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help for ^geivars^
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Generalized Entropy inequality indices, with sampling variances
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^geivars^ varname [ [^fw^eights] ^if^ exp ^in^ range]

Examples
--------
. ^geivars hhincome [w=hhsize]^
. ^geivars hhincome, if sex==1^

Description
-----------

^geivars^ estimates several inequality indices commonly
used by economists, together with their asymptotic sampling
variances. Unit record (`micro' level) data are required.

The inequality indices estimated are members of the single parameter
Generalized Entropy class GE(a) for a = -1, 0, 1, 2. The indices
differ in their sensitivities to differences in different parts
of the distribution. The more positive a is, the more sensitive
GE(a) is to income differences at the top of the distribution; the
more negative a is the more sensitive it is to differences at the
bottom of the distribution. GE(0) is the mean logarithmic deviation,
GE(1) is the Theil index, and GE(2) is half the square of the
coefficient of variation.

The formulae for the sampling variances are taken directly from
Cowell (1989).  His formulae were derived assuming that the income
receiving units ('households') are treated as a random sample from a
bivariate distribution of income and a household weight variable
(e.g. household size). We require estimates of income inequality amongst
all persons in the household population. I.e. in effect there is a
random sample of households with 'self weighting' by household size,
where the weights are similar to Stata's fweights.  Thus the variance
formulae do not also adjust for the effects of complex survey design
features (stratification and clustering) -- formulae for this case are
rather complicated and the subject of current research. These problems
do not arise, of course, if the data are unweighted.

Derivation of the formulae for the asymptotic variances use the
result that the GE(a) indices can be written as functions of sample
moments. For further details, see Cowell (1989).

^geivars^ output includes the estimates of the four indices, and three
sets of variance estimates for each corresponding to different
informational assumptions.  V0 is the variance in the case where both
mean 'income' and 'household size' are known. V1 (= V0 + delta1) is the
variance in the case where the former is not known, and V2 (= V1 + delta2)
is the variance in the case where both are unknown and estimated from the
sample.  In each case the asymptotic 't' ratio = GE(a)/sqrt[V(a)] and
associated p-value are also reported.

Author
------
Stephen P. Jenkins <stephenj@@essex.ac.uk>
Institute for Social and Economic Research
University of Essex, Colchester CO4 3SQ, U.K.

NB minor fix in February 2001 so that compatible with Stata 7.
[Still runs with Stata 5 and Stata 6.]

References
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Cowell, F.A. (1989), "Sampling variance and decomposable
inequality measures", Journal of Econometrics, 42,
27-41.
Cowell, F.A. (1995), Measuring Inequality, second edition,