Everyone has heard of the Big Mac Index, the Misery Index, even the Shoe Thrower Index. But the Book Cooking Index? This latest addition to the compendium of oddly named yet extremely fascinating "indices" is based around the statistical irregularity known as Benford's law, according to which within sets of numbers that span orders of magnitude, the distribution of first digits is strikingly regular: numbers beginning in 1 occur about 30% of the time, those beginning in 2 about 18% of the time, falling to roughly 5% of the time for the number 9. Specifically, as noted by the keenly observant Jialan Wang of Washington University in St. Louis, "there are more numbers in the universe that begin with the digit 1 than 2, or 3, or 4, or 5, or 6, or 7, or 8, or 9. And more numbers that begin with 2 than 3, or 4, and so on. This relationship holds for the lengths of rivers, the populations of cities, molecular weights of chemicals, and any number of other categories." The most curious application of this law resides in the field of corporate fraud, "because deviations from the law can indicate that a company's books have been manipulated." Here is where things get interesting for fraudulent corporate America: the inquisitive Wang "downloaded quarterly accounting data for all firms in Compustat, the most widely-used dataset in corporate finance that contains data on over 20,000 firms from SEC filings" and "used a standard set of 43 variables that comprise the basic components of corporate balance sheets and income statements." Her results were, in a word, startling.
As she observes, "here are the distribution of first digits vs. Benford's law's prediction for total assets and total revenues."
In other words, corporate numbers behaving precisely as predicted by a purely statistical law!
But here is where it gets truly strange. In her words, "Deviations from Benford's law have increased substantially over time, such that today the empirical distribution of each digit is about 3 percentage points off from what Benford's law would predict. The deviation increased sharply between 1982-1986 before leveling off, then zoomed up again from 1998 to 2002. Notably, the deviation from Benford dropped off very slightly in 2003-2004 after the enactment of Sarbanes-Oxley accounting reform act in 2002, but this was very tiny and the deviation resumed its increase up to an all-time peak in 2009."
Said otherwise, the chronological change in this parameter allows some starting conclusions. As the Economist notes, "This regularity has been used to identify cases of fraud in public documents. Someone cooking books is likely to choose numbers somewhat randomly, generating a distribution of digits far more uniform than Benford's law would predict. Any divergence that shows up sets off alarm bells in those looking for funny business."
Visually presented, it looks as follows:
Yes, book-cooking, in the purest statistical "correlation does not imply causation" has hit an all time high! Granted we doubt any regulator, least of all the corrupt criminal SEC, whose own Benford's law chart would look like a lie detector test hooked up to Jamie Dimon during a conference all, would admit this evidence in a court of law, but deluded investors who believe that corporation are being always truthful with data and reporting should probably be aware that that is certainly not the case. Per the Economist: "As Ms Wang notes, this isn't decisive proof of misbehaviour. It is suggestive, however, of the possibility that systematic number-fudging has been on the upswing in recent decades. Moreover, it's an excellent use of clever statistical analysis to provide a new perspective on an economic question."
Considering the surge in white collar criminality over the past 3 decades, especially among financial firms, and the fact that this "index" is now at all time highs, we would hardly be as politically correct about the issue as the Economist.
But that's us.
As for next steps, "What types of firms, and what kind of executives drive the greatest deviations from Benford's law? Does this measure do well in predicting known instances of fraud? How much of these deviations are driven by government deregulation, changes in accounting standards, and traditional measures of corporate governance?" All these are fascinating questions that Wang will answer in the immediate future and we will advise readers when she does.
For now, the take home message is this: if it appears that "they" are lying to you, "they" probably are. Just run a numerical regression analysis to prove it.