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Panel-data multinomial logit model

Highlights

  • Panel-data modeling of unordered categorical outcomes

  • Random-effects estimator

    • Choice of covariance structure for random effects
      • Independent
      • Identity
      • Shared
      • Exchangeable
      • Unstructured
  • Conditional fixed-effects (FE) estimator

    • Permutation subsets lessen curse of dimensionality
  • Bayesian estimation

  • Robust, cluster–robust, and bootstrap standard errors

  • Support for complex survey data

The multinomial logit (MNL) model is a popular method for modeling categorical outcomes that have no natural ordering—outcomes such as occupation, political party, or restaurant choice.

In longitudinal/panel data, we observe a sequence of outcomes over time. Say that we observe restaurant choices made by individuals each week. Do you think that restaurant choices are independent from week to week? Probably not. Someone who likes Italian food is likely to choose an Italian restaurant multiple times. These choices are driven by underlying personal preferences and characteristics, some of which are not observed.

Stata's new xtmlogit command fits random-effects and conditional fixed-effects MNL models for categorical outcomes observed over time.

To fit a random-effects multinomial logit model, we can type

. xtset subject
. xtmlogit restaurant age

and estimate the standard multinomial logit coefficients accounting for time-invariant subject-specific characteristics by including random effects specific to each outcome level.

With the command above, the random effects are assumed to be normally distributed and independent across outcome levels (restaurant choices), but several variance–covariance structures are supported, including a completely unrestricted covariance:

. xtmlogit restaurant age, covariance(unstructured)

If you suspect that subject-specific effects might be correlated with age, you can use a conditional fixed-effects estimator to account for this:

. xtmlogit restaurant age, fe


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.

. use https://www.stata-press.com/data/r17/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.


Additional resources

Learn more in the Stata Longitudinal-Data/Panel-Data Reference Manual.

Fit Bayesian fixed-effects and random-effects MNL models using the bayes prefix.



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