help bineq

Title

bineq -- Bidimensional inequality indices and associated hypothesis tests

Syntax

bineq
varname1 varname2
[if] [in]
,
[alpha(#)
beta(#)
type(index_type)
level(#)]

bineqdiff
estname1 estname2
,
[restrict(matname)
null(matname)]

Description

bineq is a module to analyze multidimensional inequality in the joint distribution of two different dimensions of wellbeing. Abul Naga, Shen and Yoo (2016) summarizes
the underlying methods. To facilitate discussion, suppose that the two dimensions of interest are income and health as in the study of Zhong (2009).
bineq can be used to:

(1) estimate Tsui's (1995) bidimensional Atkinson-Kolm-Sen (AKS) or Mean Log Deviation (MLD) inequality index. Each index is a scalar that measures inequality
in the joint distribution of income and health.

(2) estimate Abul Naga and Geoffard's (2006) decomposition of the bidimensional AKS (MLD) index into two unidimensional AKS (MLD) indices, one measuring inequality in income
and the other measuring inequality in health. In case of the bidimensional AKS index, this decomposition also produces a measure of association between income and health;
Zhong (2009) argues that this association measure can be interpreted as an index of income-related inequality in health.

{3} use the method of Abul Naga, Shen and Yoo (2016) to obtain an asymptotic covariance matrix, hence standard errors, to carry out hypothesis tests on (1) and (2) using
Wald-type statistics.

The covariance matrix and standard errors of the disaggregated indices, (2), are computed by applying the analytic formula of Abul Naga, Shen and Yoo (2016, p.140). The standard error of
the overall index, (1), is then computed by using this covariance matrix and nlcom.

bineqdiff is a postestimation command that tests for systematic difference between two sets of bineq estimates. The test statistic is based on
equation (22) of Abul Naga, Shen and Yoo (2016, p.140) and can be used, for example, to test for the urban-rural difference in inequality structure.
estname1 (estname2) refers to the name under which the first (second) set of estimates was stored using estimates store.

Options for bineq

alpha(#) is the weight on varname1 in the social welfare function. See Section 1 of Abul Naga, Shen and Yoo (2016). A larger
absolute value of this weight reflects greater aversion to inequality in varname1. For the AKS index, changing this weight affects both overall (i.e. bidimensional) and
disaggregated (i.e. unidimensional) indices. For the MLD index, changing this weight only affects the overall index. The default for the AKS index is alpha(-1);
the default for the MLD index is alpha(1).

beta(#) is the weight on varname2 in the social welfare function. See Section 1 of Abul Naga, Shen and Yoo (2016). A larger
absolute value of this weight reflects greater aversion to inequality in varname2. For the AKS index, changing this weight affects both overall (i.e. bidimensional) and
disaggregated (i.e. unidimensional) indices. For the MLD index, changing this weight only affects the overall index. The default for the AKS index is beta(-1);
the default for the MLD index is beta(1).

type(index_type) is either aks (to estimate the AKS index) or mld (to estimate the MLD index). The default is type(aks).

level(#) sets confidence level. The default is level(95).

Options for bineqdiff

restrict(matname) supplies the name of the matrix of linear restrictions, or "A" in equation (22) of Abul Naga, Shen and Yoo (2016). The default for the AKS index is
a 3-by-3 identity matrix; the default for the MLD index is a 2-by-2 identity matrix.

For the AKS index, one may set A = [1, 0, 0 \ 0, 1, 0 \ 0, 0, 1] to test for between-group differences in all three disaggregated indices;
A = [1, 0, 0 \ 0, 1, 0] to test for between-group differences in unidimensial AKS indices; A = [1, 0, 0 \ 0, 0, 1] to test for between-group differences
in the unidimensional AKS index of varname1 and the association measure; A = [0, 1, 0 \ 0, 0, 1] to test for between-group differences in the
unidimensional AKS index of varname2 and the association measure; A = [1, 0, 0] to test for between-group difference in the unidimensional AKS index of varname1;
A = [0, 1, 0] to test for between-group difference in the unidimensional AKS index of varname2; and A = [0, 0, 1] to test for between-group difference in the association measure.

For the MLD index, one may set A = [1, 0 \ 0, 1] to test for between-group differences in all two disaggregated indices; A = [1, 0] to test for
between-group difference in the MLD index of varname1; and A = [0, 1] to test for between-group difference in the MLD index of varname2.

null(matname) supplies the name of the row vector of the null hypotheses, or the transpose of "eta" in equation (22) of Abul Naga, Shen and Yoo (2016).
The default for the AKS index is a 1-by-3 vector of zeroes; the default for the MLD index is a 1-by-2 vector of zeroes. In both cases, therefore, the default null
states that there is no between-group difference.

When specifying a non-zero null hypothesis concerning unidimensional AKS or MLD indices, note that bineqdiff expects the user to specify the null
in terms of between-group differences in unidimensional "equality" indices, even though bineq reports unidimensional "inequality" indices: see Section 1 of
Abul Naga, Shen and Yoo (2016) for the algebraic relationship between equality and inequality indices.

Examples

Suppose that each row in the data set is an observation on a particular person. Assume further that there are three variables in the data set:
income that records each person's income, health that records each peson's health, and urban that equals 1 for people who live in urban areas and
0 for people who live in rural areas.

The AKS and MLD analysis of inequality in income and health can be respectively executed as follows.

. bineq income health

. bineq income health, type(mld)

The rest of this document illustrates the use of bineqdiff after estimating the AKS indices. Note, however, that bineqdiff can also be used
after estimating the MLD indices. Suppose now that the objective of the analysis is to test for the urban-rural difference in inequality structure.
This can be achieved by using bineqdiff, once area-specific inequality indices have been obtained as follows.

. bineq income health if urban == 1

. est store urban

. bineq income health if urban == 0

. est store rural

To test the null that there is no urban-rural difference in all disaggregated indices, bineqdiff can be executed under its default settings.

. bineqdiff urban rural

To test the null that there is no urban-rural difference in the unidimensional index of health and the association measure (i.e. Zhong's (2009)
measure of income-related inequality in health), a linear restriction matrix A should be specified and supplied as follows.

. matrix A = [0, 1, 0 \ 0, 0, 1]

. bineqdiff urban rural, restrict(A)

Using bootstrap and/or bsample in conjunction with bineq and bineqdiff may improve the joint hypothesis test's size accuracy in finite samples.
The Monte Carlo experiment of Abul Naga, Shen and Yoo (2016) reports substantial size accuracy gains from using percentile-t bootstrapping that
compares the asymptotic test statistics produced by bineqdiff against bootstrapped critical values.

Stored results

bineq stores the following in e():

Scalars

e(N) number of observations

e(alpha) # in alpha(#)

e(beta) # in beta(#)

Macros

e(cmd) bineq

e(alphavar) varname1

e(betavar) varname2

e(type) index_type in type(index_type)

Matrices

e(b) row vector that reports disaggregated indices

e(V) asympotic covariance matrix for disaggregated indices

Functions

e(sample) marks the estimation sample

bineqdiff stores the following in e():

Scalars

r(chi2) chi-squared test statistic

r(df) degrees of freedom for the test statistic

r(p) p-value for the test statistic

Matrices

r(restrict) contents of matrix matname in restrict(matname)

r(null) contents of matrix matname in null(matname)

References

Abul Naga, R. H., Geoffard, P.-Y. 2006. Decomposition of bivariate inequality indces by attributes. Economics Letters 90: 362-367.

Abul Naga, R. H., Shen, Y. and Yoo, H. I. 2016. Joint hypothesis tests for multidimensional inequality indices. Economics Letters 141: 138-142.

Tsui, K. Y. 1995. Multidimensional generalizations of the relative and absolute inequality indices: the Atkinson-Kolm-Sen Approach. Journal of Economic Theory 67: 251-265.

Zhong, H. 2009. A multivariate analysis of the distribution of individual's welfare in China: What is the role of health? Journal of Health Economics 28: 1062-1070.

Author

Dr. Hong Il Yoo

Durham University Business School

Durham University

Durham, UK

h.i.yoo@durham.ac.uk

Also see

Help: bootstrap, bsample