Inequality as a summary measure

Example 4.1 of Mize and Han (2025)

A nominal variable such as race-ethnicity has no single marginal effect: with four categories there are six pairwise differences in predictions. The ME inequality statistic summarizes them as one number – the average absolute difference in the outcome across the categories. This first example uses a linear regression of hourly wages on a four-category race-ethnicity variable in the 2021 GSS.

use "https://tdmize.github.io/data/data/cda_gss", clear
(cda_gss.dta |  GSS 1972-2021 CDA - Categorical Data Analysis | date created 2023)
keep if year == 2021
(64,814 observations deleted)
drop if missing(wages, race4)
(2,234 observations deleted)

. 
regress wages i.race4 c.age i.woman
      Source |       SS           df       MS      Number of obs   =     1,714
-------------+----------------------------------   F(5, 1708)      =     37.11
       Model |  49208.2413         5  9841.64825   Prob > F        =    0.0000
    Residual |  452993.938     1,708  265.218933   R-squared       =    0.0980
-------------+----------------------------------   Adj R-squared   =    0.0953
       Total |  502202.179     1,713   293.17115   Root MSE        =    16.286

------------------------------------------------------------------------------
       wages | Coefficient  Std. err.      t    P>|t|     [95% conf. interval]
-------------+----------------------------------------------------------------
       race4 |
      Black  |     -3.440      1.351   -2.546   0.011       -6.090      -0.790
      Other  |      6.664      1.705    3.910   0.000        3.321      10.007
   Hispanic  |     -3.766      1.294   -2.911   0.004       -6.304      -1.228
             |
         age |      0.255      0.029    8.697   0.000        0.197       0.312
             |
       woman |
      Women  |     -6.095      0.793   -7.690   0.000       -7.649      -4.540
       _cons |     14.729      1.528    9.640   0.000       11.732      17.725
------------------------------------------------------------------------------
meinequality race4
ME Inequality Estimates (N = 1714)

                         |   Estimate   Std. err.           z       P>|z| 
-------------------------+-----------------------------------------------
ME Inequality            |                                               
                   race4 |      4.904       0.860       5.702       0.000

Racial-ethnic groups’ wages differ by close to $5 per hour, on average. The default statistic is weighted: each pairwise comparison counts in proportion to the share of the sample in the two groups compared. The unweighted version gives every comparison the same weight:

meinequality race4, unweighted
ME Inequality Estimates (N = 1714)

                         |   Estimate   Std. err.           z       P>|z| 
-------------------------+-----------------------------------------------
Unwgt. ME Inequality     |                                               
                   race4 |      5.788       1.042       5.558       0.000
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