help for ineqdeco, ineqdec0Stephen P. Jenkins (revised February 2021)

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, aweights, pweights, and iweights
are allowed; see 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. (If you have grouped (banded) data, you should not use this program
unless you know what you're doing. On estimation of inequality indices from grouped
data, see e.g. Cowell and Mehta, 1983.) 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. Bootstrapped standard errors for weighted
estimates can be derived using svy bootstrap and Philippe Van Kerm's
rhsbsample on SSC. 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.
(ssc install svygei_svyatk), (ssc install 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 <s.jenkins@lse.ac.uk>

London School of Economics

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.

Cowell, F.A. and Mehta, F. 1983.
The estimation and interpolation of inequality measures.
Review of Economic Studies 49: 273-290.

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.