at IGM.

Introduction for Gene Fama

On behalf of the American Finance Association and the University of Chicago Graduate School of Business, it is an honor and a pleasure to introduce Gene Fama. This talk is being videotaped for the AFA history project, so we speak for the ages.

Gene will tell us how the efficient-markets hypothesis developed. I’d like to say a few words about why it’s so important. This may not be obvious to young people in the audience, and Gene will be too modest to say much about it.

“Market efficiency” means that asset prices incorporate available information about values. It does not mean that orders are “efficiently” processed, that prices “efficiently” allocate resources, or any of the other nice meanings of “efficiency.” Why should prices reflect information? Because of competition and free entry. If we could easily predict that stock prices will rise tomorrow, we would all try to buy today. Prices would rise today until they reflect our information.


This seems like a pretty simple “theory,” hardly worth all the fuss. Perhaps you expect general relativity, lots of impenetrable equations. Gene is more like Darwin, and the efficient markets hypothesis is more like evolution. Both evolution and efficient markets are elegant, simple, and powerful ideas that organized and energized vast empirical projects, and that’s the true measure of any theory. Without evolution, natural history would just be a collection of curious facts about plants and animals. Without the efficient markets hypothesis, empirical finance would just be a collection of Wall-Street anecdotes, how-I-got-rich stories, and technical-trading newssheets.

Efficient-market theory and empirical work are also a much deeper intellectual achievement than my little story suggests. There are plenty of hard equations. It took nearly a century to figure out the basic prediction of an efficient market, from Bachelier’s random walk to the consumption Euler equation (price equals conditionally expected value, discounted by marginal utility growth). It took hard work and great insight to account for risk premiums, selection biases, reverse causality, and endogenous variables, and to develop the associated statistical procedures.

Efficient-markets empirical work doesn’t check off easy “predictions.” It typically tackles tough anomalies, each of which looks superficially like a glaring violation of efficiency, and each endorsed by a cheering crowd of rich (or perhaps lucky?) traders. It’s not obvious that what looks like an inefficiently low price is really a hidden exposure to systematic risk. It took genius to sort through the mountains of charts and graphs that computers can spit out, to see the basic clear picture.

Efficient-market predictions can be beautifully subtle and unexpected. One example: In an efficient market, expert portfolio managers should do no better than monkeys throwing darts. That’s a remarkable prediction. Experts are better than amateurs in every other field of human endeavor: Tiger Woods will beat you at golf; you should hire a good house painter and a better tax lawyer. The prediction is even more remarkable for how well it describes the world, after we do a mountain of careful empirical work.

That empirical work consists, fundamentally, of applying scientific method to financial markets. Modern medicine doesn’t ask old people for their health secrets. It does double-blind clinical trials. To this, we owe our ability to cure many diseases. Modern empirical finance doesn’t ask Warren Buffett to share his pearls of investment wisdom. We study a survivor-bias-free sample of funds sorted on some ex-ante visible characteristic, to separate skill from luck, and we correct for exposure to systematic risk. To this we owe our wisdom, and maybe, as a society, a lot of wealth as well.

This point is especially important now, in a period of great financial turbulence. It’s easy to look at the latest market gyration and opine, “Surely markets aren’t efficient.” But that’s not how we learn anything of lasting usefulness. Efficient markets taught us to evaluate theories by their rejectable predictions and by the numbers; to do real, scientific, empirical work, not to read newspapers and tell stories.

Efficient markets are also important to the world at large, in ways that I can only begin to touch on here. The assurance that market prices are in some sense basically “right” lies behind many of the enormous changes we have seen in the financial and related worlds, from index funds, which have allowed for wide sharing of the risks and rewards of the stock market, to mark-to-market accounting, quantitative portfolio evaluation and benchmarking, and modern risk management.

With 40 years’ hindsight, are markets efficient? Not always, and Gene said so in 1970. For example, prices rise on the release of inside information, so that information, though known by someone, was not reflected in the original price. More recently, I think we have seen evidence that short-sales constraints and other frictions can lead to informationally-inefficient prices.

This is great news. Only a theory that can be proved wrong has any content at all. Theories that can “explain” anything are as useless as “prices went down because the Gods are angry.”

Gene went on, arguing that no market is ever perfectly efficient, since no market is perfectly competitive and frictionless. The empirical question has always been to what degree a given phenomenon approaches an unanattainable ideal.

Still, the answer today is much closer to “yes” than to “no” in the vast majority of serious empirical investigations. It certainly is a lot closer to “yes” than anyone expected in the 1960s, or than the vast majority of practitioners believe today. There are strange fish in the water, but even the most troublesome are surprisingly small fry. And having conquered 157 anomalies with patient hard work, many of us can be excused for suspecting that just a little more work will make sense of the 158th.

However, empirical finance is no longer really devoted to “debating efficient markets,” any more than modern biology debates evolution. We have moved on to other things. I think of most current research as exploring the amazing variety and subtle economics of risk premiums – focusing on the “joint hypothesis” rather than the “informational efficiency” part of Gene’s 1970 essay.

This is also great news. Healthy fields settle debates with evidence and move on to new discoveries. But don’t conclude that efficient markets are passé. As evolution lies quietly behind the explosion in modern genetics, markets that are broadly efficient, in which prices quickly reflect information, quietly underlie all the interesting things we do today. This is the best fate any theory can aspire to.

Gene will talk about the history of efficient markets. People expect the wrong things of history as they expect overly complex “theory.” No lone genius ever thought up a “hypothesis,” went out to “test” it, and convinced the world with his 2.1 t-statistic. Theory and empirical work develop together, ideas bounce back and forth between many people, the list of salient vs. unimportant facts shifts, and evidence, argument and, alas, age gradually change people’s minds. This is how efficient markets developed too, as Gene has always graciously acknowledged. Gene’s two essays describe the ideas, but much less of this process. It was an amazing adventure, and historians of science should love this story. Ladies and Gentlemen, please welcome Gene Fama to tell us about it.

A brief parable of over-differencing

The Grumpy Economist has sat through one too many seminars with triple differenced data, 5 fixed effects and 30 willy-nilly controls. I wrote up a little note (7 pages, but too long for a blog post), relating the experience (from a Bob Lucas paper) that made me skeptical of highly processed empirical work.

The graph here shows velocity and interest rates.  You can see the nice sensible relationship.

(The graph has an important lesson for policy debates. There is a lot of puzzling why people and companies are sitting on so much cash. Well, at zero interest rates, the opportunity cost of holding cash is zero, so it’s a wonder they don’t hold more. This measure of velocity is tracking interest rates with exactly the historical pattern.) 

But when you run the regression, the econometrics books tell you to use first differences, and then the whole relationship falls apart. The estimated coefficient falls by a factor of 10, and a scatterplot shows no reliable relationship.  See the the note for details, but you can see in the second graph  how differencing throws out the important variation in the data. 

The perils of over differencing, too many fixed effects, too many controls, and that GLS or maximum likelihood will jump on silly implications of necessarily simplified theories are well known in principle. But a few clear parables might make people more wary in practice.  Needed: a similarly clear panel-data example.