Within-person effects

Correlated random effects (Mundlak) logit models

With panel data, the effect of a variable that changes over time can come from differences between people (people who are married compared with people who are not) or from changes within a person (the same person when married compared with when not married). A random-effects model mixes the two.

Fixed-effects models estimate the within-person effect, but marginal effects after them have known problems, especially for categorical outcomes. The correlated random effects (Mundlak) specification gives the same within-person estimate: add each time-varying predictor’s person mean to a random-effects model. The coefficient on the predictor is then the within-person effect.

The data are those of the panel models example: depressive symptoms among adults aged 51 and older in a 30% sample of households from the Health and Retirement Study, 2010 to 2020. Marital status and age change over time, so each gets its person mean.

use "https://tdmize.github.io/data/data/hrs_si_2025", clear
keep if year >= 2010 & age >= 51
(169,991 observations deleted)
keep if mod(floor(hhidpn/1000), 10) < 3
(76,874 observations deleted)
generate depressed = cesd >= 4 if !missing(cesd)
(1,821 missing values generated)
drop if missing(depressed, marstatB, age, woman, race4cat, collegeB)
(1,933 observations deleted)
xtset hhidpn year
Panel variable: hhidpn (unbalanced)
 Time variable: year, 2010 to 2020, but with gaps
         Delta: 1 unit
bysort hhidpn: egen mean_marstatB = mean(marstatB)
bysort hhidpn: egen mean_age = mean(age)
quietly xtlogit depressed i.marstatB c.age i.woman i.race4cat i.collegeB, re intpoints(30)
estimates store re
quietly xtlogit depressed i.marstatB c.mean_marstatB c.age c.mean_age ///
    i.woman i.race4cat i.collegeB, re intpoints(30)
estimates store cre

The within-person effect

In the correlated random effects model (cre), the marginal effect of being married holds the person’s mean fixed: it is the within-person effect, the change in the probability of depressive symptoms for the same person when married compared with when not married. The random-effects model (re) gives the usual estimate, and the Difference row tests whether the two differ.

mecompare marstatB, models(re cre)
NOTE: mecompare clusters the standard errors on the highest-level group of the multilevel or panel 
> model(s), so they will differ from the models' own. Fit every model without vce(robust); mecompar
> e supplies the clustering.

Predicting: Default prediction for each model
Marginal effects and cross-model differences (N_re=31545) (N_cre=31545)

                                 |  ME #   Estimate  Robust SE      P>|z|
---------------------------------+---------------------------------------
marstatB                         |                                       
    currently m - not curren     |                                       
                              re |     1     -0.096      0.006      0.000
                             cre |     2     -0.066      0.011      0.000
                      Difference |     3     -0.030      0.009      0.001

In the random-effects model, being married is associated with a 9.6 percentage point reduction in the probability of depressive symptoms. Examining only within-person variation, the effect of being married is a 6.6 percentage point reduction. The random-effects estimate is 3.0 points larger (p < .01) because it pools both between- and within-person variation, shown next.

The between-person effect

cochange() lets the person mean change along with marital status. With start() and end() it changes from 0 (not married at any interview) to 1 (married at every interview), so the second row is the between-person effect: the difference between people who are married at every interview and people who are married at none.

mecompare marstatB, models(cre) cochange(mean_marstatB) ///
    start(mean_marstatB=0) end(mean_marstatB=1)
Predicting: Pr(depressed=1)
Marginal effects (N_cre=31545)

                                 |  ME #   Estimate         SE      P>|z|
---------------------------------+---------------------------------------
marstatB                         |                                       
    currently m - not curren     |                                       
                             cre |     1     -0.066      0.009      0.000
---------------------------------+---------------------------------------
marstatB with co-change          |                                       
                             cre |     2     -0.110      0.007      0.000

NOTE: marstatB with co-change: marstatB 0 to 1; mean_marstatB 0 to 1.
metest 2 - 1
                                 |  estimate         se     pvalue 
---------------------------------+--------------------------------
             cochange - marstatB |    -0.045      0.012      0.000

Within a person, being married is associated with a 6.6 percentage point reduction in the probability of depressive symptoms; between people, it is associated with an 11.0 percentage point reduction. metest shows the between-person effect is 4.5 points larger (p < .001): people who are married at every interview differ from people who are married at none by more than the same person differs when married and when not.

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