Variance estimation for Generalized Entropy and Atkinson inequality indices: the complex survey data...
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Transcript of Variance estimation for Generalized Entropy and Atkinson inequality indices: the complex survey data...
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
Generalized Entropy Advantage: subgroup decomposability
for
transfer sensitivity
Classes of inequality measures satisfying the axioms
Atkinson index Advantage: welfare interpretation
Gini coefficient Advantage: most well-known inequality
index
inequality aversion
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 esti-mates!
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. 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
What 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 don‘t apply to Gini coefficient.
Overview 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
The 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 )
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 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 [pw=xwgt], psu(hh_id)
svyset [pw=xwgt], psu(PSU_id) strata(STRATA_id)
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 house-hold 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 Number of obs = 12939Strata: strata Number of strata = 169PSU: psu 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 .1300143MLD | .1020797 .00495919 20.58 0.000 .0923599 .1117996Theil | .1027892 .0058706 17.51 0.000 .091283 .1142954GE(2) | .1201693 .00962991 12.48 0.000 .101295 .1390436GE(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 Number of obs = 12939Strata: strata Number of strata = 169PSU: psu 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 .0546477A(1) | .0970424 .00447794 21.67 0.000 .0882658 .105819A(1.5) | .1434968 .00616915 23.26 0.000 .1314055 .1555881A(2) | .1908923 .00804946 23.71 0.000 .1751157 .206669A(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 Number of obs = 12939Strata: strata Number of strata = 169PSU: psu Number of PSUs = 1004 Population size = 31487411Subpop: female, subpop. size = 16499055---------------------------------------------------------------------------Index | Estimate Std. Err. z P>|z| [95% Conf. Interval]---------+-----------------------------------------------------------------GE(-1) | .112828 .00573308 19.68 0.000 .1015914 .1240646MLD | .0994741 .00471331 21.10 0.000 .0902362 .1087121Theil | .0998958 .00543287 18.39 0.000 .0892476 .110544GE(2) | .1151464 .00877057 13.13 0.000 .0979564 .1323364GE(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 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)
Empirical illustration (ctd.)
Index Germany Great Britain
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
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 ac-counted for
Conclusions
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.)
Selected 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