Variance estimation for
Generalized Entropy and
Atkinson inequality indices:
the complex survey data case
Martin Biewen (Goethe University Frankfurt)
Stephen Jenkins (University of Essex)
Presentation at 4th German Stata User Group Meeting, Mannheim, 31 March 2006
Inequality indices: measures of
the dispersion of a distribution


Imposition of a small number of axioms substantially
restricts functional form that indices may have
Axioms for
 Anonymity
 Scale invariance
 Replication invariance
 Normalization
 Principle of Transfers: mean preserving spread
in increases
Classes of inequality measures
satisfying the axioms
for

Generalized Entropy
 Advantage: subgroup decomposability
transfer sensitivity
Classes of inequality measures
satisfying the axioms


Atkinson index
 Advantage: welfare interpretation
inequality aversion
Gini coefficient
 Advantage: most well-known inequality index
Estimation of inequality indices


These indices are routinely calculated by many
analysts …
 The most commonly-used programs among Stata
users are ineqdeco and inequal7 (available
using ssc)
But only rarely do analysts report estimates of the
associated sampling variances (or SEs) of the estimates!
Estimation of inequality indices


Analytical derivations to date have omitted some
important situations (and indices)
 Most derivations assume i.i.d. observations (cf.
survey clustering or other sample dependencies!),
and don‘t consider probability weighting (cf. stratification!)
 The methods that do exist are not ‘well known’
Lack of available software
 But cf. geivars (Cowell (1989), linearization
methods; i.i.d. assumptions) and ineqerr
(bootstrap), both available using ssc
What we provide





Estimates of indices and associated sampling variances for all members of the GE and Atkinson classes,
while also …
Accounting for clustering and stratification, and for
the i.i.d. case
Analytical results (see our paper) and new Stata
programs (version 8.2): svygei and svyatk
Based on Taylor-series linearization methods combined with a result from Woodruff (JASA, 1971).
Results don‘t apply to Gini coefficient.
Overview of analytical derivation




Write estimator of each index as a function of population totals (involves sums over clusters, weights etc.)
(Taylor-series approximation) Variance of each estimator can be approximated by variance of 1st order
‘residual’
As is, each expression is not easily calculated …
But (Woodruff): reversing order of summation in
‘residual’ → estimation is equivalent to derivation of a
sampling variance of a total estimator for which one
can apply standard svy methods
The programs:
svygei and svyatk
svygei varname [if exp] [in range]
[,alpha(#) subpop(varname) level(#)
Calculations for
(use alpha(#) option to chose one
other than )
svyatk varname [if exp] [in range]
[,epsilon(#) subpop(varname) level(#)
Calculations for
(use epsilon(#) option to chose one other than


)
Where, of course, the data have first been svyset.
How data are organised, and described using svyset
is of crucial importance …
Survey data set-up for estimation
of inequality among individuals
1) Observation unit is person; sampling unit is household; all persons in
each household attributed with the equivalised income of the household to which they belong; individual sample weight available (‘xwgt’)
but no information about PSU or strata:
svyset [pw=xwgt], psu(hh_id)
2) As 1), except also know PSU and strata information (includes allowance
for within-household correlation):
svyset [pw=xwgt], psu(PSU_id) strata(STRATA_id)
3) Observation unit is household; sampling unit is household;
weight (‘xhhwgt’)= household sample weight
household size;
no information about PSU or strata
svyset [pw=xhhwgt]
→ i.i.d. case
Illustration




German Socio-Economic Panel (GSOEP), wave 18
data (2001) used as a cross-section
12,939 individuals in 5,195 households; 1004 PSUs
(‘psu’), 169 strata (‘strata’)
Equivalized (‘square-root equivalence scale’) post-tax
post-benefit household income (‘eq’)
Each individual attributed with the equivalised income
of her household (→ ‘clustering’ within households)
 Even if survey does not include PSU and strata
identifiers, you should account for this (use household identifier as PSU variable)
Generalized Entropy indices
. ssc install svygei_svyatk
. version 8.2
. svyset [pweight=xwgt], psu(psu) strata(strata)
. svygei eq
Complex survey estimates of Generalized Entropy inequality indices
pweight: xwgt
Strata: strata
PSU: psu
Number of obs
= 12939
Number of strata = 169
Number of PSUs
= 1004
Population size = 31487411
--------------------------------------------------------------------------Index
| Estimate
Std. Err.
z
P>|z|
[95% Conf. Interval]
---------+----------------------------------------------------------------GE(-1)
| .1179647
.00614786
19.19
0.000
.1059151
.1300143
MLD
| .1020797
.00495919
20.58
0.000
.0923599
.1117996
Theil
| .1027892
.0058706
17.51
0.000
.091283
.1142954
GE(2)
| .1201693
.00962991
12.48
0.000
.101295
.1390436
GE(3)
| .1713159
.02301064
7.45
0.000
.1262159
.2164159
---------------------------------------------------------------------------
Atkinson indices
. svyset [pweight=xwgt], psu(psu) strata(strata)
. svyatk eq
Complex survey estimates of Atkinson inequality indices
pweight: xwgt
Strata: strata
PSU: psu
Number of obs
= 12939
Number of strata = 169
Number of PSUs
= 1004
Population size = 31487411
--------------------------------------------------------------------------Index
| Estimate
Std. Err.
z
P>|z|
[95% Conf. Interval]
---------+----------------------------------------------------------------A(0.5)
| .0496963
.0025263
19.67
0.000
.0447448
.0546477
A(1)
| .0970424
.00447794
21.67
0.000
.0882658
.105819
A(1.5)
| .1434968
.00616915
23.26
0.000
.1314055
.1555881
A(2)
| .1908923
.00804946
23.71
0.000
.1751157
.206669
A(2.5)
| .2432834
.01237288
19.66
0.000
.219033
.2675338
---------------------------------------------------------------------------
Subpopulation option
. gen female = sex==2
. svygei eq, subpop(female)
Complex survey estimates of Generalized Entropy inequality indices
pweight: xwgt
Strata: strata
PSU: psu
Number of obs
Number of strata
Number of PSUs
Population size
=
=
=
=
12939
169
1004
31487411
Subpop: female, subpop. size = 16499055
--------------------------------------------------------------------------Index
| Estimate
Std. Err.
z
P>|z|
[95% Conf. Interval]
---------+----------------------------------------------------------------GE(-1)
|
.112828
.00573308
19.68
0.000
.1015914
.1240646
MLD
| .0994741
.00471331
21.10
0.000
.0902362
.1087121
Theil
| .0998958
.00543287
18.39
0.000
.0892476
.110544
GE(2)
| .1151464
.00877057
13.13
0.000
.0979564
.1323364
GE(3)
| .1596125
.02029283
7.87
0.000
.1198392
.1993857
---------------------------------------------------------------------------
Empirical illustration in our paper



GSOEP income data for 2001 (same as used here)
British Household Panel Survey for 2001 (9,979 individuals in 4,058 households; 250 PSUs, 75 strata)
Results:
 Inequality larger in Britain than in Germany, for
all indices, and difference is statistically significant
 z-ratios (index  SE) vary from 7.5 to 23.9 (DE)
and 5.1 to 31.9 (GB), being smallest for top-sensitive indices and largest for middle-sensitive indices
 Although sample larger in Germany, z-ratios are
not always smaller (→ different sample designs)
Empirical illustration (ctd.)
Index
Germany
Est.
Great Britain
Std.
z-rat. Est.
Std.
z-rat.
GE(-1) .11796 .00614
19.19 .31329 .03751
8.35
MLD
.10207 .00496
20.58 .17420 .00608 28.64
Theil
.10278 .00587
17.51 .16769 .00755 22.19
GE(2)
.12016 .00963
12.48 .21164 .01868 11.33
reject
Empirical illustration (ctd.)

Effects of different assumptions about survey design
on sampling variance estimates?
 For each index, the estimated standard error is
larger if one accounts for survey clustering and
stratification (unsurprising), but …
 Results suggest that accounting for survey design
features per se have little (additional) effect on
variance estimates as long as the replication of
incomes within multi-person households is accounted for
Conclusions

Researchers now have the means to estimate sampling variances for most of the inequality indices in
common use, accomodating a range of potential
assumptions about design effects
Topics for future research:
 GE indices are additively decomposable by population subgroup (→ ineqdeco): extend results here to
the components of decompositions
 Extend results to Gini coefficient and other measures
based on order-statistics (Lorenz curves etc.)
Selected references

Biewen, M. and Jenkins S.P. (2006): Estimation of
Generalized Entropy and Atkinson indices from complex survey data, forthcoming in: Oxford Bulletin of
Economics and Statistics


Cowell, F.A. (2000): Measurement of inequality, in
A.B. Atkinson and F. Bourguignon (eds), Handbook
of Income Distribution, Vol. 1, Elsevier, Amsterdam
Woodruff, R.S. (1971): A simple method for approximating the variance of a complicated estimate, Journal of the American Statistical Association, 66, 411-4