Benford’s Law

Benford’s Law describes the phenomenon that in a natural set of numbers, the likelihood the first digit is a one is higher than the likelihood that the first digit is two, and so for till nine which has the lowest likelihood of being the first digit. This is best displayed as a logarithmically, decreasing probability distribution. Although Benford’s Law is most typically used to detect various forms of tax and financial fraud, it is not just tax fraud that fits the distribution. National population, river lengths, stock market volumes, volcanic crater radii, electricity bills, universal physical constants, and even music (the length of notes) all follow Benfords Law. Evidently, Benford’s Law is a mind-bending phenomenon and its applications are wide-ranging. For instance, it can be employed to track the validity of economic reports from developing countries in order to inform future economic policy.

Notably, there was a study that did such an analysis on China, Brazil, India, Mexico, Indonesia, and Turkey. These countries were chosen because of their relatively high nominal GDP and availability of FTSE industry data. FTSE data encompasses the national stock returns of the following ten industries: basic material, consumer goods, consumer services, financial, health care, industrials, oil and gas, technology, telecommunications, and utilities. The researchers then computed the monthly log return of each industry (in US dollar) for each country and then engaged in minor data manipulation to, for instance, remove abnormal repetitions of numbers, then plotted the results against the Benford’s Law curve. Additionally, they used the chi-square test to calculate a quantitative difference between the stock return data and Benford’s Law values. Comparing the stock return probability curve to Benford’s Law curve it is abundantly clear that, for the most part, the stock return data does not follow Benford’s Law.

The researches add the caveat that other accounting test should be done on this data before deeming the data faked and it is important to that not all naturally occurring data sets conform to the law, nevertheless, for the sake of argument, this insight will take the results at face value– countries are fabricating economic data. Governments of and industries based in underdeveloped countries have a variety of motivations for such information. For example, a dictatorial government may desire to present alternative financial reports in order to hide corrupt or oppressive behavior from the rest of the world or its citizens; or, perhaps a company reports fraudulent numbers regarding oil prices with the intention of raising oil prices or covering up huge losses. Ultimately, there are many reasons for fraud on such a national scale to occur. It is important to find a way to detect such fabrications because if international or domestic economic policy is based on untrue data, the policy will be ineffective. Economic policy is already complicating enough without the addition of falsified data on which to base decisions. Therefore more robust investigations of countries’ financial reports could lead to more reliable information and potentially more effective economic policy in developing nations.

References

Shi, J., Ausloos, M., & Zhu, T. (2018). Benford’s law first significant digit and distribution distances for testing the reliability of financial reports in developing countries. Physica A: Statistical Mechanics and Its Applications, 492, 878–888. ScienceDirect. https://doi.org/10.1016/j.physa.2017.11.017