Reformulating a problem based on the symmetry of interactive effects?

[Jose Alcides Santos]

Dears,

I am conducting a study on social mobility in Brazil using a survey of 2014. The study uses for social origin a typology of social class based on employment. The social destination is analyzed in terms of income from all sources. An original question of the research is: is the effect of social origin lower among the more educated individuals? Would higher education equalize or neutralize the effect of social origin, whether these advantages or disadvantages of origin? To answer this question I have estimated interactive effects between the origin and education of the children from 27 to 66 years old.
I performed semi-elasticity estimates using class as a categorical variable and income as a continuous variable. However, I question whether make sense to rephrase the terms of this original research question for the purpose of using elasticity. I take into account that the effects of interactions are symmetrical, that is, the effect of education by class origin is equivalent to the effect of the class origin by education. In view of this symmetry of interactive effects, the original question could be formulated in this equivalent way: Is the effect of education lower in all class origins (advantageous and disadvantageous) among more educated individuals?
The results with interactive effects were estimated using the margins (option: eyex) in the form of elasticity between education and children's income conditional on social origin (i.class##c.education ). The results were stratified by 8, 10 and 15 year of complete education. This was the estimated model and the margins command:

svy, subpop(id66): glm income  i.class##c.education   i.cohort  i.state [where he lived at 15 years]  i.sex  i.collor,  family(gamma) link(log)
margins tipbr5_d, eyex (education)  at(education  = (8 11 15))

This was the outputs:

Expression   : Predicted mean income, predict()
ey/ex w.r.t. : education  
1._at        : education        =           8
2._at        : education        =          11
3._at        : education        =          15
-----------------------------------------------------------------------------------
                  |            Delta-method
                  |      ey/ex   Std. Err.      z    P>|z|     [95% Conf. Interval]
------------------+----------------------------------------------------------------
education         |
     _at#class    |
       1#1. top   |   1.304128   .0498687    26.15   0.000     1.206387    1.401869
  1#2. skilled    |   1.158187   .0617412    18.76   0.000     1.037177    1.279198
  1#3. few_assets |   .8176008   .0213492    38.30   0.000     .7757571    .8594446
  1#4. worker     |    .866426   .0257348    33.67   0.000     .8159866    .9168653
  1#5. destitute  |   .7345084   .0242293    30.31   0.000     .6870198     .781997
       2#1. top   |   1.793176   .0685695    26.15   0.000     1.658783     1.92757
  2#2. skilled    |   1.592508   .0848941    18.76   0.000     1.426118    1.758897
  2#3. few_assets |   1.124201   .0293552    38.30   0.000     1.066666    1.181736
  2#4. worker     |   1.191336   .0353854    33.67   0.000     1.121982     1.26069
  2#5. destitute  |   1.009949   .0333153    30.31   0.000     .9446522    1.075246
       3#1. top   |    2.44524   .0935038    26.15   0.000     2.261976    2.628504
  3#2. skilled    |   2.171601   .1157647    18.76   0.000     1.944706    2.398496
  3#3. few_assets |   1.533002   .0400298    38.30   0.000     1.454545    1.611459
  3#4. worker     |   1.624549   .0482528    33.67   0.000     1.529975    1.719122
  3#5. destitute  |   1.377203     .04543    30.31   0.000     1.288162    1.466244
-----------------------------------------------------------------------------------

It is observed that in all class origins the effects are higher in the higher levels of schooling, that is, the higher education does not equalize or neutralize the effect of the origin. On the contrary, it seems to increase the effect of origin.

[Phil Bromiley]

You didn't get a quick response. It might be because you don't actually pose a question. A shorter, more direct posting would also help.

You might look at marginsplot after your margins statement. I might also look at straight predictions rather than ey/ex.

[Jose Alcides Santos]

Thanks for the reply.
In the study I had to use elasticity or semi-elasticity with the purpose of the result not being affected by the income value in different cohorts.
As explain by Cameron and Trivendi in Microeconometrics Using Stata. “Elasticities can be more useful than MEs [marginal effects], because they are scale-free measures”.
My question was whether the reformulation of the problem would be equivalent. I believe so. I should present the two results that seem complementary by changing only the angle of vision of the problem.