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