Thursday, March 5, 2009

Did math fail us?

By and large, mainstream economists didn't expect the events of the past 6 months. Many missed the housing bubble; many thought the losses would be contained. In diagnosing this failure, some have focused on the role of "simplistic" or "unrealistic" mathematical models used by economists and financiers to understand how markets work. This month's issue of Wired Magazine, for example, talks about David Li's "Gaussian Copula" function, ominously described as "The Formula That Killed Wall Street":

"His method was adopted by everybody from bond investors and Wall Street banks to ratings agencies and regulators. And it became so deeply entrenched—and was making people so much money—that warnings about its limitations were largely ignored.

Then the model fell apart. Cracks started appearing early on, when financial markets began behaving in ways that users of Li's formula hadn't expected. The cracks became full-fledged canyons in 2008—when ruptures in the financial system's foundation swallowed up trillions of dollars and put the survival of the global banking system in serious peril.

David X. Li, it's safe to say, won't be getting that Nobel anytime soon. One result of the collapse has been the end of financial economics as something to be celebrated rather than feared. And Li's Gaussian copula formula will go down in history as instrumental in causing the unfathomable losses that brought the world financial system to its knees."

But was it really the math that let us down? While there are many reasons to doubt the usefulness of the Gaussian Copla, the real failing was one of implementation. After all, math--as used in economics--is just a tool, to be used or misused as people see fit.

Mario Livio, the noted astrophysicist and author of "Is G-d a Mathematician?", argues that while not everything in economics can be successfully modeled with math, there is no reason to stop using it altogether:

"One major reason for the difficulty in making predictions in economics is the fact that many variables of the world of economics — the psychology of the masses, to name one — do not naturally lend themselves to quantitative analysis. Consequently, some crucial aspects cannot be, at least at present, adequately represented in any model.

A second problem arises from the fact that the predictive value of any theory relies on the constancy of the underlying relationships among the different variables. In other words, one needs some assurance that under repeated, completely specified states of, say, consumers, employers, banks, trade unions and so on, the same probability for a given outcome is guaranteed to follow. In the absence of such guarantees, as one critic of mathematical economics has put it, "resembling a science is different from being a science."

Does this mean that we should give up on mathematical economics? In my very humble opinion, absolutely not. Recall that physics, also, was not considered mathematical in Aristotle's time. Yet physics advanced to the point where mathematics is at its very core. The fact that at the moment success in economic forecasts is limited should not impede research in mathematical economics any more than the failure to predict the precise number of spots on the skin of a person with measles should limit medical research into vaccines."

Mathematical models provide useful abstractions that help isolate the effect of specific variables. These models can help clarify our thinking about economic phenomena and provides the field with a consistent vocabulary for debating important questions.

The real story, suggested toward the end of the Wired article, is that when we have a lack of respect for what the models can and can't tell us--when we confuse the model with reality--the whole thing can blow up in our faces.

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