Inequality indices, with optional decomposition by subgroup
ineqdeco varname [weights] [if exp] [in range] [, bygroup(groupvar) welfare summarize ]
ineqdec0 varname [weights] [if exp] [in range] [, bygroup(groupvar) welfare summarize ]
fweights and aweights are allowed; see help weights.
Description
ineqdeco and ineqdec0 estimate a range of inequality and related indices commonly used by economists, plus decompositions of a subset of these indices by population subgroup. Inequality decompositions by subgroup are useful for providing inequality profiles at a point in time, and also for analyzing secular trends using shift-share analysis. Unit record (`micro' level) data are required. Observations with values of varname less than or equal to zero are excluded from calculations using ineqdeco. By contrast, calculations using ineqdec0 do not exclude these observations: values of varname less than or equal to zero are valid (unless otherwise excluded using the if or in options). As a consequence, the portfolio of indices that is estimated by ineqdec0 is restricted. See below for details.
Inequality indices estimated by ineqdeco are: members of the single parameter Generalized Entropy class GE(a) for a = -1, 0, 1, 2; the Atkinson class A(e) for e = 0.5, 1, 2; the Gini coefficient, and the percentile ratios p90 / p10 and p75 / p25. Also presented are related summary statistics such as subgroup means and population shares. Optionally presented are indices related to the Atkinson inequality indices, namely equally-distributed-equivalent income Yede(e), social welfare indices W(e), and the Sen welfare index: see below for details. The indices estimated by ineqdec0 are the percentile ratios p90/p10 and p75/p25, GE(2) = half the squared coefficient of variation, the Gini coefficient, and Sen's welfare index.
The inequality indices differ in their sensitivities to income differences in different parts of the distribution. The more positive a is, the more sensitive GE(a) is to income differences at the top of the distribution; the more negative a is, the more sensitive it is to differences at the bottom of the distribution. GE(0) is the mean logarithmic deviation, GE(1) is the Theil index, and GE(2) is half the square of the coefficient of variation. The more positive e > 0 (the 'inequality aversion parameter') is, the more sensitive A(e) is to income differences at the bottom of the distribution. The Gini coefficient is most sensitive to income differences about the middle (more precisely, the mode).
For textbook reviews of inequality measurement from the perspective of economists, see Cowell (1995) or Jenkins (1991). See also Cowell (2000). On the characterization of Generalized Entropy indices, and their subgroup decomposability properties, see e.g. Shorrocks (1984) and references therein. On the Atkinson indices, see Atkinson (1970). The decomposition of Atkinson indices is discussed by Blackorby et al. (1981). For extensive empirical illustrations of inequality index decomposition, see inter alia Jenkins (1995) who also applies the decomposition of inequality trends proposed by Mookherjee and Shorrocks (1982). Cowell and Jenkins (1995) compare decompositions based on Generalized Entropy and Atkinson indices. The welfare indices estimated here are discussed by Sen (1976), and Jenkins (1997) who also provides empirical illustrations.
groupvar must take non-negative integer values only. To create such a variable from an existing variable, use the egen function group. By default, observations with missing values on groupvar are excluded from calculations when the bygroup option is specified. If you wish to include them, create a new variable with the egen function group and use its missing option. The egen function group is also useful for multi-way decompositions. E.g. for a decomposition by sex and region, create a new groupvar defining sex-region combinations by specifying sex and region in group(varlist).
Bootstrapped standard errors for the estimates of the indices can be derived using bootstrap. Standard errors derived using linearization methods can be calculated for GE(a) using svygei, for A(e) using svyatk, and for the Gini using svylorenz.
Technical details
Consider a population of persons (or households ...), i = 1,...,n, with income y_i, and weight w_i. Let f_i = w_i / N, where N = SUM w_i. (In what follows all sums are over all values of whatever is subscripted.) When the data are unweighted, w_i = 1 and N = n.
Arithmetic mean income is m. Suppose there is an exhaustive partition of the population into mutually-exclusive subgroups k = 1,...,K.
The Generalized Entropy class of inequality indices is given by
GE(a) = [1 / (a (a - 1)] { [SUM f_i (y_i / m)^a] - 1 }, a != 0 and a != 1,
GE(1) = SUM f_i (y_i / m) log(y_i / m),
GE(0) = SUM f_i log(m / y_i).
Each GE(a) index can be additively decomposed as
GE(a) = GE_W(a) + GE_B(a)
where GE_W(a) is Within-group Inequality and GE_B(a) is Between-Group Inequality and
GE_W(a) = SUM [v_k^(1-a)] . [s_k^a] . GE_k(a)
where v_k = N_k / N is the number of persons in subgroup k divided by the total number of persons (subgroup population share), and s_k is the share of total income held by k's members (subgroup income share). (Strictly speaking, v_k is the sum of the weights in subgroup k divided by the sum of the weights for the full estimation sample.)
GE_k(a), inequality for subgroup k, is calculated as if the subgroup were a separate population, and GE_B(a) is derived assuming every person within a given subgroup k received k's mean income, m_k.
Define the equally-distributed-equivalent income
Yede(e) = [SUM f_i (y_i)^(1-e)]^(1 / (1 - e)), e > 0 and e != 1,
= exp( SUM f_i . log y_i ), e = 1.
The Atkinson indices are defined by
A(e) = 1 - Yede(e) / m.
These indices are decomposable (but not additively decomposable):
A(e) = A_W(a) + A_B(a) - A_W(a) . A_B(a)
where
A_W(a) = 1 - [SUM (v_k) . Yede_k / m] and
A_B(a) = 1 - Yede / [SUM v_k. Yede_k ].
Social welfare indices are defined by
W(e) = [Yede(e)^(1-e)] / (1 - e), e > 0, e != 1;
W(1) = log Yede(1).
Each of these welfare indices is an increasing function of a `generalized mean of order (1 - e)', Yede(e). All the welfare indices are additively decomposable:
W(e) = SUM v_k W_k(e).
The Gini coefficient is given by
G = 1 + (1 / N) - [2/(m . N^2)] [SUM (N - i + 1) y_i]
where persons are ranked in ascending order of y_i. The Gini coefficient (and the percentile ratios) cannot be written as the sum of a term summarizing within-group inequality and a term summarizing between-group inequality.
Sen's (1976) welfare index is given by
S = m(1 - G).
Options
bygroup(groupvar) requests inequality decompositions by population subgroup, with subgroup membership summarized by groupvar.
welfare requests calculation of equally-distributed-equivalent incomes and welfare indices in addition to the inequality index calculations.
summarize requests presentation of summary, detail output for varname.
Saved results
r(p5), r(p10), r(p25) Percentiles p5, p10, p25, r(p50), r(p75), r(p90) p50, p75, p90, r(p95) p95
r(p90p10), r(p75p25) Percentile ratios p90/p10, p75/p25, r(p25p50), r(p10p50) p25/p50, p10/p50, r(p90p50), r(p75p50) p90/p50, p75/p50
r(gem1), r(ge0), GE(a), for a = -1, 0, 1, 2 r(ge1), r(ge2) r(gini) Gini coefficient
r(ahalf), r(a1), r(a2) A(e), for e = 0.5, 1, 2
r(mean), r(sd), r(Var) mean, standard deviation, variance r(min), r(max) minimum, maximum r(N), r(sumw) Number of observations, sum of weights
If the welfare option is specified, also saved are:
r(edehalf), r(ede1) Yede(e) for e = 0.5, 1, 2 r(ede2)
r(whalf), r(w1) W(e) for e = 0.5, 1, 2, and r(w2), r(wgini) Sen's welfare measure
If the bygroup option is specified, also saved are:
r(within_gem1) GE_W(a), for a = -1, 0, 1, 2 r(within_ge0) r(within_ge1) r(within_ge2) r(between_gem1) GE_B(a), for a = -1, 0, 1, 2 r(between_ge0) r(between_ge1) r(between_ge2)
r(within_ahalf) A_W(a), for e = 0.5, 1, 2 r(within_a1) r(within_a2) r(between_ahalf) A_B(a), for e = 0.5, 1, 2 r(between_a1) r(between_a2)
r(gem1_k), r(ge0_k) GE_k(a), for a = -1, 0, 1, 2, and r(ge1_k), r(ge2_k) each subgroup k, where the values of k correspond to the values of groupvar in the estimation sample. See r(levels) below.
r(ahalf_k), r(a1_k) A_k(a), for a = 0.5, 1, 2, and r(a2_k) each subgroup k
r(gini_k) Gini for each subgroup k
r(mean_k), r(lambda_k) subgroup mean (m_k), and relative mean (m_k/m) r(lgmean_k) subgroup log mean, log(m_k) r(theta_k) subgroup income share, s_k r(sumw_k) subgroup sum of weights r(v_k) subgroup population share, v_k
r(levels) macro containing the set of values of groupvar (the number of unique values = K)
If the welfare option is specified, also saved are:
r(whalf_k), r(w1_k) W_k(a), for a = 0.5, 1, 2, and r(w2_k) each subgroup k
r(edehalf_k), r(ede1_k) Yede_k(a), for a = 0.5, 1, 2, and r(ede2_k), r(wgini_k) Sen's welfare measure, for each subgroup k
For the convenience of users of earlier versions of these programs, a selected set of estimates is also saved in global macros, as follows.
S_9010, S_7525 Percentile ratios p90/p10, p75/p25
S_im1, S_i0, S_i1, S_i2 GE(a), for a = -1, 0, 1, 2
S_gini Gini coefficient
S_ahalf, S_a1, S_a2 A(e), for e = 0.5, 1, 2
Examples
. ineqdeco x [aw = wgtvar]
. ineqdec0 x [aw = wgtvar]
. ineqdeco x, by(famtype) w
. ineqdeco x if sex==1, w s
. // bootstrapped standard errors for Gini in Stata version 8
. preserve
. keep if x > 0 & x < .
. version 8: bootstrap "ineqdeco x" gini = r(gini), reps(100)
. restore
. // bootstrapped standard errors for Gini in Stata version 9
. preserve
. keep if x > 0 & x < .
. bootstrap gini = r(gini), reps(100): ineqdeco x
. restore
. // multi-way decomposition
. egen sexXregion = group(sex region)
. ineqdeco x, by(sexXregion)
Further examples are provided in the downloadable materials accompanying the presentation by Jenkins (2006).
Author
Stephen P. Jenkins <stephenj@essex.ac.uk> Institute for Social and Economic Research University of Essex, Colchester CO4 3SQ, U.K.
Acknowledgements
For comments and suggestions, I am grateful to Philippe Van Kerm and Nick Cox. Thanks also to Johannes Giesecke and Austin Nichols who drew attention to a bug in the subgroup decomposition calculations that arose when if/in qualifiers led to no observations in one of more of the by groups. Austin provided code to fix the bug.
References
Atkinson, A.B. 1970. On the measurement of inequality. Journal of Economic Theory 2: 244-63.
Blackorby, C., Donaldson, D., and Auersperg, M. 1981. A new procedure for the measurement of inequality within and between population subgroups. Canadian Journal of Economics 14: 665-85.
Cowell, F.A. 1995. Measuring Inequality. Hemel Hempstead: Prentice-Hall/Harvester-Wheatsheaf.
Cowell, F.A. 2000. Measurement of inequality. In Handbook of Income Distribution Volume 1, eds A.B. Atkinson and F. Bourguignon. Amsterdam: Elsevier Science, 59-85.
Cowell, F.A. and Jenkins, S.P. 1995. How much inequality can we explain? A methodology and an application to the USA. Economic Journal 105: 421-430.
Jenkins. S.P. 1991. The measurement of income inequality. In L. Osberg (ed.) Economic Inequality and Poverty: International Perspectives. Armonk, NY: M.E. Sharpe.
Jenkins, S.P. 1995. Accounting for inequality trends: decomposition analyses for the UK, 1971-86. Economica 62: 29-63.
Jenkins, S.P. 1997. Trends in real income in Britain: a microeconomic analysis. Empirical Economics 22: 483-500.
Jenkins, S.P. 2006. Estimation and interpretation of measures of inequality, poverty, and social welfare using Stata. Presentation at North American Stata Users' Group Meetings 2006, Boston MA. http://econpapers.repec.org/paper/bocasug06/16.htm.
Mookherjee, D. and Shorrocks, A. 1982. A decomposition analysis of the trend in UK inequality. Economic Journal 92: 886-992.
Sen, A.K. 1976. Real national income. Review of Economic Studies 43: 19-39.
Shorrocks, A.F. 1984. Inequality decomposition by population subgroups. Econometrica 52: 1369-88.
Also see
inequal7 if installed; sumdist if installed; svylorenz if installed; svygei if installed; svyatk if installed; povdeco if installed.