# On mathematical aspects of the most recent Nobel prize in economics winners' work

Can somebody briefly introduce the mathematical aspects, in particular, those related to math finance, of the three economists who were just awarded this year's Nobel Memorial Prize in Economic Science?

According to NYtimes:

The three economists, who worked independently, were described as collectively illuminating the workings of financial markets by showing that stock and bond prices move unpredictably in the short term but with greater predictability over longer periods.

This seems very related to math finance: all kinds of random noises, yet in the long run, may not deviate too much from, say, their rational means. Many thanks for the comments, introductions and ideas!

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My personal feeling is that if this very same question were asked by a well-known high-ranked MO user, many people would find it interesting, may be simply suggesting that it become Community Wiki because no "correct" answer may exist. – Filippo Alberto Edoardo Oct 15 '13 at 5:29
I take it that those who have voted to close as off-topic have actually looked at the mathematics these economists have used and are satisfied that it holds no research interest. – Gerry Myerson Oct 15 '13 at 5:29
Fama and Shiller's research doesn't have much mathematical content, but Hansen's has some. I think the question got put on hold so quickly because the phrasing makes it sound more off-topic than it is. – arsmath Oct 15 '13 at 5:43
@Gerry: I didn't vote to close, but it's a well-established view that questions essentially asking for someone to write a blog post are off topic here, even if the requested topic does have research interest. – Mark Meckes Oct 15 '13 at 14:08
@PiyushGrover: You're right that there isn't universal agreement or completely consistent implementation. That's why I characterized it as a well-established "view" --- some people feel that way, others don't, and for many it depends on the circumstances. – Mark Meckes Oct 15 '13 at 18:28

Just to add a little something to arsmath's very good answer: The mathematics in Fama's main idea that returns are impredictable are indeed simple, and moreover, not due to him. What Fama did is a huge empirical study to support that claim. For the mathematical argument itself, which is simple but not absolutely trivial, in other words which has some mathematical content, I recommend any interested reader this very nice article of Samuelson (1965), Proof that properly anticipated prices fluctuate randomly.

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The mathematical content of Fama and Shiller's research is pretty minimal. In the short run, returns are unpredictable, while in the long run it seems to be predictable. The short run unpredictability is an important input to mathematical finance, since it helps motivate the geometric Brownian motion model for returns, but Fama and Shiller's research is straightforward statistics.

Hansen's work is more mathematical. There is a long tradition in statistics of fitting models by matching moments. For example, you can fit a Gaussian to data by matching the mean and standard deviation. To do this, you pick as many moments as you have parameters to fit. Hansen's proposed the generalized method of moments, where you choose more moments than parameters, and you test statistically for the ability of the model to fit all of the moments simultaneously. Because of estimation error, it won't match all the moments exactly, but with some error, but we can test whether that error is "too large" for the model. For example, a Gaussian has skewness and excess kurtosis of exactly zero, while in a sample these will not be exactly zero. With some work you can adapt this into a test for Gaussianity, such as described in this paper by Richardson and Smith.

This is particularly interesting for economics, because one class of models that economists consider are infinite-horizon rational-expectations models. These models assume that agents solve an infinite-horizon planning problem. Prices in the model enter as a kind of Lagrange multiplier. (The details can be found in Stokey and Lucas, Recursive Methods in Economic Dynamics.) These models lead to an analogue of Euler-Lagrange equations in the calculus of variations (economists call these Euler equations). For example, suppose there is a stock that pays a random dividend $D_t$ at every time $t$. Then under a large class of rational-expectations models, the return $R_t$ on the stock satisfies an equation of the form $$E_t( m R_t ) = 1,$$ where $m$ depends on the specific model. The Generalized Method of Moments allows you to test whether this equation holds in the data.

To step back from the economics a bit, the mathematical interest is this: a large-class of infinite-horizon stochastic optimization problems will produce an expectational equation of the form $$E_t F(x_t, x_{t+1}).$$ for some $F$. If $x_t$ is some observable data, then we can use GMM test whether the data comes from the optimization problem. So Hansen's work is in the intersection of the calculus of variations and statistics.

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"In the short run, returns are unpredictable, while in the long run it seems to be predictable." Erm... This is what I was told by a sales agent at Fidelity Investments in pretty much the same words when I opened an account with them back in 1995. I guess they got Nobel for something more interesting than this triviality. Once you seem to be familiar with the works, can you explain clearly what exactly Fama and Shiller did? – fedja Oct 15 '13 at 10:58
Fama's research was in the 70s, while Shiller's is in the 80s, so it could be ultimately the sales agent's source. The case for the short-term unpredictability is pretty simple: the correlation of today's returns with tomorrow's returns is pretty close to zero. There are many more sophisticated tests, but they all point to the same thing. – arsmath Oct 15 '13 at 11:14
Fama co-wrote many other important papers. He and co-authors introduced the "event study", where you look at the impact of an event on stock prices. There's a standard model that relates individual stock to the market, the CAPM, that in the 70s he and Bliss provided the strongest empirical evidence for. This is ironic, because in the 90s he and French provided the strong empirical evidence against, and produced a replacement in the form of the Fama-French three-factor model. – arsmath Oct 15 '13 at 11:19
Thanks! So, the Nobel prize committee in economics just kept faithful to its habit of awarding the prize 30 years after the work is done and 20 years after it becomes "common knowledge". Now things fit together much better :-). – fedja Oct 15 '13 at 14:38
@Gerry Myerson Not quite. Higgs theory predicted a new particle but until it had been detected, it was just a mere (though very plausible) speculation like some other theories (say, the one that predicted proton decay). The price was awarded as soon as the theory got the experimental confirmation. However, the stock market hasn't confirmed anything about Fama and Shiller theories in the last five years that couldn't be readily observed and verified before 1990, say. – fedja Oct 16 '13 at 1:27