DIFFERENCE BETWEEN POINT ESTIMATES AND DISTRIBUTIONS

Let us apply this analysis to how planners make the mistakes they make, and why deficits tend to be worse than planned:

FIGURE 36. The gap between predictions and reality: probability distribution of outcomes from costs of projects in the minds of planners (top) and in reality (bottom). In the first graph they assume that the costs will be both low and quite certain. The graph on the bottom shows outcomes to be both worse and more spread out, particularly with higher possibility of unfavorable outcomes. Note the fragility increase owing to the swelling left tail.

This misunderstanding of the effect of uncertainty applies to government deficits, plans that have IT components, travel time (to a lesser degree), and many more. We will use the same graph to show model error from underestimating fragility by assuming that a parameter is constant when it is random. This is what plagues bureaucrat-driven economics (next discussion).

Appendix II (Very Technical): WHERE MOST ECONOMIC MODELS FRAGILIZE AND BLOW PEOPLE UP

When I said "technical" in the main text, I may have been fibbing. Here I am not.

The Markowitz incoherence: Assume that someone tells you that the probability of an event is exactly zero. You ask him where he got this from. "Baal told me" is the answer. In such case, the person is coherent, but would be deemed unrealistic by non-Baalists. But if on the other hand, the person tells you "I estimated it to be zero," we have a problem. The person is both unrealistic and inconsistent. Something estimated needs to have an estimation error. So probability cannot be zero if it is estimated, its lower bound is linked to the estimation error; the higher the estimation error, the higher the probability, up to a point. As with Laplace's argument of total ignorance, an infinite estimation error pushes the probability toward ½.

We will return to the implication of the mistake; take for now that anything estimating a parameter and then putting it into an equation is different from estimating the equation across parameters (same story as the health of the grandmother, the average temperature, here "estimated" is irrelevant, what we need is average health across temperatures). And Markowitz showed his incoherence by starting his "semi-nal" paper with "Assume you know E and V" (that is, the expectation and the variance). At the end of the paper he accepts that they need to be estimated, and what is worse, with a combination of statistical techniques and the "judgment of practical men." Well, if these parameters need to be estimated, with an error, then the derivations need to be written differently and, of course, we would have no paper—and no Markowitz paper, no blowups, no modern finance, no fragilistas teaching junk to students.… Economic models are extremely fragile to assumptions, in the sense that a slight alteration in these assumptions can, as we will see, lead to extremely consequential differences in the results. And, to make matters worse, many of these models are "back-fit" to assumptions, in the sense that the hypotheses are selected to make the math work, which makes them ultrafragile and ultrafragilizing.

Simple example: Government deficits.

We use the following deficit example owing to the way calculations by governments and government agencies currently miss convexity terms (and have a hard time accepting it). Really, they don't take them into account. The example illustrates:

(a) missing the stochastic character of a variable known to affect the model but deemed deterministic (and fixed), and

(b) F, the function of such variable, is convex or concave with respect to the variable.

Say a government estimates unemployment for the next three years as averaging 9 percent; it uses its econometric models to issue a forecast balance B of a two-hundred-billion deficit in the local currency. But it misses (like almost everything in economics) that unemployment is a stochastic variable. For this illustration, take unemployment to vary by 1 percentage point on either side of the central estimate, with equal weight on the two outcomes. We can calculate the effect of the error with the following:

• Unemployment at 8%, Balance B(8%) = -75 bn (improvement of 125 bn)
• Unemployment at 9%, Balance B(9%)= -200 bn
• Unemployment at 10%, Balance B(10%)= -550 bn (worsening of 350 bn)

The concavity bias, or negative convexity bias, from underestimation of the deficit is -112.5 bn, since ½ {B(8%) + B(10%)} is approximately -312 bn, not -200 bn. The exact equal-weight average is minus three hundred twelve and a half billion. This is the exact case of the inverse philosopher's stone.

FIGURE 37. Nonlinear transformations allow the detection of both model convexity bias and fragility. Illustration of the example: histogram from Monte Carlo simulation of government deficit as a left-tailed random variable simply as a result of randomizing unemployment, of which it is a concave function. The method of point estimate would assume a Dirac stick at -200, thus understating both the approximate expected deficit (-312) and the possibility of a much worse balance. (From Taleb and Douady, 2012).