## 21 December 2009 – Chart of the Week

Chart of the Week: Benford’s Law and forecasting economic data

Back in 1938, Frank Benford noticed an odd statistical pattern in the first digit of all kinds of data sets. He found that about a third of the time, whatever the data, it began with the digit 1. Whether it was a 1, or in the 10s, or in the 100s, or whatever, on average it was the leading digit 30.1% of the time. The digit 2 is the next in line, followed by 3, and so on, tapering off until the leading digit reaches 9.

“Benford’s Law,” also known as the “first digit law,” has a rich history in mathematics that is neatly summarized in Wikipedia and in the J-Walk blog. It appears in everything from death rates, to river lengths, to stock prices … and to economic data.

For instance, over the past decade, the odds that the change in initial jobless claims begins with a “1” is 31.9%, a “2” is 19.5%, a “3” is 13.1%, whereas a “9” is just 4.0%. The same pattern appears over and over again in the data, from consumer confidence to the ISM survey to the trade balance, wherever the numbers are distributed logarithmically in the real world.

Economists appear to follow a slightly different pattern. Based on forecasts tracked by Bloomberg back to 1999, the number 1 remains the most popular leading digit for Wall Street economists. However, while the distribution generally slopes down from there, for some reason the number 5 pops up with unnatural frequency. The same hump shows up in payrolls and new homes sales, among other data sets. Forecasters appear to particularly favor starting their estimate with a 5 in the case of initial jobless claims.

An easy do-it-yourself forecast-o-meter would be to take the consensus number, see if it starts with a higher digit, and then round up to the nearest power of 10. If it starts with a lower number, round down. According to Bloomberg, on Wednesday the consensus expects personal income to go up 0.5%. The smart money could be for at least 1.0%.

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