```-------------------------------------------------------------------------------
help for ineqfac                                Stephen P. Jenkins (March 2009)
-------------------------------------------------------------------------------

Inequality decomposition by factor components, following Shorrocks (1982a, b)

ineqfac facvars [weights] [ if exp] [in range ] [, i2 stats
total(newvarname) ]

fweights and aweights are allowed; see help weights.

Description

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 the set of factors

facvars = {factor_1  factor_2  ...  factor_F},

define the variable totvar such that for each observation in the data
set,

F
totvar = SUM (factor_f).
f=1

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", s_f:

s_f = rho_f*sd(factor_f)/sd(totvar)

where rho_f is the correlation between factor_f and totvar, and sd(.) is
the standard deviation. Equivalently, s_f is the slope coefficient from
the regression of factor_f on totvar.  Observe that for each observation,

F
SUM (s_f) = 1.
f=1

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

Shorrocks shows that choice of the decomposition rule is an issue
independent of that concerning which index is used to summarise
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 re-write s_f as:

s_f = rho_f*[m(factor_f)/m(totvar)]*[CV(factor_f)/CV(totvar)]

where m(.) is the mean, and CV(.) is the coefficient of variation =
sd(.)/m(.). If m(factor_f) > 0 for every f, then also

s_f = rho_f*[m(factor_f)/m(totvar)]*[I2(factor_f)/I2(totvar)]^(.5)

where I2(.) = half the squared coefficient of variation (a member of the
Generalised Entropy class of inequality measures).

Thus total inequality can be written in terms of the factor correlations
with total income, the factor shares in total income =
m(factor_f)/m(totvar), and the factor inequalities (summarized using CV
or I2).

ineqfac reports the estimates for each factor component of:  s_f, S_f =
s_f*CV(totvar), m(factor_f)/m(totvar), CV(factor_f), and
CV(factor_f)/CV(totvar), plus, optionally, the correlations, means and
standard deviations of the factor components and totvar. Optionally,
inequality may be summarised using I2 rather than CV as long as
m(factor_f) > 0, for every f.

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 might include labour income, income from
investments and pensions, cash transfers, etc. See Shorrocks (1982b) and
Jenkins (1995) for examples. ineqfac may also be used to summarise and
compare the riskiness of portfolios of wealth holdings: s_f has the same
form as the "beta coefficient" used in finance analysis.

Options

stats provides the means, standard deviations, and correlations of the
factor components and totvar.

total(newvarname) creates a new variable equal to the sum of the factor
components for each observation.

i2 summarises inequality using I2 rather than CV. The option is available
only if every factor mean is positive.

Saved results

r(mean_total), r(sd_total),                     mean, standard deviatio
> n, variance and CV for Total
r(var_total), r(cv_total)

r(sf_f), r(mean_f),                             proportionate inequalit
> y contribution, mean,
r(sd_f), r(var_f), r(cv_f)                      standard deviation, var
> iance, CV, and
r(share_f)                                      share of total (mean_f/
> mean_total), for each factor, f.
"f" is replaced by the
> factor's varname.

r(N)                                            Number of non-missing o
> bservations used

r(nfactor)                                      Number of factor variab
> les

r(varlist)                                      The names of the factor
>  variables

Example

. ineqfac labour invest transfer, stats

. ineqfac labour invest transfer, stats total(grossi)

. generate negtax = -tax

. ineqfac labour invest transfer negtax, stats total(disposablei)

Author

Stephen P. Jenkins <stephenj@essex.ac.uk>
Institute for Social and Economic Research
University of Essex, Colchester CO4 3SQ, U.K.

Acknowledgement

This is an update of ineqfac, first written for Stata version 5.  Andreas
Peichl beta-tested the revised code.

References

Jenkins, S.P. 1995. Accounting for Inequality Trends: Decomposition
Analyses for the U.K., 1971-1986.  Economica 62: 29-63.

Shorrocks, A.F. 1982a. Inequality Decomposition by Factor Components.
Econometrica 50: 193-212.

Shorrocks, A.F. 1982b. The Impact of Income Components on the
Distribution of Family Incomes.  Quarterly Journal of Economics 98:
311-326.

Also see

```