## Look out for No. 1

In the late 1990s, eurozone wannabes squeezed and stretched to meet the criteria for accession, including low inflation and government deficits, and moderate levels of debt. The criteria were somewhat irksome, especially for an economy such as Greece, but nevertheless the Greeks seemed to comply.

Eventually, it became clear that the Greek numbers did not quite add up. Eurostat, the European statistics agency, has complained about “widespread misreporting of deficit and debt data” from the Greek authorities. In 2006, eyebrows were raised when Greece’s GDP jumped 25 per cent overnight thanks to a statistical revision that sought to incorporate prostitution and money laundering, among other industries. In late 2009, the incoming prime minister announced that the deficit was more like 12.5 per cent of GDP than 3.7 per cent.

Had its economic statistics been more rigorously reported, it seems unlikely that Greece would have made it into the eurozone. But could the anomalies have been spotted at the time? Perhaps so.

I’ve written about Benford’s Law before: it’s a statistical regularity that often occurs in “real” data but not in manipulated numbers. Now four researchers have published a paper using Benford’s Law to examine Greek macroeconomic data. (Perhaps the origin of the paper should not be a surprise: it’s by Bernhard Rauch, Max Göttsche, Gernot Brähler and Stefan Engel, and it’s published in the German Economic Review.)

Benford’s Law was discovered in 1881 by the astronomer Simon Newcomb, and then again by Frank Benford, a physicist at General Electric, in 1938. The law is a curious one: it predicts the frequency of the first digits of a collection of numbers. For example, measure the lengths of the world’s rivers, and see how many of the digits begin with “one” (184 miles; 1,543 miles) versus “three” (3,022 miles) or “nine” (985 miles). Newcomb and Benford discovered that the first digit is usually a “one” – fully 30 per cent of the time, over six times more common than an initial “nine”. And the result is true whether one counts the numbers on the front page of The New York Times or leafs through baseball statistics.

Nobody seems sure why so much data has the Benford distribution. We do know that exponential growth produces it. To move from a GDP of one billion Flainian Pobble Beads (a unit of currency in The Hitchhiker’s Guide to the Galaxy) to two billion Flainian Pobble Beads requires cumulative growth of 100 per cent, which will take a while. But to move from a GDP of 9 billion to 10 billion Flainian Pobble Beads requires only 10 per cent growth. Benford distributions are, uniquely, scale-invariant – in other words, if one measures GDP in dollars instead of Pobble Beads, the Benford property remains.

Manipulated data often fail to satisfy Benford’s Law. A manager who must submit receipts for expenses over £20 may end up filing claims for lots of £18 and £19 expenses – and the data will then contain too many ones, eights and nines. A forensic accountant can easily check this, and while not an infallible check (fraudster Bernard Madoff filed Benford-compatible monthly returns), it’s an indicator of possible trouble.

Which brings us back to the data Greece submitted to the European statistics agency. According to Rauch and his colleagues, Greek data are further from the Benford distribution than that of any other European Union member state. Romania, Latvia and Belgium also have abnormally distributed data, while Portugal, Italy and Spain have a clean bill of health.

Would a Benford-style analysis have helped spot Greece’s problems? In principle, yes. In practice, one wonders whether politics would have trumped statistics. A shame: according to Benford’s Law, Greece’s data were particularly odd in 2000, just before it joined the euro.

*Also published at ft.com.*

^{th}of September, 2011 • Undercover Economist • Comments off

## 10 Comments

David White says:Great title! Your readers might also find the following link interesting (BBC Radio 4 Programme on Benford’s Law)

http://www.bbc.co.uk/radio4/science/further5.shtml

10^{th}of September, 2011John Mulenga says:I’d like to see this test applied to the reported economic growth in my country Zambia. I’m no economist but the growth seems at odds with the high inflation, weakening currency and “investors” who employ foreigners and expatriate their profits (which are poorly taxed). Maybe you could look at Zambia some time?

10^{th}of September, 2011Glenn says:I teach a unit on Benford’s law, and I have used voting information. Does anyone know how I could get my hands on some of the data sets mentioned in the article? I would love to use some real data as an exercise in class!

12^{th}of September, 2011Charles Arthur says:The other thing about Benford’s Law is that it is base-invariant. (If you think about it, aliens with 8 fingers on each hand would discover B’s Law too.)

Although it would be a bit difficult to test it for base 2. But it’s one of my favourite mathematical discoveries.

12^{th}of September, 2011Cedric Middlebourne says:Very enjoyable and informative

13^{th}of September, 2011D Jones says:Hi Tim,

Thanks for a very interesting article.

Would it really have prevented the euro crisis, however?

A big part of the problem is surely the unprecedented scale (since 1929) of the financial crisis. It may have been unwise to let Greece in for political reasons – but surely the consequences would not have been so catastrophic had there not been such a huge failure in the global financial system? And even if Greece had been kept out – the enormity of the credit crunch means surely we would still be dealing with the problems of Ireland and Portugal instead?

Thanks again.

15^{th}of September, 2011Martyn Wilson says:Oops! 9 billion to 10 billion is *not* 10% growth. More like 11.1 (and a bit more)

17^{th}of September, 2011Steve Jones says:@Charles Arthur

It’s easy enough to test for in base 2 – just convert to a different base and test that. Of course you need sufficient bits to make it a sensible test in the higher base, but that’s just another way of saying the numbers have to span a sufficient range whatever the base.

17^{th}of September, 2011Owen Smith says:Readers may be interested in http://testingbenfordslaw.com/ – a collaborative excercise to gather & display large datasets, comparing the leading digit distribution to that predicted by Benford’s law.

The results still amaze me every time I see them 🙂

18^{th}of September, 2011LJ says:Can you provide any proof that Benford’s law is scale invariant? I’ve always understood Benford’s law to be base invariant, but definitely not scale invariant, as it often emerges from mapping logarithmically distributed data onto a different scale.

Great post, I’d never thought about using it forensically as you describe.

19^{th}of September, 2011