Cowell, Frank

Fiorio, Carlo V.

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   Rethinking Inequality Decomposition: Comment  Frank A. Cowell∗ and Carlo V. Fiorio†   *London School of Economics  †University of Milan and Econpubblica          DARP 82      The Toyota Centre February 2006      Suntory and Toyota International         Centres for Economics and Related        Disciplines        London School of Economics        Houghton Street        London WC2A 2A              (+44 020) 7955 6674 Distributional Analysis Research Programme  The Distributional Analysis Research Programme was established in 1993 with funding from the Economic and Social Research Council. It is located within the Suntory and Toyota International Centres for Economics and Related Disciplines (STICERD) at the London School of Economics and Political Science. The programme is directed by Frank Cowell. The Discussion Paper series is available free of charge. To subscribe to the DARP paper series, or for further information on the work of the Programme, please contact our Research Secretary, Leila Alberici on:   Telephone:  UK+20 7955 6674 Fax:   UK+20 7955 6951 Email:  l.alberici@lse.ac.uk Web site:  http://sticerd.lse.ac.uk/DARP                          © Authors: Frank Cowell and Carlo Fiorio. All rights reserved. Short sections of text, not to exceed two paragraphs, may be quoted without explicit permission provided that full credit, including © notice, is given to the source.  Rethinking Inequality Decomposition:CommentFrank A. Cowell and Carlo V. Fiorioy6th December 2005Decomposition analysis of inequality is important for understanding themain determinants of inequality and for policy analysis. The traditionalapproach to the subject was based purely on the analysis of the mathematicalproperties of inequality indices and is open to the criticism that the formalrequirements for exact decomposition are perhaps too demanding for somepractical applications. Recent applied work has reawakened interest in in-equality decomposition by focusing on the use of regression-based approachesto avoid some of the restrictions of the traditional methods.Morduch and Sicular (2002) have suggested a specic regression-basedmethod for decomposing inequality. They claim that their method possessesthree main advantages: (a) it yields an exact allocation of contributions to theidentied variables, (b) it is general, in that it can be employed with a varietyof inequality indices and decomposition rules, (c) it is associated with a simpleprocedure for deriving standard errors and condence intervals for estimatedcomponents of inequality. In this comment we argue that these points are notLondon School of Economics and STICERD. Address: Houghton Street, London WC2A2AE, UK. email: f.cowell@lse.ac.ukyUniversity of Milan and Econpubblica. Address: DEAS, via Conservatorio, 7. 20133Milan, Italy. email: carlo.orio@unimi.it1convincing and, in some respects, wrong. We rst summarize the approachtaken by Morduch and Sicular (2002) to introduce the regression-based meth-odology for inequality decomposition; we then examine the three main claimsin detail.1 The Morduch and Sicular (2002) decompos-ition methodMorduch and Sicular (2002) focus on natural decomposition rules for in-equality decomposition, which pertain to inequality indices that can be writtenas a weighted sum of incomes (Shorrocks, 1982):I(y) =nXi=1ai(y)yi (1)where yi is the income of person i, n is the number of persons in the popula-tion, y := (y1; :::; yn) ; and ai(y) is a weighting factor. If each yi is the sumof component incomes yki coming from K di¤erent sources (such as pension,employment income, transfers and so on)yi =KXk=1ykithen the proportional contribution of source k to overall inequality can bedened assk :=Pni=1 ai(y)ykiI(y)(2)In the regression-based inequality decomposition proposed by Morduch andSicular (2002), primary attention is given to the data generating process thatled to a particular distribution of income, as in Oaxaca (1973) and Blinder2(1973). Assuming that the income generating process isy = X +  (3)where y is an n 1 vector, X is an nM matrix of individual and householdcharacteristics (age, education, household size, residence, etc.),  is a M  1vector of coe¢ cients and  is an n  1 vector of residuals, a sample of obser-vations {yi, xi, i = 1; 2; :::n} can be used to estimate the model. The vector can be interpreted as the e¤ect (or price) of the independent variables onincome. Using (3), per capita income of household i is then represented as:yi =MXm=1bmxmi +bi (4)where bm is the OLS coe¢ cient estimate and bi is the OLS residual forhousehold i. By analogy with (2), shares attributable to the characteristicm = 1; :::;M take the form:sm := bmPni=1 ai(y)xmiI(y)(5)This decomposition might be applied to any inequality index that can be writ-ten as a weighted sum of incomes.Let us now discuss the three main claims for this model.2 ExactnessMorduch and Sicular (2002) state that [this method] yields an exact alloca-tion of contributions to the identied variables.In the traditional inequalitydecomposition literature a decomposition method is exact if it can express in-3equality as an additive function of incomes or income shares without residual.For instance, while all members of the class of Generalized Entropy indices areexactly decomposable, the Gini index is not: for arbitrary partitions of thepopulation there will be a residual term that is di¢ cult to interpret unam-biguously (among many others, see Cowell, 1980; Shorrocks, 1980; Cowell andKuga, 1981; Pyatt, 1976; Lambert and Aronson, 1993).The method is an exact decomposition only of by, not of y, which is ofprimary interest. In fact, as y = bx+ b, the b residual is always present, andthe decomposition is clearly not exact. Moreover, as the application to Chinesedata shows, in some cases the residual term accounts to a very large share ofthe inequality index: in two cases out of four (the Theil-T and the AlternativeCV) the decomposition leaves about 90% of total inequality unexplained!13 GeneralityMorduch and Sicular (2002) consider that the method is general in that itcan be employed with di¤erent inequality indices and decomposition rules.Although it is true that one can run a regression with whatever random vari-able available, it is not true that any regression is a good model. In this casethe dependent variable is a complex composite (per capita household income,where income comprises both labor and non-labor income) and it is estimatedusing a linear relationship of household characteristics. It is hardly surpris-ing that the R2 of this regression is lower than 45% and that all inequalitydecompositions present very large residuals.This method also reveals substantial variability across indices. For in-stance, while the net e¤ect2 of an additional year of education is to reduce per1See Residual regressionline in Table 2, p.103.2The net e¤ect of education is given by the sum of the e¤ects of the variables average4capita household income inequality by about 50% using the Theil-T index, itincreases it using the Gini index. Morduch and Sicular (2002) motivate thedi¤erence in outcome between the two indices pointing out the fact that thedecomposition of the Gini index does not fulll the Corollary of the UniformAddition Property (CUAP), which states that if a component of total income(say, income type k) is positive and equally-distributed its contribution tototal income inequality, sk, must be negative. However, it is not clear whethersatisfaction of CUAP is a merit or not.If it is a merit, and consequently the Theil-T should be preferred to the Giniindex in this kind of empirical analysis, then it is di¢ cult to be satised witha decomposition method, which leaves over 90% of inequality unexplained.Moreover, this property is not satised by the decomposition rule proposedby Shorrocks (1982), which is the unique one to be invariant to the choice ofinequality measure. In fact, Shorrocks (1982) showed that the unique decom-position rule, invariant to the inequality index used, is:sk :=cov(yk;y)var(y)(6)where yk = fyki ; i = 1; :::ng. Factor contributions can be either positive ornegative, depending on the factor providing a disequalizing or equalizing con-tribution. Using Shorrockss rule (6) the contribution sk is negative only ifthere is a negative correlation between total income and income componentk, as it is often the case for taxes or transfers. The suggested decompositionrule is then just one more in a large class of possible decomposition rules.However, unless valid reasons are provided for constraining the set of poten-tial decomposition rules, preferably to the point where inequality index caneducation of adults and education squared. The same approach is taken also by Fields (2003).5be decomposed in only one way, the inequality contribution assigned to anyincome source can vary arbitrarily, depending on the choice of decompositionrule. This turns the calculation of inequality contributions into a meaninglessexercise...(Shorrocks, 1983, p. 315).Large residuals and highly variable results are most likely due to the inabil-ity of a single equation model to explain the complexity of per capita householdincome. A multiple equation income-generating model such as that of Bour-guignon et al. (2001) or semi-parametric methods, such as that proposed byDiNardo et al. (1996), would be more successful although clearly more complexto implement.4 EstimationThirdly Morduch and Sicular (2002) consider that the method is associatedwith a simple procedure for deriving standard errors and condence intervalsfor the estimated components of inequality. Since the decomposition (5) islinear in the parameters, they claim that the standard error of sk, (sk), iseasily estimated using the standard error of bk, (bk), which comes as anoutput in regression analysis:(sm) = (bm) P ai(y)xmiI(y)(7)However, equation (7) would be correct if and only if (Pai(y)xmi =I(y)) wasnot stochastic, which is clearly untrue (recall (3)). Hence, the correct standarderror is:(sm) = bmPni=1 ai(y)xmiI(y)(8)6which might be quite complicated to compute, as it requires the use of thebootstrap or non-trivial asymptotic distribution (see for example Cowell, 1989;Cowell and Flachaire, 2002).ReferencesBlinder, A. S. (1973). Wage discrimination: Reduced form and structuralestimates. Journal of Human Resources, 8(4):436455.Bourguignon, F., Fournier, M., and Gurgand, M. (2001). Fast developmentwith a stable income distribution: Taiwan, 1979-94. Review of Income andWealth, 47(2):139163.Cowell, F. and Kuga, K. (1981). Additivity and the Entropy Concept: An Ax-iomatic Approach to Inequality Measurement. Journal of Economic Theory,25:131143.Cowell, F. A. (1980). On the structure of additive inequality measures. Reviewof Economic Studies, 47:52131.Cowell, F. A. (1989). Sampling variance and decomposable inequality meas-ures. Journal of Econometrics, 42(1):2741.Cowell, F. A. and Flachaire, E. (2002). Sensitivity of inequality measures toextreme values. Distributional Analysis Discussion Paper, 60, STICERD,LSE.DiNardo, J., Fortin, N. M., and Lemieux, T. (1996). Labor Market Insititutionsand the Distribution of Wages, 1973-1992: a Semiparametric Approach.Econometrica, 64:10011044.7Fields, G. S. (2003). 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