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help for ineqrbd         Carlo V. Fiorio and Stephen P. Jenkins (December 2008)
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Regression-based inequality decomposition, following Fields (2003)

ineqrbd depvar rhsvars [weights] [ if exp] [in range ] [, i2 stats
noconstant noregression fields ]

fweights and aweights are allowed; see help weights.

Description

ineqbrd performs regression-based decomposition of the inequality in
depvar into the contributions accounted for by each of the rhsvars.  The
formulae used are those proposed by Fields (2003) which, in turn, are
closely related to those proposed by Shorrocks (1982) for
non-regression-based decomposition of inequality by income source.

Consider the linear model for each observation

(1) yvar = b0 + b1*X1 + b2*X2 + ... + bK*XK + residual

where the Xf, for f = 1,...,K are the variables included in rhsvars, and
the bf are the corresponding regression coefficients that are estimated
by OLS.

The linear model can be rewritten as

(2) yvar = b0 + Z1 + Z2 + ... + ZK + residual

where each Zf, for each f = 1,...,K is a `composite' variable, equal to
the product of an estimated regression coefficient and a variable. For
inequality decomposition calculations, the value of b0 is irrelevant as
it is constant for every observation.

Alternatively, one may look at predicted yvar (`yhat')

(3) yhat = b0 + Z1 + Z2 + ... + ZK

in which case there is no residual term.

Neglecting the constant, equations (2) and (3) are of exactly the same
form as the equation used by Shorrocks (1982) when deriving rules for
inequality decomposition by factor components.  (E.g. total income is the
sum of labour earnings, income from savings and other assets, private and
public transfers. How much inequality in total income is attributable to
each of these factors?) Shorrocks proved that a set of arguably
persuasive axioms led to a unique additive and exact decomposition rule,
with one term for each factor. The decomposition rule did not depend on
the choice of measure summarizing inequality in total income.

Fields (2003) exploited the parallel with the factor decomposition case,
and applied the Shorrocks decomposition rule to relate inequality in
predicted yvar to contributions from each of the composite variables.
Alternatively, one may apply the decomposition rule to the inequality of
yvar itself, in which case there is also a decomposition term
corresponding to the residual.  See Cowell and Fiorio, 2006.

ineqrbd provides a regression-based Shorrocks-type decomposition of a
variable labelled "Total", where Total is defined as yvar, unless the
fields option is used in which case Total refers to predicted yvar. In
either case, the contribution to inequality in Total of each term is
labelled "s_f" in the output.

Options

fields implies decomposition of predicted yvar rather than of yvar.

noregression suppresses reporting of the OLS regression equation used to
derive the composite variables and residual.

noconstant excludes the intercept term from the regression.

stats provides the means, standard deviations, and correlations, of
Total, the residual (unless the fields option is used), and the
composite variables Zf, f = 1,...,K. Results for the composite
variables are ordered in the same order as the underlying variables
are ordered in rhsvars.

i2 summarises inequality using half the squared coefficient of variation
(the Generalized Entropy measure I2), rather than the coefficient of
variation (CV).  Observe that that inequality may be negative, e.g.
because the mean of a composite variable may be negative.

Saved results

r(total)                            contains predicted yvar if fields optio
> n
used; else contains  yvar

r(mean_tot), r(sd_tot), r(cv_tot)   mean, standard deviation, CV for Total

r(sf_Z0), r(mean_Z0),               proportionate inequality contribution,
> mean,
r(sd_Z0), r(cv_Z0)                  standard deviation, CV for the residual
> .
r(sf_Z0) is not reported if fields opti
> on used.

r(sf_Zf), r(mean_Zf),               proportionate inequality contribution,
> mean,
r(sd_Zf), r(cv_Zf)                  standard deviation, CV for each of vari
> ables
in rhsvars, where "f" is an integer
1,..., K, indicating the order in which
>  entered in
rhsvars.
r(varlist)                          contains yvar rhsvars
r(yvar)                             contains yvar
r(xvars)                            contains rhsvars

Example

. sysuse auto

. ineqrbd price trunk weight length foreign

Authors

Carlo Fiorio <carlo.fiorio@unimi.it>
Department of Economics, Business and Statistics
University of Milan, Via Conservatorio, 7
20122 Milan, Italy

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

Acknowledgement

ineqrbd uses code from ineqfac by S.P. Jenkins.  Thanks to Andreas Peichl
and Nico Pestel who discovered bugs in earlier versions and provided
code for squashing them.

References

Cowell, F.A. and Fiorio, C.V. 2006. Rethinking Inequality Decomposition:
Comment.  Distributional Analysis Research Programme Working Paper
82.  London: STICERD, London School of Economics.
http://sticerd.lse.ac.uk/dps/darp/DARP82.pdf

Fields, G. 2003. Accounting for Income Inequality and Its Change:  A New
Method, with Application to the Distribution of Earnings in the
United States', Research in Labor Economics.

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

Also see

ineqfac if installed

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