Re: st: Computing the Gini or another inequality coefficient from a limited number of data points

[Daniel Feenberg <feenberg@nber.org>]

On Fri, 10 Feb 2012, Jen Zhen wrote:

> Dear list members,
>
> I would like to compute a measure of income inequality similar to the
> Gini index. I do not know everyone's income, so need to make an
> approximation.
>
> (1)
> For the 5 most recent years, I know for 6 income brackets how many
> individuals there are and their joint income, hence also the average
> income in the bracket. For the full-fledged Gini index I would need to
> know the area under the curve which shows the cumulative income
> against the cumulative number of tax payers (to visualize what I mean,
> look e.g. at the 2nd figure here:
> http://en.wikipedia.org/wiki/Gini_index).
> Now I believe that with the information I have I don't know the entire
> curve but I know only 7 points on it (the six points mentioned plus
> the origin). So I think I can approximate the said area if I simply
> assume that between the 7 points the line is straight, but that will
> systematically underestimate the true degree of inequality. So I'm
> wondering if there is a sensible way to smooth the curve and hence get
> a better approximation?
>
> (2)
> For the 5 earliest years unfortunately I know only the number of
> individuals in each bracket but not their joint income. So my idea was
> that I would regress the mean income in each bracket on a 3rd-order
> function in the year to see how it develops in the 5 latest years and
> use this to predict/estimate the mean income for each bracket in the 5
> earlier years, then use the procedure described in (1). A simpler
> alternative would be to just use the midpoint of each bracket, but I
> guess this would be less good.
>
> Does this procedure sound sensible? Or is there a better way to
> compute inequality from these data?

You could assume an income distribution function, such as log-normal, 
poisson, or even Gini and solve for the parameters using the available 
data. We do this in "Income Inequality and the Incomes of Very High-Income 
Taxpayers: Evidence from Tax Returns"

  http://www.nber.org/chapters/c10880

with a Poisson. The Poisson is a two parameter distribution, so we solve 
for the parameters within an income bracket using only the 2 breakpoints 
that define the bracket. That way there is no need to extrapolate beyond 
the observed data, or coerce data everywhere in the income distribution 
to a small number of parameters estimated over the whole distribution.

Daniel Feenberg
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