Title: Variance estimation for Generalized Entropy and Atkinson inequality indices: the complex survey data case
1Variance 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
2Inequality 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
3Classes of inequality measures satisfying the
axioms
for
- Generalized Entropy
- Advantage subgroup decomposability
transfer sensitivity
4Classes of inequality measures satisfying the
axioms
- Atkinson index
- Advantage welfare interpretation
- Gini coefficient
- Advantage most well-known inequality index
inequality aversion
5Estimation 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
esti-mates!
6Estimation 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 dont consider probability weighting (cf.
strati-fication!) - 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
7What we provide
- Estimates of indices and associated sampling
varian-ces 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
com-bined with a result from Woodruff (JASA,
1971). - Results dont apply to Gini coefficient.
8Overview of analytical derivation
- Write estimator of each index as a function of
popula-tion totals (involves sums over clusters,
weights etc.) - (Taylor-series approximation) Variance of each
esti-mator 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
9The programs svygei and svyatk
- svygei varname if exp in range ,alpha()
subpop(varname) level() - svyatk varname if exp in range ,epsilon()
subpop(varname) level() - Where, of course, the data have first been
svyset. - How data are organised, and described using
svyset is of crucial importance
Calculations for
(use alpha() option to chose one other than
)
Calculations for
(use epsilon() option to chose one other than
)
10Survey 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
house-hold to which they belong individual
sample weight available (xwgt) but no
information about PSU or strata - 2) As 1), except also know PSU and strata
information (includes allowance for
within-household correlation) - 3) Observation unit is household sampling unit
is household - weight (xhhwgt) household sample weight
household size - no information about PSU or strata
svyset pwxwgt, psu(hh_id)
svyset pwxwgt, psu(PSU_id) strata(STRATA_id)
svyset pwxhhwgt
? i.i.d. case
11Illustration
- 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
house-hold identifier as PSU variable)
12Generalized Entropy indices
- . ssc install svygei_svyatk
- . version 8.2
- . svyset pweightxwgt, psu(psu) strata(strata)
- . svygei eq
- Complex survey estimates of Generalized Entropy
inequality indices -
- pweight xwgt
Number of obs 12939 - Strata strata
Number of strata 169 - PSU psu
Number of PSUs 1004 -
Population size 31487411 - --------------------------------------------------
------------------------- - Index Estimate Std. Err. z
Pgtz 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
13Atkinson indices
- . svyset pweightxwgt, psu(psu) strata(strata)
- . svyatk eq
- Complex survey estimates of Atkinson inequality
indices -
- pweight xwgt
Number of obs 12939 - Strata strata
Number of strata 169 - PSU psu
Number of PSUs 1004 -
Population size 31487411 - --------------------------------------------------
------------------------- - Index Estimate Std. Err. z
Pgtz 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 - --------------------------------------------------
-------------------------
14Subpopulation option
- . gen female sex2
- . svygei eq, subpop(female)
- Complex survey estimates of Generalized Entropy
inequality indices -
- pweight xwgt
Number of obs 12939 - Strata strata
Number of strata 169 - PSU psu
Number of PSUs 1004 -
Population size 31487411 - Subpop female, subpop. size 16499055
- --------------------------------------------------
------------------------- - Index Estimate Std. Err. z
Pgtz 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
15Empirical illustration in our paper
- GSOEP income data for 2001 (same as used here)
- British Household Panel Survey for 2001 (9,979
indi-viduals 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-sensi-tive indices and largest for
middle-sensitive indices - Although sample larger in Germany, z-ratios are
not always smaller (? different sample designs)
16Empirical illustration (ctd.)
Index Germany Germany Germany Great Britain Great Britain Great Britain
Index Est. 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
17Empirical 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
ac-counted for
18Conclusions
- Researchers now have the means to estimate
samp-ling 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
popula-tion 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.)
19Selected references
- Biewen, M. and Jenkins S.P. (2006) Estimation of
Generalized Entropy and Atkinson indices from
com-plex 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
approxi-mating the variance of a complicated
estimate, Jour-nal of the American Statistical
Association, 66, 411-4