------------------------------------------------------------------------------- help forineqrbdCarlo V. Fiorio and Stephen P. Jenkins (December 2008) -------------------------------------------------------------------------------

Regression-based inequality decomposition, following Fields (2003)

ineqrbddepvarrhsvars[weights] [ifexp] [inrange] [,i2statsnoconstantnoregressionfields]

fweights andaweightsare allowed; see help weights.

Description

ineqbrdperforms regression-based decomposition of the inequality indepvarinto the contributions accounted for by each of therhsvars. 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 + residualwhere 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 + residualwhere 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 + ... + ZKin 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

notdepend 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 yvarto contributions from each of the composite variables. Alternatively, one may apply the decomposition rule to the inequality ofyvaritself, in which case there is also a decomposition term corresponding to the residual. See Cowell and Fiorio, 2006.

ineqrbdprovides a regression-based Shorrocks-type decomposition of a variable labelled "Total", where Total is defined asyvar, unless thefieldsoption is used in which case Total refers topredicted yvar. In either case, the contribution to inequality in Total of each term is labelled "s_f" in the output.

Options

fieldsimplies decomposition ofpredicted yvarrather than ofyvar.

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

noconstantexcludes the intercept term from the regression.

statsprovides the means, standard deviations, and correlations, of Total, the residual (unless thefieldsoption 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 inrhsvars.

i2summarises 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 resultsr(total) contains

predicted yvariffieldsoptio > n used; else containsyvarr(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

fieldsopti > 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 inrhsvars. r(varlist) containsyvarrhsvarsr(yvar) containsyvarr(xvars) containsrhsvars

Example

. sysuse auto

. ineqrbd price trunk weight length foreign

AuthorsCarlo 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

ineqrbduses code fromineqfacby S.P. Jenkins. Thanks to Andreas Peichl and Nico Pestel who discovered bugs in earlier versions and provided code for squashing them.

ReferencesCowell, 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.

Econometrica50: 193-212.

Also seeineqfac if installed