Variance estimation for Generalized Entropy and Atkinson inequality indices: the complex survey data case

1
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
2
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

3
Classes of inequality measures satisfying the
axioms
for

• Generalized Entropy
• Advantage subgroup decomposability

transfer sensitivity
4
Classes of inequality measures satisfying the
axioms

• Atkinson index
• Advantage welfare interpretation
• Gini coefficient
• Advantage most well-known inequality index

inequality aversion
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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!

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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 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

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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 dont apply to Gini coefficient.

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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

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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
)
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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 pwxwgt, psu(hh_id)
svyset pwxwgt, psu(PSU_id) strata(STRATA_id)
svyset pwxhhwgt
? i.i.d. case
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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)

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Generalized 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

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Atkinson 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
• --------------------------------------------------
  -------------------------

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Subpopulation 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

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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)

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Empirical 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
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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

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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.)

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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