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help for ^ineqdeco5^
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Inequality indices, with decomposition by subgroup
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^ineqdeco5^ varname [^[^fweights aweights^]^ ^if^ exp ^in^ range]
[, ^by^group^(^groupvar^)^ ^w^ ^s^umm]
^ineqdeco5^ is for use with Stata versions 5 to 8.1. For versions 8.2 onwards,
use ^ineqdeco^.
Options
-------
^by^group^(^groupvar^)^ requests inequality decompositions by population
subgroup, with subgroup membership summarized by groupvar.
^w^ requests calculation of equally-distributed-equivalent incomes and
welfare indices in addition to the inequality index calculations.
^s^umm requests presentation of ^summary, detail^ output for varname.
Saved results (global macros)
-----------------------------
S_9010, S_7525 Percentile ratios p90/p10, p75/p25
S_im1, S_i0, S_i1, S_i2 GE(a), for a = -1, 0, 1, 2 (defined below)
S_gini Gini coefficient
S_ahalf, S_i1, S_a2 A(e), for e = 0.5, 1, 2 (defined below)
Examples
--------
. ^ineqdeco5 x [w=wgtvar]^
. ^ineqdeco5 x, by(famtype) w^
. ^ineqdeco5 x if sex==1, w s^
Description
-----------
^ineqdeco5^ estimates a range of inequality and related indices commonly
used by economists, plus decompositions of a subset of these indices by
population subgroup. Inequality decompositions by subgroup are useful
for providing inequality `profiles' at a point in time, and for analyzing
secular trends using shift-share analysis. Unit record (`micro' level) data
are required.
Inequality indices estimated are: members of the single parameter
Generalized Entropy class GE(a) for a = -1, 0, 1, 2; the Atkinson class
A(e) for e = 0.5, 1, 2; the Gini coefficient, and the percentile ratios
p90/p10 and p75/p25. Also presented are related summary statistics such as
subgroup means and population shares. Optionally presented are indices
related to the Atkinson inequality indices, viz equally-distributed-
equivalent income Yede(e), social welfare indices W(e), and the Sen
welfare index: see below for details.
The inequality indices differ in their sensitivities to income 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 more positive e>0 (the 'inequality aversion parameter')
is, the more sensitive A(e) is to income differences at the bottom of
the distribution. The Gini coefficient is most sensitive to income
differences about the middle (more precisely, the mode).
Detailed description
--------------------
Consider a population of persons (or households ...), i = 1,...,n,
with income y_i, and weight w_i. Let f_i = w_i/N, where
i=n
N = SUM(w_i). When the data are unweighted, w_i = 1 and N = n.
i=1
Arithmetic mean income is m. Suppose there is an exhaustive partition of
the population into mutually-exclusive subgroups k = 1,...,K.
The Generalized Entropy class of inequality indices is given by
_ _
| _ _ |
1 | | i=n | |
GE(a) = ------- | | SUM (f_i)(y_i/m)^^a]| - 1 |, a~=0, a~=1
a(a-1) | | i=1 | |
| - - |
- -
i=n
GE(1) = SUM (f_i)(y_i/m)[log(y_i/m)]
i=1
i=n
GE(0) = SUM (f_i)[log(m/y_i)].
i=1
Each GE(a) index can be additively decomposed as
GE(a) = GE_W(a) + GE_B(a)
where GE_W(a) is Within-group Inequality and GE_B(a) is Between-Group
Inequality.
k=K
GE_W(a) = SUM [(v_k)^^(1-a)].[(s_k)^^a].GE_k(a)
k=1
where v_k = N_k/N is the number of persons in subgroup k divided by the
total number of persons (subgroup population share), and s_k is the
share of total income held by k's members (subgroup income share).
GE_k(a), inequality for subgroup k, is calculated as if the subgroup were a
separate population, and GE_B(a) is derived assuming every person within
a given subgroup k received k's mean income, m_k.
Define the equally-distributed-equivalent income
_ _
| _ _ |^^[1/(1-e)]
| | i=n | |
Yede(e) = | | SUM (f_i)(y_i)^^(1-e]| | , e>0, e~=1
| | i=1 | |
| - - |
- -
i=n
= SUM (f_i).[log(y_i) ], e=1.
i=1
The Atkinson indices are defined by
A(e) = 1 - [Yede(e)/m].
These indices are decomposable (but not additively decomposable):
A(e) = A_W(a) + A_B(a) - [A_W(a)].[A_B(a)]
where k=K
A_W(a) = 1 - [SUM (v_k).(Yede_k)/m]
k=1
and k=K
A_B(a) = 1 - (Yede)/[SUM (v_k).(Yede_k)/m].
k=1
Social welfare indices are defined by
W(e) = {[Yede(e)]^^(1-e)}/(1-e), e>0, e~=1
W(1) = log[Yede(1)].
Each of these indices is an increasing function of a `generalized mean
of order (1-e)'. All the welfare indices are additively decomposable:
k=K
W(e) = SUM (v_k).[W_k(e)].
k=1
The Gini coefficient is given by
i=n
G = 1 + (1/N) - [2/(m.N^^2)][SUM (N-i+1)(y_i)]
i=1
where persons are ranked in ascending order of y_i. The Gini coefficient
(and the percentile ratios) are not properly decomposable by subgroup into
within- and between-group inequality components.
Sen's (1976) welfare index is given by:
S = m(1-G).
Author
------
Stephen P. Jenkins
Institute for Social and Economic Research
University of Essex, Colchester CO4 3SQ, U.K.
NB minor fixes in February 2001:
(i) Made compatible with Stata 7
(NB still runs with Stata 5 and Stata 6.)
(ii) bug fix for Gini with fweights (minor).
References
----------
Atkinson, A.B. (1970) "On the measurement of inequality",
Journal of Economic Theory, 2, 244-63.
Blackorby, C., Donaldson, D., and Auersperg, M. (1981),
"A new procedure for the measurement of inequality
within and between population subgroups", Canadian
Journal of Economics, XIV, 665-85.
Cowell, F.A. (1995), Measuring Inequality, second edition,
Prentice-Hall/Harvester-Wheatsheaf, Hemel Hempstead.
Jenkins, S.P. (1991), "The measurement of income inequality", in
L. Osberg (ed.), Economic Inequality and Poverty: International
Perspectives, Armonk NY, M.E. Sharpe.
Jenkins, S.P. (1995) "Accounting for inequality trends: decomposition
analyses for the UK, 1971-86", Economica, 62, 29-63.
Jenkins, S.P. (1997), "Trends in real income in Britain: a microeconomic
analysis", Empirical Economics, 22, 483-500.
Sen, A.K. (1976) "Real national income", Review of Economic Studies, 43,
19-39.
Shorrocks, A.F. (1984), "Inequality decomposition by population
subgroups", Econometrica, 52, 1369-88.
Also see
--------
^inequal^ (sg30: STB-23) if installed; ^rspread^ (sg31: STB23) if installed
^povdeco^ if installed; ^sumdist^ if installed
^inequal2^ (http://fmwww.bc.edu/RePEc/bocode/i) if installed;
^ineqerr^ [STB-51: sg115] if installed