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Does Benford’s Law hold in economic research and forecasting?

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Abstract

First and higher order digits in data sets of natural and socio-economic processes often follow a distribution called Benford’s law. This phenomenon has been used in business and scientific applications, especially in fraud detection for financial data. In this paper, we analyse whether Benford’s law holds in economic research and forecasting. First, we examine the distribution of regression coefficients and standard errors in research papers, published in Empirica and Applied Economics Letters. Second, we analyse forecasts of GDP growth and CPI inflation in Germany, published in Consensus Forecasts. There are two main findings: The relative frequencies of the first and second digits in economic research are broadly consistent with Benford’s law. In sharp contrast, the second digits of Consensus Forecasts exhibit a massive excess of zeros and fives, raising doubts on their information content.

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Notes

  1. Even if that is the case, attempts to replicate the studies mostly fail. McCullough et al. (2006) analysed more than 150 articles from the Journal of Money, Credit, and Banking, but were able to reproduce the results in less than 10 percent of the cases.

  2. For specific digits, e.g. whether there is an excess of fives, the standard normal statistic \( {\text{T}}_{\text{d}} = {{\sqrt {{\text{N}}_{\text{d}} } \left( {{\text{h}}_{\text{d}} - {\text{p}}_{\text{d}} } \right)} \mathord{\left/ {\vphantom {{\sqrt {{\text{N}}_{\text{d}} } \left( {{\text{h}}_{\text{d}} - {\text{p}}_{\text{d}} } \right)} {\sqrt {{\text{p}}_{\text{d}} (1 - {\text{p}}_{\text{d}} )} }}} \right. \kern-\nulldelimiterspace} {\sqrt {{\text{p}}_{\text{d}} (1 - {\text{p}}_{\text{d}} )} }} \) can be used to check whether the observed frequency significantly deviates from its theoretical value.

  3. A closely related statistic is Nigrini’s (1996a,b) distortion factor.

  4. Recently, Tam Cho and Gaines (2007) proposed the Euclidean distance as a measure to characterize the deviation from the Benford distribution. This measure is independent of the sample size, however, it is lacking a statistical foundation.

  5. The results for third digits have been evaluated as well (overall showing a very good agreement with Benford’s law) but are not reported due to space limitations. An analysis of higher-order digits (which are more likely to be uniformly distributed) is impeded by insufficient digits in most published papers.

  6. For both journals also the possible sequences of the first and second digits (e.g. 14, 73, 86) have been analysed. The results, which are not reported here, show no clear pattern, neither regarding the tendencies of tests (which rejects more often) nor the effects of sample size.

  7. Based on an analysis of the realised CPI growth rates for Germany (10/1989–07/1994).

  8. The Chi2 statistic for the second digits is 11.61 (p-value: 0.24) and for the “first digit after the decimal point” 5.61 (p-value: 0.82).

  9. One promising approach has been proposed by Berlemann and Nelson (2005). They introduce a small-scale experimental stock market which yields the (mean) forecast of inflation rate as well as a likelihood measure for different inflation scenarios. The main idea is to use the market as the best instrument to uncover and aggregate private information.

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Authors and Affiliations

  1. Research Centre Deutsche Bundesbank, Wilhelm-Epstein-Strasse 14, 60431, Frankfurt am Main, Germany

    Stefan Günnel & Karl-Heinz Tödter

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  1. Stefan Günnel
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  2. Karl-Heinz Tödter
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Correspondence to Karl-Heinz Tödter.

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The paper was written while the first author was a visiting researcher at the Research Centre of the Deutsche Bundesbank. The views expressed in this paper are those of the authors and do not necessarily reflect those of the Deutsche Bundesbank. We thank two anonymous referees and the editor for helpful comments and suggestions.

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Günnel, S., Tödter, KH. Does Benford’s Law hold in economic research and forecasting?. Empirica 36, 273–292 (2009). https://doi.org/10.1007/s10663-008-9084-1

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