Random-effects panel-data modeling with endogenous selection
Two-level multilevel models with endogenous selection
Advanced inferences
Inference statistics
Expected means and probabilities
Marginal effects and contrasts
Average structural functions (ASFs)
More ...
Conditional analysis—specify values of all covariates
Test whether selection matters
Population-averaged—values of specified covariates
Inferences and plots over groups
Heckman selection models adjust for bias when some outcomes are missing not at random. Imagine modeling income. The problem is that income is observed only for those who work. Missingness is not random.
Stata fits Heckman selection models and can fit them with panel (two-level) data.
You want to fit the model
\( y_{it} = x_{it}\beta + \alpha_{i} + \varepsilon_{it} \)
where \(y_{it}\) is sometimes missing. The equation that determines which \(y_{it}\) are not missing is
\( S_{it} = 1(z_{it}\gamma + v_{i} + u_{it} > 0) \)
In these equations, \(\alpha_{i}\), \(\varepsilon_{it}\), \(v_{i}\), and \(u_{it}\) will not be estimated. Their correlations with each other, however, will be estimated along with \(\beta\) and \(\gamma\).
The above model can be fit even though income is not observed for everyone and even if their employment status changes over time.
Why fit a selection model? Because it is possible that people who work and whose income is therefore observed systematically differ from those who do not, and those differences are for unobserved reasons.
For instance, if more productive people work, their income will be higher than those who do not work. Or, if income of the less productive is lower, they might need to work more. Allowing for selection allows for either of the above alternatives and other alternatives too. After estimation, we can test whether selection matters.
We have fictional data on 8,000 individuals from 2011 to 2018. Among the variables are income, which is observed only for those who work. We worry that unobservables might lead to biased results.
To fit the selection model, we must model income and the probability of working. We model probability of working as a function of experience, age, region of the county, and whether the person has college or technical college training.
We fit the model
. xtheckman income c.age##c.age i.training#(c.exp##c.exp),
select(working = age exp i.region i.training)
If you are new to Stata, things like c.age##c.age mean to include age and age squared in the model. The "c." means continuous. The "i." in i.training and i.region means categorical variable and indicates the categories are to be included in the model.
The results are
Random-effects regression with selection Number of obs = 8,000
Selected = 7,235
Nonselected = 765
Group variable: id Number of groups = 1,000
Obs per group:
min = 8
avg = 8.0
max = 8
Integration method: mvaghermite Integration pts. = 7
Wald chi2(6) = 13011.86
Log likelihood = -28748.805 Prob > chi2 = 0.0000
| Coefficient Std. err. z P>|z| [95% conf. interval] | ||||
| wage | ||||
| age | .0841345 .05193 1.62 0.105 -.0176465 .1859154 | |||
| c.age#c.age | -.0006552 .0006167 -1.06 0.288 -.0018638 .0005534 | |||
| training# | ||||
| c.exp | ||||
| 0 | .1497442 .1897593 0.79 0.430 -.2221772 .5216656 | |||
| 1 | 3.198649 .2042991 15.66 0.000 2.798231 3.599068 | |||
| _cons | 7.744222 1.223347 6.33 0.000 5.346506 10.14194 | |||
| working | ||||
| age | .0083258 .001507 5.52 0.000 .005372 .0112795 | |||
| exp | .069833 .007423 9.41 0.000 .0552843 .0843818 | |||
| region | ||||
| 2 | .1683876 .0653623 2.58 0.010 .0402798 .2964953 | |||
| 3 | .0286791 .0630488 0.45 0.649 -.0948944 .1522525 | |||
| 4 | .0476718 .0639092 0.75 0.456 -.0775879 .1729315 | |||
| 5 | .0054477 .0621657 0.09 0.930 -.1163948 .1272901 | |||
| 1.training | .8223611 .0596781 13.78 0.000 .7053942 .939328 | |||
| _cons | .3367662 .0905615 3.72 0.000 .1592688 .5142635 | |||
| var(e.wage) | 81.07829 2.142513 76.98594 85.38819 | |||
| corr(e.wor~g, | ||||
| e.wage) | -.5812249 .0615897 -9.44 0.000 -.6892935 -.447854 | |||
| var(wage[id]) | 19.93373 1.405619 17.36067 22.88815 | |||
| var( | ||||
| working[id]) | .0989163 .0262816 .0587635 .1665052 | |||
| corr( | ||||
| working[id], | ||||
| wage[id]) | .258161 .1038653 2.49 0.013 .045996 .4480403 | |||
The first panel in the results reports the income equation.
The second panel reports the working (selection) equation.
After that are reported three variances and two correlations. The correlations are of interest.
| correlation | estimate | SE | ||
| corr(e.working. e.income) | -0.58 | 0.06 | ||
| corr(working[id], income[id]) | 0.26 | 0.10 | ||
The first correlation is the correlation of the residuals in the income and working (selection) equation, the correlation of \(\varepsilon_{it}\) and \(u_{it}\).
The second is the correlation of random effects and unobservables that do not change over time, or the correlation of \(\alpha_{i}\) and \(v_{i}\).
Selection was an issue if either of these correlations are significant. Both are.
Read more about Heckman selection models for panel data in the Stata Longitudinal-Data/Panel-Data Reference Manual; see [XT] xtheckman.
