help for ineqrbdCarlo V. Fiorio and Stephen P. Jenkins (revised February 2021)

Regression-based inequality decomposition, following Fields (2003)

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

fweights, aweights, pweights, and iweights
are allowed; see 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 option
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 option used.

r(sf_Zf), r(mean_Zf), proportionate inequality contribution, mean,
r(sd_Zf), r(cv_Zf) standard deviation, CV for each of variables
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>

University of Milan

Stephen P. Jenkins <s.jenkins@lse.ac.uk>

London School of Economics

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