Oaxaca Blinder Decomposition Interaction

[Tim Sulls]

Hello, everyone. Can you help me in interpreting the results of this oaxaca blinder decomposition?

I have been reading various threads here and related literature and either cannot find a clear interpretation of the standard output or it is too technical and I do not follow.

use http://fmwww.bc.edu/RePEc/bocode/o/oaxaca.dta
oaxaca lnwage educ exper tenure, by(female) noisily

------------------------------------------------------------------------------
      lnwage | Coefficient  Std. err.      z    P>|z|     [95% conf. interval]
-------------+----------------------------------------------------------------
overall      |
     group_1 |   3.440222   .0174874   196.73   0.000     3.405947    3.474497
     group_2 |   3.266761   .0218522   149.49   0.000     3.223932    3.309591
  difference |   .1734607    .027988     6.20   0.000     .1186052    .2283163
  endowments |   .0852798    .015693     5.43   0.000     .0545222    .1160375
coefficients |    .082563   .0255804     3.23   0.001     .0324263    .1326996
 interaction |    .005618    .010966     0.51   0.608    -.0158749    .0271109
-------------+----------------------------------------------------------------
endowments   |
        educ |   .0510912    .012239     4.17   0.000     .0271031    .0750792
       exper |   .0254173   .0088089     2.89   0.004     .0081522    .0426824
      tenure |   .0087714   .0086201     1.02   0.309    -.0081238    .0256665
-------------+----------------------------------------------------------------
coefficients |
        educ |  -.0640559   .1193498    -0.54   0.591    -.2979772    .1698654
       exper |  -.0397237     .04058    -0.98   0.328    -.1192591    .0398117
      tenure |   .0420986   .0270201     1.56   0.119    -.0108598    .0950571
       _cons |   .1442439   .1340957     1.08   0.282    -.1185788    .4070667
-------------+----------------------------------------------------------------
interaction  |
        educ |  -.0033202   .0062283    -0.53   0.594    -.0155274    .0088871
       exper |  -.0063464   .0066988    -0.95   0.343    -.0194758     .006783
      tenure |   .0152845   .0101496     1.51   0.132    -.0046084    .0351774
------------------------------------------------------------------------------

Specifically, if exper was significant across all three sections: endowment, coefficient, and interaction. How would we describe this? Similarly, what does it mean for any one variable to be significanrt in one (or more) section(s) but not others?

[George Ford]

https://ete-online.biomedcentral.com/articles/10.1186/s12982-021-00100-9

you can cut it to a two-way decomposition using the option "omega", especially since the interaction terms are all insignificant.

[George Ford]

This shows what's happening for two style of models.

clear all
sysuse auto, clear

oaxaca mpg weight displacement, by(foreign) pooled noisily
reg mpg weight displacement foreign
matrix P = e(b)

tabstat weight displacement foreign, by(foreign) save
matrix Ax = r(Stat1)
matrix Bx = r(Stat2)

di "group_1" _col(20) P[1,1]*Ax[1,1] + P[1,2]*Ax[1,2] + P[1,3]*Ax[1,3] + P[1,4]
di "group_2" _col(20) P[1,1]*Bx[1,1] + P[1,2]*Bx[1,2] + P[1,3]*Bx[1,3] + P[1,4]
di "difference" _col(20) (P[1,1]*Ax[1,1] + P[1,2]*Ax[1,2] + P[1,3]*Ax[1,3] + P[1,4])-( P[1,1]*Bx[1,1] + P[1,2]*Bx[1,2] + P[1,3]*Bx[1,3] + P[1,4]) 
di "explained" _col(20) P[1,1]*(Ax[1,1]-Bx[1,1]) + P[1,2]*(Ax[1,2]-Bx[1,2])
di "unexplained" _col(20) P[1,3]*(Ax[1,3] - Bx[1,3])


oaxaca mpg weight displacement, by(foreign) noisily
reg mpg weight displacement if ~foreign
matrix A = e(b)
reg mpg weight displacement if  foreign
matrix B = e(b)

di "group_1" _col(20) A[1,1]*Ax[1,1] + A[1,2]*Ax[1,2] + A[1,3]
di "group_2" _col(20) B[1,1]*Bx[1,1] + B[1,2]*Bx[1,2] + B[1,3]
di "difference" _col(20) (A[1,1]*Ax[1,1] + A[1,2]*Ax[1,2] + A[1,3])-(B[1,1]*Bx[1,1] + B[1,2]*Bx[1,2] + B[1,3]) 
di "endowments" _col(20) B[1,1]*(Ax[1,1]-Bx[1,1]) + B[1,2]*(Ax[1,2]-Bx[1,2])
di "coefficients" _col(20) Bx[1,1]*(A[1,1]-B[1,1]) + Bx[1,2]*(A[1,2]-B[1,2]) + (A[1,3]-B[1,3])
di "interaction" _col(20) (A[1,1]-B[1,1])*(Ax[1,1]-Bx[1,1]) + (A[1,2]-B[1,2])*(Ax[1,2]-Bx[1,2])

[Tim Sulls]

This is very helpful. If the interactions were significant, what does that mean? For example, if tenure is significant in all three buckets I am struggling to know what that means.

[George Ford]

Nobody explains it very well, do they? I usually stick to the two-way as it is straightforward to interpret.

In the twoway, you have single coefficient vector that measures everything (matrix P above).

In the three way, the coefficients are allowed to differ between the groups (matrix A and B above). Since both the X and the Beta differ between groups (or may), the interaction term measures the "return" to endowments and may indicate a "double ad(disad)vantage" or whether the effects offset each other. Depending on the reference group, these interaction effects are part of the endowment or part of the coefficients (unexplained) effects (you can switch the reference group and see the effects change). (See paper above). If the X are the same (or close to it), then the interaction effect is small, even if the Betas differ a lot. The difference in coefficients, then, isn't really measuring much. If the X differ a lot, then the interaction effect can be large if the Betas differ a lot, or small if they differ little. So, if you have a large spread in X and Beta, then the interaction effect is large and shows an advantage/disadvantage in that particular X. I generally think that the interaction effects are properly part of the "unexplained" effect. So, run it pooled, then not (threeway twice, including swap in the second model) to see how it affects the endowment and coefficient effect sizes.

[Tim Sulls]

George, I cannot thank you enough. For someone trying to learn this on their own your words are quite useful.

I now get where it all comes from but the explanation is still tricky for me. Sorry if I missed something in your brilliant response.

If we take tenure as significant then are my interpretations correct?

1. Endowment: Men and women have similar tenure distributions and little of the wage difference is accounted for by actual gender differences in tenure
2. Coefficient: The wage difference is more pronounced at higher levels of tenure
3. Interaction (I am still least sure here): The gender differences is increasing as tenure increases (i.e., a squared term).

[FernandoRios]

Hi Tim,
The interpretation of the interaction is always difficult because it could go in directions that you wouldn't expect.

Say, for example, that all coefficients are positive, and that men have the advantage of both characteristics and coefficients.
- Endowment effect: Men earn more because they have higher education
- Coefficient effect: Men earn more because they get paid more for each year of education
- Interaction: They also earn more because each additional year of education has a higher return (dX * dB)

When the signs flip, its a bit harder to explain the interaction. Probably because of this, most papers only use 2 way decomposition.
F

[Tim Sulls]

Thanks F. Just to be 100% clear when you say "most use the 2 way" is it different code to do this than

 
 oaxaca lnwage educ exper tenure, by(female) noisily

or do you mean that is why they do not interpret the interaction?

[FernandoRios]

Yep, you use either w(0) or w(1) option.
In fact, while OB is great to get the full decomposition, if you are just starting to learn about it, I would highly recommend to do the decomposition by hand first. even if its with some Toy data. That way you know exactly what is happening behind the black box, and will make it easier to explain.

Fernando

[Tim Sulls]

Awesome and I have been doing by hand so thanks for confirming tip! When would you use w(0) versus w(1)?

[FernandoRios]

would depend on the question in hand.
Are you measuring gaps explained by endowments from group1 perspective? or from group 2 perspective?
Again, doing it by hand first is the best way to go

[Tim Sulls]

group == 1 (with == 0 as reference)

[George Ford]

#3 is "by hand"