st: shift-share analysis (counterfactual) of inequality changes by subgroups using ineqdeco

[Nicola Pensiero <nicola.pensiero@gmail.com>]

Dear STATA users,
my question is related to the shift-share analysis (counterfactual) of
inequality changes by subgroups using the module ineqdeco. The problem
is that the estimated counterfactual Theil (GE(1)) value is negative,
which probably means that the implementation of the formula is wrong
in some ways.

I have households groups with a given income measured in two years
(1977 and 2010).

Given the quantities household type mean and population share of group
x, I ask what  would have happened to inequality (within and between
groups) in year 2010 if the subgroup population share, v_k was the one
of year 1977.


My implementation of the counterfactual exercise is as follows.


I saved the subgroup population share v_x from year 1977

ineqdeco income1977, by (group)
foreach x of num 1/30 {
local v_`x' = r(v_`x')
}

Then I saved the subgroup income share s_x and GE_k for year 2010.

ineqdeco income2010, by (group)

foreach x of num 1/30 {
local s_`x' = r(theta_`x')
}

foreach x of num 1/30 {
local GE_alpha1_`x' = r(ge1_`x')
}


Finally, I used the subgroup population share from year 1977 and the
subgroup income share and GE_k from year 2010 in the calculation of
Theil index GE(1).

foreach num of numlist 1/30 {
local my_GE`num' = `v_`num''^0 * `s_`num''^1 * `GE_alpha1_`num'' +
`v_`num''^0 * `s_`num''^1* log(`s_`num''^1)
}

local my_GEglobal = `my_GE1' +........ `my_GE30'

Thank you, Nicola Pensiero

PS: the index should be decomposed in its within and between
components as follows:

GE = SUM [v_k^(1-a)] . [s_k^a] . GE_k(a) + GE_Between(a)
where GE_B(a) is derived assuming every person within a given subgroup
k received k's mean income, m_k.

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