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help for ^ineqfac^                             (STB-48: sg104)
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Inequality decomposition by factor components
---------------------------------------------

^ineqfac^ facvars [weight] [^if^ exp] [^in^ range]
			[, ^s^tats ^tot^al^(^totvar^)^ ^i2^ ]  

^fweight^s and ^aweight^s are allowed; see help @weights@.
			

Description
 -----------

^ineqfac^ provides an exact decomposition of the inequality of total
income into inequality contributions from each of the factor components
of total income. More specifically, given 
	facvars = {factor_1  factor_2  ...  factor_F}, 
define the variable totvar such that for each observation in the data set,
                              F
	            totvar = SUM (factor_f).
                             f=1
Shorrocks (1982a) proved that there was a unique 'decomposition rule' 
for which inequality in totvar across observations could be expressed 
as the sum of inequality contributions from each of the factor components, 
and which also satisfied some other basic axioms.  The decomposition rule 
is the "proportionate contribution of factor f to total inequality", s_f:

           s_f = rho_f*sd(factor_f)/sd(totvar)
                   
where rho_f is the correlation between factor_f and totvar, and sd(.) is the
standard deviation. (Equivalently, s_f is the slope coefficient from the
regression of factor_f on totvar.)   Observe that for each observation,
                          F
	                 SUM (s_f) = 1.
                         f=1

Factor components with a positive value for s_f make a disqualizing
contribution to inequality in total income; factor components with negative
s_f values make an equalizing contribution.

Shorrocks shows that choice of the decomposition rule is an issue independent 
of that concerning which index is used to summarise inequality.  However there 
happens to be a nice link with the case in which inequality is measured using 
the coefficient of variation, for one can also re-write s_f as:

            s_f = rho_f*[m(factor_f)/m(totvar)]*[CV(factor_f)/CV(totvar)]
		
		= rho_f*[m(factor_f)/m(totvar)]*[I2(factor_f)/I2(totvar)]^^(.5)

where m(.) is the mean, and CV(.) is the coefficient of variation = 
sqrt{[sd(.)/m(.)]^^2}, and I2(.) = half the squared coefficient of variation 
(a member of the Generalised Entropy class of inequality measures).

Thus total inequality can be written in terms of the factor correlations with
total income, the factor shares in total income = m(factor_f)/m(totvar), and 
the factor inequalities (summarized using CV or I2).

^ineqfac^ reports the estimates for each factor component of: 
s_f, S_f = s_f*CV(totvar), m(factor_f)/m(totvar), CV(factor_f), and
CV(factor_f)/CV(totvar), plus, optionally, the correlations, means 
and standard deviations of the factor components and totvar. Optionally
inequality is summarised using I2 rather than CV.

^ineqfac^ was designed as a tool for income distribution analysis in the case
where the current sample contains observations on income components for each 
of a set of income receiving units (e.g. families, households, persons).  
In this case, facvars might include labour income, income from investments 
and pensions, cash transfers, etc. See Shorrocks (1982b) and Jenkins (1995) 
for examples. ^ineqfac^ may also be applied to summarise and compare the 
riskiness of portfolios of wealth holdings: s_f has exactly the same form as 
the "beta coefficient" used in finance analysis.


Options
-------

^stats^ provides the means, standard deviations, and correlations of
        the factor components and totvar.
^total(^totvar^)^ creates a new variable, totvar, equal to the sum of
                    the factor components for each observation.
^i2^   summarises inequality using I2 rather than CV.


Example
-------

   . ^ineqfac labour invest transfer, s^  
   . ^ineqfac labour invest transfer, s i2^  
   . ^ineqfac labour invest transfer, total(grossi)^  
   . ^ge negtax = -tax^
   . ^ineqfac labour invest transfer negtax, total(dispi)^  


Author
------

      Stephen P. Jenkins 
      Institute for Social and Economic Research
      University of Essex, Colchester CO4 3SQ, U.K.
      stephenj@@essex.ac.uk


References
----------

Jenkins, S.P. (1995), 'Accounting for Inequality Trends: Decomposition 
	Analyses for the U.K., 1971-1986', Economica, 62, 29-63.
Shorrocks, A.F. (1982a), 'Inequality Decomposition by Factor Components', 
	Econometrica, 50, 193-212.
Shorrocks, A.F. (1982b), 'The Impact of Income Components on the Distribution
	of Family Incomes', Quarterly Journal of Economics, 98, 311-326.


Also see
--------

    STB: STB-48 sg104