Think of a Number: A Theory of Rational Forecasting
MAY 21, 2009 by ANTHONY DE JASAY
Recently, an economist gained notoriety and filled his appointment diary with lucrative conferences by having some of his forecasts for U.S. economic data, made two years ago and looking quite eccentric at the time, come gloriously true. This random event inspires me to put forward the sketch of a theory of rational forecasting.
Suppose that 500 of the most distinguished academic, industry, and Wall Street economists are polled for their best guess of the U.S. unemployment rate and the Dow Jones Industrial Average 12 months from now. As each has a reputation to preserve, none will stick his neck out with an outlandish forecast that has but a tiny probability of coming out right—even though it would earn him a jackpot if it did. Therefore, the forecasts of the 500 will cluster in a narrow range of, say, 7–11 percent for the unemployment rate and between 7500 and 9500 for the Dow.
What is left for the 250,000 other, less-distinguished economists to do to gain fame and fortune? They too can offer forecasts and might put them on some record. If they place them in the cluster and the actual outcome is in the cluster, they remain unremarked and neither gain nor lose anything. If they go way outside the cluster and the outcome is in the cluster, nobody will remember the wrong forecast made a year earlier. They will again gain nothing and lose nothing. If their forecast is in the cluster and the actual outcome is way outside it, they will be in the good company of their 500 more-distinguished fellows and will again remain unremarked.
There is, therefore, a single rational forecasting mode for our undistinguished economist to adopt. Let him think of a number for the unemployment rate and one for the Dow Jones index—say 23 percent and 4000, respectively. He can easily draft a scenario for the next 12 months full of horrors and glitches that would make the forecast numbers plausible. The probability that either one of his numbers will turn out right is very small and that both will turn out right is even smaller. As we have seen, if both his numbers are wrong, he is no worse off. But if one is right, he is richly rewarded, while if both his numbers are right, he gets riches beyond the dreams of avarice (RBDA).
In slightly more formal terms, he has access to a positive-sum game against nature. The worst payoff is zero and the best is RBDA. The sum is necessarily positive, for no matter how infinitesimally small the probability of a really horrible scenario, it cannot be zero. If he is rational, in the sense that his choice maximizes the expected value of the outcome, our economist must forecast horrors.
It would seem that many are in fact engaged in this positive-sum game. Of course, the more that do so, the more the horror story becomes a self-fulfilling prophecy. This calls to mind Blaise Pascal, the seventeenth-century French philosopher who taught that it was a command of reason to be God-fearing, for the reward, eternal salvation, had infinite value and however small one judged the probability that there was a God, the probability-adjusted value of infinity was still infinity. One might add that if one did not fear God, the smallest probability that He existed would put the expected value of eternal damnation at minus infinity.
There are, to my knowledge, no records showing how many people became God-fearing under the weight of Pascal’s somewhat brazen appeal to canny calculation. Nor do we know how many blood-curdling economic forecasts are the result of career planning rather than sincere professional conviction. What we do know, though, is that such forecasts are the best method of deepening the gloom, frightening the credulous, and making the worst more probable.