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

 From Nicola Pensiero To statalist@hsphsun2.harvard.edu Subject Re: st: shift-share analysis (counterfactual) of inequality changes by subgroups using ineqdeco Date Wed, 13 Jun 2012 23:04:14 +0200

```Thank you Austin Nichols,
what I mean is to write the GE(1) index (theil index) distinguishing
its within and between parts
The Theil index GE(1) can be additively decomposed as
GE(1)=GE_W(a) + GE_B(a)
GE(1) = SUM [v_k^(1-a)] . [s_k^a] . GE_k(a)+ GE_B(a)

My implementation of the within part is:

=`v_`num''^0 * `s_`num''^1 * `GE_alpha1_`num''

where a=1, because Theil index assumes that a=1, which implies taking
v locals to the zero power.

My implementation of the between part is
`v_`num''^0 * `s_`num''^1* log(`s_`num''^1)

However, there is something odd about my implementation of the
decomposition which results in inadmissible outcome values.

Nicola Pensiero

> Can you clarify what you mean by this?
>
> 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)
> }
>
> Are you really taking v locals to the zero power?
> This reduces to s_i*G_i + s_i*ln(s_i).
>
> On Wed, Jun 13, 2012 at 4:51 AM, Nicola Pensiero
> <nicola.pensiero@gmail.com> wrote:
>> 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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```