Generalized Entropy inequality indices, with sampling variances ---------------------------------------------------------------
^geivars^ varname [ [^fw^eights] ^if^ exp ^in^ range] Examples -------- . ^geivars hhincome [w=hhsize]^ . ^geivars hhincome, if sex==1^
^geivars^ estimates several inequality indices commonly used by economists, together with their asymptotic sampling variances. Unit record (`micro' level) data are required.
The inequality indices estimated are members of the single parameter Generalized Entropy class GE(a) for a = -1, 0, 1, 2. The indices differ in their sensitivities to differences in different parts of the distribution. The more positive a is, the more sensitive GE(a) is to income differences at the top of the distribution; the more negative a is the more sensitive it is to differences at the bottom of the distribution. GE(0) is the mean logarithmic deviation, GE(1) is the Theil index, and GE(2) is half the square of the coefficient of variation.
The formulae for the sampling variances are taken directly from Cowell (1989). His formulae were derived assuming that the income receiving units ('households') are treated as a random sample from a bivariate distribution of income and a household weight variable (e.g. household size). We require estimates of income inequality amongst all persons in the household population. I.e. in effect there is a random sample of households with 'self weighting' by household size, where the weights are similar to Stata's fweights. Thus the variance formulae do not also adjust for the effects of complex survey design features (stratification and clustering) -- formulae for this case are rather complicated and the subject of current research. These problems do not arise, of course, if the data are unweighted.
Derivation of the formulae for the asymptotic variances use the result that the GE(a) indices can be written as functions of sample moments. For further details, see Cowell (1989).
^geivars^ output includes the estimates of the four indices, and three sets of variance estimates for each corresponding to different informational assumptions. V0 is the variance in the case where both mean 'income' and 'household size' are known. V1 (= V0 + delta1) is the variance in the case where the former is not known, and V2 (= V1 + delta2) is the variance in the case where both are unknown and estimated from the sample. In each case the asymptotic 't' ratio = GE(a)/sqrt[V(a)] and associated p-value are also reported.
Author ------ Stephen P. Jenkins <stephenj@@essex.ac.uk> Institute for Social and Economic Research University of Essex, Colchester CO4 3SQ, U.K.
NB minor fix in February 2001 so that compatible with Stata 7. [Still runs with Stata 5 and Stata 6.]
References ---------- Cowell, F.A. (1989), "Sampling variance and decomposable inequality measures", Journal of Econometrics, 42, 27-41. Cowell, F.A. (1995), Measuring Inequality, second edition, Prentice-Hall/Harvester-Wheatsheaf, Hemel Hempstead.
Also see -------- ^inequal^ (sg30: STB-23) if installed; ^rspread^ (sg31: STB23) if installed; ^ineqdeco^ if installed.