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Extracting economic data from Pareto law distributions

4TH PART

Thursday 24 November 2016, by Bernard Zimmern

One of the biggest obstacle to getting reliable results from economic surveys such as the world known Survey of Consumer Finances of the US Federal Reserve is the dispersion of data from one survey to the other, particularly the averages, because of the Pareto law distribution of economic figures.

The quantities we seek to measure such as income, assets, jobs created, etc. are economic quantities and most are governed by this law, also named the 80/20 law.

It is known since the late nineteenth century, through the work of Wilfred Pareto, that income is distributed according to a frequency distribution law that bears his name: the Pareto distribution.

Scholars such as Benoît Mendelbrot showed that Pareto was a universal law, perhaps more universal than the Gauss law, as it governs the frequency of words in the language, the size of galaxies or cities, Nile floods, stock prices, wealth, incomes, companies sizes, etc. It is indeed the great law governing the distribution of most concepts used by economists.

Though very universal, it is a law that is least understood and mastered.

As Nobel prize Paul Krugman stated it in 1995 :

“We have to say that the rank-size rule is a major embarrassment for economic theory: one of the strongest statistical relationships we know, lacking any clear basis in theory ».

To evaluate the significance of using a Pareto law on averages, lets take the example of the average income of active entrepreneurs in the US as extracted from the 2013 SCF

We know that the average measured through a sample stands, for 95% of samples, around the real mean + or - 2 σ, σ being the standard deviation of the variable.

If the distribution has a standard deviation from the mean value equal to the mean value (which is not uncommon in an observed Pareto distribution), with 900 observations, which is the case of the number of respondents beong active entrepreneurs in the survey, if the population is governed by Gauss’s law, the standard deviation of the mean value would be 3,3% of the mean (1 divided by square root of 900, i.e.1/30= 3,3%).

The real stand deviation we have measured is 12,5%, around 4 times larger.

The reason is that the Pareto distribution gives high probabilities to exceptional values, which are somehow distorting the results.

And that is why it complicates statisticians life (and why Pareto is rarely taught in statistical courses).

Indeed, averages are a favorite economic tool of economists, more than medians, because averages are additional and multiplicational whereas medians are not. But averages are strongly influenced by rare cases, extreme values.

Mathematically, the variance of the mean of a Pareto distribution with exponent less than 2 (see below) and not limited, is infinite whereas it is finished in most other law distributions even with extreme values, such as the exponential distribution, the Poisson distribution, Gibrat’s law, etc.

The world had the opportunity to measure the consequence of the Pareto law during the 2008 crisis: the phenomenal growth of the derivatives market was based on the idea that bringing together different assets in "securitized" packages, would limit the risks of losses incurres by the package; but the calculations were done, assuming the assets included in the package had Gaussian risk distributions, this leading to sums with much lower probabilities of extreme values than if assumed Paretian. This has been described fully in "The black swan" (Nassim Taleb).

One reason for doing so was that computers easily add Gauss variables, not Pareto’s.

IV. 1. CAN WE GIVE ANY MEANING TO THE FACT A VARIABLE FOLLOWS A PARETO DISTRIBUTION?

The great mathematician Paul Lévy has given in the early 1930 the proof that the Pareto law is as well a law of chance as the Gauss law.
He has indeed proven that there are 3 laws having the property that the sum of random variables following a given law belongs to the same random distribution law ; those « stable » laws are the Gauss’s law, the Poisson law and the Pareto law. It is therefore not surprising to find it in every corner of the economy where so many factors intermingle.

It is interesting to find that the investment by worker made by entrepreneurs who created their business [1] is a Pareto distribution. Below the Pareto distribution from the 2013 SCF survey.

To prove a distribution is Paretian, the simplest method is to graph the cumulated frequency, starting by the higher values of the variable, against the variable itself. When using the logarithme of cumulated frequency against the logarithme of the variable, as the probability cumulative function is P(X>x)= (xm/x)α, we have Log P= Constante (xmα) – α Log x.
The graph gives a straight line of negative slope –α. Here α= 1,35.
It is usual that the straight line has some scattered values on the right side (the outliers, corresponding to the highest values) and a flat end on the left side, that has little weight as the increase in cumulative frequency is small if not negligible.

To prove a distribution is Paretian, the simplest method is to graph the cumulated frequency, starting by the higher values of the variable, against the variable itself. When using the logarithme of cumulated frequency against the logarithme of the variable, as the probability cumulative function is P(X>x)= (xm/x)α, we have Log P= Constante (xmα) – α Log x.
The graph gives a straight line of negative slope –α. Here α= 1,35.
It is usual that the straight line has some scattered values on the right side (the outliers, corresponding to the highest values) and a flat end on the left side, that has little weight as the increase in cumulative frequency is small if not negligible.

This lead to the conclusion that investment par employee is a highly random variable all the more since those data do not include those who failed. Creating jobs is very hazardous.

But being a random variable does not mean there is no statistical relationship between income by entrepreneurs and the number of their employees.
Thre is one showing that when creating a business and investing to create jobs, the entrepreneurs hope to increase income and wealth.

One of the link between entrepreneur income and employment is the link between employment and the profit of the business. This is shown below, with both coordinates being the Log10 of the variable.

IV. 2 (to follow)

Footnotes

[1Answering Yes to question X 3103 (do you own a business), X 3104 (are you active in this business) and answering Yes to the subquestion 2 in X 3108( how do you became active entrepreneur ; subquestion 2 is « by creating it »)

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