## R2 and the crow’s triangular flight

from** Lars Syll**

In many statistical and econometric studies **R ^{2}** is used to measure goodness of fit – or more technically, the fraction of variance ”explained” by a regression.

But it’s actually a rather weird measure. As eminent mathematical statistician **David Freedman **writes:

The math is fine, but the concept is a little peculiar … Let’s take an example. Sacramento is about 78 miles from San Francisco, as the crow flies. Or, the crow could fly 60 miles East and 50 miles North, passing near Stockton at the turn. If we take the 60 and 50 as exact, Pythagoras tells us that the squared hypotenuse in the triangle is

60

^{2}+ 50^{2}= 3600 + 2500 = 6100 miles^{2}.With “explained” as in “explained variance”, the geography lesson can be cruelly summarized. The area – squared distance – between San Francisco and Sacramen-to is 6100 miles

^{2}, of which 3600 is explained by East …The theory of explained variance boils down to Pythagoras’ theorem on the crow’s triangular flight. Explainig the area between San Francisco and Sacra-mento by East is zany, and explained variance may not be much better.

Your saying “explained variance boils down to Pythagoras’ theorem” makes as much sense as saying “mathematics boils down to numbers”.

The problem with econometrics has little to do with the correctness of the mathematics (you may agree), but rather to do with the understanding or interpreting the mathematics. Econometrics has been mostly useless for economics largely because most economists have mindlessly applied statistical packages to data, not really understanding their “black box” tools and drawing invalid conclusions or making false claims, including causality.

According to PGM Swann, most studies are statistically insignificant, when the signal-to-ratio criterion is applied to non-normal errors.

“Your saying ‘explained variance boils down to Pythagoras’ theorem’ makes as much sense as saying ‘mathematics boils down to numbers’ ”.

Exactly. And that is precisely what many people since 1740 still believe mathematics boils down to, not realising that the implications of Pythagorian points, lines and right angles, Al Khorismi’s decimal numerals, Cartesian coordinates, Napier’s logarithmic measures etc, is what mathematics is all about: NOT commercial accountancy. To take these for granted is to risk catastropic misconceptions like ‘infinite flat earth economics’ and the belief – mistaking merely indicative graphs of ‘supply’ and ‘demand’ for reality – that prices necessarily rise and fall with demand.