Nested logit models and mediation/attenuation

Example 6.2 of Mize, Doan, and Long (2019)

One of the most common cross-model questions is whether an effect changes when controls are added, such as in tests of mediation/attenuation. Here the effect of a college degree on the probability of being very happy is compared across four nested logits: the bivariate model, then a full set of controls, then wages, then occupational prestige. Unlike logit coefficients, marginal effects – here, changes in the predicted probability – can be compared across models without the problem of rescaling.

use "https://tdmize.github.io/data/data/gss_cme", clear
(gss_cme.dta | GSS 1972 - 2016 Weighted | 2018-07-10)
drop if year < 2000
(38,116 observations deleted)
drop if employed != 1
(9,556 observations deleted)
drop if missing(vhappy, college, wages, occprest, age, married, parent, woman, conserv, reltrad)
(5,578 observations deleted)

. 
quietly logit vhappy i.college, vce(robust)
estimates store m1
. 
quietly logit vhappy i.college i.married i.parent i.woman i.conserv ///
              i.reltrad i.year c.age##c.age, vce(robust)
estimates store m2
. 
quietly logit vhappy i.college c.wages i.married i.parent i.woman i.conserv ///
              i.reltrad i.year c.age##c.age, vce(robust)
estimates store m3
. 
quietly logit vhappy i.college c.wages c.occprest i.married i.parent i.woman ///
              i.conserv i.reltrad i.year c.age##c.age, vce(robust)
estimates store m4
mecompare college, models(m1 m2 m3 m4)
Predicting: Pr(vhappy)

Marginal effects across models (N_m1=9216) (N_m2=9216) (N_m3=9216) (N_m4=9216)

                                 |  ME #   Estimate  Robust SE      P>|z|
---------------------------------+---------------------------------------
college                          |                                       
    College Deg - No Col Deg     |                                       
                              m1 |     1      0.072      0.010      0.000
                              m2 |     2      0.060      0.011      0.000
                              m3 |     3      0.036      0.011      0.001
                              m4 |     4      0.019      0.012      0.103

Use metest to calculate the cross-model tests. Here, whether the effect diminishes in each subsequent model.

metest, clear
metest 1 - 2, add
                                 |  estimate         se     pvalue 
---------------------------------+--------------------------------
college                          |                                
                         m1 - m2 |     0.012      0.004      0.003 
metest 2 - 3, add
                                 |  estimate         se     pvalue 
---------------------------------+--------------------------------
college                          |                                
                         m1 - m2 |     0.012      0.004      0.003 
                         m2 - m3 |     0.024      0.004      0.000 
metest 3 - 4, add
                                 |  estimate         se     pvalue 
---------------------------------+--------------------------------
college                          |                                
                         m1 - m2 |     0.012      0.004      0.003 
                         m2 - m3 |     0.024      0.004      0.000 
                         m3 - m4 |     0.017      0.004      0.000

With two models, the Difference row is printed directly:

mecompare college, models(m1 m2)
Predicting: Pr(vhappy)

Marginal effects and cross-model differences (N_m1=9216) (N_m2=9216)

                                 |  ME #   Estimate  Robust SE      P>|z|
---------------------------------+---------------------------------------
college                          |                                       
    College Deg - No Col Deg     |                                       
                              m1 |     1      0.072      0.010      0.000
                              m2 |     2      0.060      0.011      0.000
                      Difference |     3      0.012      0.004      0.003
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