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Stata Technical Bulletin

ineqfac: inequality decomposition by factor components

ineqfac provides an exact decomposition of the inequality of total income into inequality contributions from each of the
factor components of total income. More specifically, given

facvars = {factor_1 factor_2 ... factor~F}

define the variable totvar such that for each observation in the dataset,

Shorrocks (1982a) proved that there was a unique ‘decomposition rule’ for which inequality in totvar across observations
could be expressed as the sum of inequality contributions from each of the factor components, and which also satisfied some
other basic axioms.

The decomposition rule is the “proportionate contribution of factor f to total inequality”, sʃ:

Sf = pʃɑ(faetor,/)/ɑ(totvar)

where pf is the correlation between factor./ and totvar, and σ(.) is the standard deviation. Equivalently, Sf is the slope
coefficient from the regression of factor./ on totvar. Observe that for each observation,

Σ^sf
f=l

Factor components with a positive value for Sf make a disequalizing contribution to inequality in total income; factor
components with negative Sf values make an equalizing contribution.

Shorrocks (1982a) shows that choice of the decomposition rule is an issue independent of that concerning which index is
used to summarize inequality. However there happens to be a nice link with the case in which inequality is measured using the
coefficient of variation, for one can also rewrite S f as

^sf ⁼ P∕[m(factor.∕)∕m(totvar)][CV(factor.∕)CV(totvar)]

or

2f ⁼ P∕[m(factor.∕)∕m(totvar)][I'2(factor.∕)∕I'2(totvar)]'⁵

where m is the mean, and CV is the coefficient of variation, and 22 is half the squared coefficient of variation, or equivalently,
GE (2) as defined earlier.

Thus total inequality can be written in terms of the factor correlations with total income, the factor shares in total income
(= m(factor.∕)∕m(totvar)), and the factor inequalities (summarized using either CV or /2).

ineqfac reports the estimates for each factor component of: Sf, Sf = s∕.GV(totvar), m(factor.∕)∕m(totvar),
GV(factor./), and GV(factor.∕)∕GV(totvar), plus, optionally, the correlations, means and standard deviations of the factor
components and totvar. Optionally, inequality is summarized using T2 rather than CV.

ineqfac was designed as a tool for income distribution analysis in the case where the current sample contains observations
on income components for each of a set of income receiving units (e.g., families, households, persons). In this case, facvars

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