Panel-data multinomial logit model

Let's see it work

We wish to find out whether individuals are more likely to be out of the labor force if they have children under the age of five in their household. We will use a (fictitious) dataset of men and women who were asked about their employment status every two years.

. webuse estatus
(Fictitious employment status data)

Here is an excerpt of the dataset, showing the employment history for three individuals:

. list id year estatus hhchild age in 22/41, sepby(id) noobs


    

    
 id   year              estatus   hhchild   age

    

    
  5   2002             Employed       Yes    38 

    
  5   2004             Employed        No    40 

    
  5   2006             Employed        No    42 

    
  5   2008             Employed        No    44 

    
  5   2010   Out of labor force        No    46 

    
  5   2012   Out of labor force        No    48 

    
  5   2014           Unemployed        No    50 

    

    
  6   2002           Unemployed       Yes    31 

    
  6   2004             Employed       Yes    33 

    
  6   2006   Out of labor force       Yes    35 

    
  6   2008           Unemployed       Yes    37 

    
  6   2010   Out of labor force       Yes    39 

    
  6   2012           Unemployed        No    41 

    

    
  7   2002   Out of labor force       Yes    33 

    
  7   2004             Employed       Yes    35 

    
  7   2006             Employed       Yes    37 

    
  7   2008   Out of labor force       Yes    39 

    
  7   2010             Employed        No    41 

    
  7   2012             Employed        No    43 

    
  7   2014             Employed        No    45 

    

The outcome of interest is employment status (estatus), which has three levels: Employed, Unemployed (but seeking employment), and Out of labor force (not seeking employment). Our predictor of interest, hhchild, indicates whether they have children under the age of five in their household at the time of the interview.

Before we can fit our model, we need to specify our panel identifier variable, id, by using xtset.

. xtset id

Panel variable: id (unbalanced)

Now we can use xtmlogit to model the probability of each employment type by hhchild while controlling for the effects of age, annual household income (hhincome), and whether a significant other was also living in the household (hhsigno). We will start with a random-effects model (the default) and use the rrr option to get exponentiated coefficients that can be interpreted as relative-risk ratios.

. xtmlogit estatus i.hhchild age hhincome i.hhsigno, rrr

Fitting comparison model ...

Refining starting values:

Grid node 0:  Log likelihood = -4504.5591
Grid node 1:  Log likelihood = -4538.6352

Fitting full model:

Iteration 0:  Log likelihood = -4504.5591
Iteration 1:  Log likelihood =  -4495.871
Iteration 2:  Log likelihood = -4490.5098
Iteration 3:  Log likelihood = -4490.4197
Iteration 4:  Log likelihood = -4490.4196

Random-effects multinomial logistic regression       Number of obs    =  4,761
Group variable: id                                   Number of groups =    800

Random effects u_i ~ Gaussian                        Obs per group:
                                                                  min =      5
                                                                  avg =    6.0
                                                                  max =      7

Integration method: mvaghermite                      Integration pts. =      7

                                                     Wald chi2(8)     = 199.25
Log likelihood = -4490.4196                          Prob > chi2      = 0.0000



           estatus 
 
        RRR   Std. err.      z    P>|z|     [95% conf. interval]

Out_of_labor_force 
 
                                                                
           hhchild 
 
                                                                
              Yes  
 
   1.579937   .1513905     4.77   0.000     1.309414    1.906349
               age 
 
   .9947946   .0065832    -0.79   0.430      .981975    1.007781
          hhincome 
 
   .9954927   .0018251    -2.46   0.014     .9919221    .9990762
             
 
                                                                
           hhsigno 
 
                                                                
              Yes  
 
   1.642859   .1550291     5.26   0.000     1.365452    1.976625
             _cons 
 
   .4949307   .1392991    -2.50   0.012     .2850836     .859244

Unemployed         
 
                                                                
           hhchild 
 
                                                                
              Yes  
 
   .9607243   .1148148    -0.34   0.737     .7601038    1.214296
               age 
 
   1.004257    .008211     0.52   0.603     .9882918     1.02048
          hhincome 
 
   .9696874   .0025722   -11.60   0.000      .964659    .9747421
             
 
                                                                
           hhsigno 
 
                                                                
              Yes  
 
   1.099323   .1310654     0.79   0.427     .8702452    1.388701
             _cons 
 
   .8078165    .280628    -0.61   0.539     .4088963    1.595924

Employed           
 
  (base outcome)                                                

            var(u1)
 
   .8573133   .1083915                      .6691459    1.098394
            var(u2)
 
   .7378532   .1388652                      .5102376    1.067008

Note: Estimates are transformed only in the first 3 equations to relative-risk
      ratios.
Note: _cons estimates baseline relative risk (conditional on zero random effects).
LR test vs. multinomial logit: chi2(2) = 227.68           Prob > chi2 = 0.0000

Note: LR test is conservative and provided only for reference.

The first two sections in the output show the relative-risk ratio estimates of our predictors with respect to the base category Employed. The last section shows the estimated variances of the random effects. By default, the random effects are uncorrelated, but their covariance structure can be changed using the covariance() option. For example, correlations between random effects can be estimated using covariance(unstructured), or each category can share a common random effect using covariance(shared).

Adjusting for age, household income, and having a significant other at home, the relative risk of being out of the labor force for individuals having at least one child under the age of 5 in the household versus having no children under 5 in the household is 1.6 times as large as the relative risk of employment (95% CI [1.3, 1.9]). To understand these effects in terms of probabilities, we can use the margins command.

. margins hhchild

Predictive margins                                       Number of obs = 4,761
Model VCE: OIM

1._predict: Pr(estatus==Out_of_labor_force), predict(pr outcome(1))
2._predict: Pr(estatus==Unemployed), predict(pr outcome(2))
3._predict: Pr(estatus==Employed), predict(pr outcome(3))



             
 
            Delta-method                                        
             
 
     Margin   std. err.      z    P>|z|     [95% conf. interval]

_predict#hhchild 
 
                                                                
           1#No  
 
   .3025675   .0131546    23.00   0.000      .276785      .32835
          1#Yes  
 
   .3912476   .0120405    32.49   0.000     .3676486    .4148466
           2#No  
 
   .1628713   .0101131    16.11   0.000     .1430501    .1826925
          2#Yes  
 
   .1398537   .0079462    17.60   0.000     .1242794    .1554279
           3#No  
 
   .5345612   .0136994    39.02   0.000     .5077108    .5614116
          3#Yes  
 
   .4688987   .0116594    40.22   0.000     .4460468    .4917507

For an individual without children, the expected probability of being out of the labor force (labeled 1#No) is 0.30, the expected probability of being unemployed (2#No) is 0.16, and the expected probability of being employed is 0.53 (3#No). We also find that individuals with children in the household increase their probability of being out of the labor force by 9 percentage points. We could see how these probabilities change by household income using an additional margins command and visualize the results using marginsplot.

. quietly margins hhchild, at(hhincome=(20(20)100))

. marginsplot, by(_predict, label("Out of labor force" "Unemployed" "Employed"))
     byopts(rows(1) title("Marginal probabilities of employment status"))
     legend(order(4 "Child under 5 at home" 3 "No child under 5 at home"))

To get separate graphs for each outcome, we used the by(_predict) option in marginsplot. The rest of the options add titles and labels.

Comparing the lines within each employment category, we see that having a child at home does not have much impact on the probability of being unemployed but does influence the decision to work or to be out of the labor force.

In the model we just fit, we used random effects to account for unobserved characteristics of the individuals in our dataset. Random-effects models require that the random effects be uncorrelated with the predictors, and the random-effects MNL model is no exception. A widely used alternative is the fixed-effects estimator. To fit our model with conditional fixed effects, we simply add the fe option.

. xtmlogit estatus i.hhchild age hhincome i.hhsigno, fe rrr
note: 80 groups (451 obs) omitted because of no variation in the outcome variable
      over time.

Computing initial values ...

Setting up 26,168 permutations:
....10%....20%....30%....40%....50%....60%....70%....80%....90%....100%

Fitting full model:

Iteration 0:  Log likelihood = -2154.4175
Iteration 1:  Log likelihood = -2154.2058
Iteration 2:  Log likelihood = -2154.2057

Fixed-effects multinomial logistic regression        Number of obs    =  4,310
Group variable: id                                   Number of groups =    720

                                                     Obs per group:
                                                                  min =      5
                                                                  avg =    6.0
                                                                  max =      7

                                                     LR chi2(8)       =  67.42
Log likelihood = -2154.2057                          Prob > chi2      = 0.0000



           estatus 
 
        RRR   Std. err.      z    P>|z|     [95% conf. interval]

Out_of_labor_force 
 
                                                                
           hhchild 
 
                                                                
              Yes  
 
   1.784236   .2237128     4.62   0.000     1.395488     2.28128
               age 
 
   .9977834   .0146507    -0.15   0.880     .9694778    1.026915
          hhincome 
 
   .9895225   .0086923    -1.20   0.231     .9726318    1.006707
                   
 
                                                                
           hhsigno 
 
                                                                
              Yes  
 
   1.658753   .1654425     5.07   0.000     1.364217    2.016878

Unemployed         
 
                                                                
           hhchild 
 
                                                                
              Yes  
 
   1.181866   .1933766     1.02   0.307     .8576197    1.628702
               age 
 
   1.004991   .0194887     0.26   0.797      .967511    1.043924
          hhincome 
 
   .9717411   .0116616    -2.39   0.017     .9491514    .9948684
                   
 
                                                                
           hhsigno 
 
                                                                
              Yes  
 
    1.11936   .1454154     0.87   0.385     .8677426    1.443939

Employed           
 
  (base outcome)                                                

The results are similar to those of the random-effects estimator. And they can be interpreted in the same way.