1 item has been added to your cart.
Sample selection arises when the sampled data are not representative of the population of interest. A classic example of sample selection is women's work participation. Suppose that we want to model the wages of women. If we consider only the sample of women who chose to work, we may end up with a sample in which the wages are too high because women who would have low wages may have chosen not to work. Of course, if the decision whether to work is random, there would be no problem with using only the sample of women who work. This is not a realistic assumption in this case. To obtain valid inference in this example, we must model the outcome, the wages, and the decision to work. We will refer to the two models as the outcome model and the participation model.
In Stata, you can use heckman to fit a Heckman selection model to continuous outcomes, heckprobit to fit a probit sample-selection model to binary outcomes, and heckoprobit to fit an ordered probit model with sample selection to ordinal outcomes. You can now simply prefix these commands with bayes: to fit the corresponding Bayesian sample-selection models.
Continuing with our example of women's work participation,
we first fit the classical Heckman sample-selection model. Below we model
both the wages and the decision to work based on the
level of education
and age. For the decision to work, we additionally include marriage
status and
number of children.
. heckman wage educ age, select(married children educ age)
Heckman selection model Number of obs = 2,000
(regression model with sample selection) Selected = 1,343
Nonselected = 657
Wald chi2(2) = 508.44
Log likelihood = -5178.304 Prob > chi2 = 0.0000
| wage | Coef. Std. Err. z P>|z| [95% Conf. Interval] | |
| wage5 | ||
| education | .9899537 .0532565 18.59 0.000 .8855729 1.094334 | |
| age | .2131294 .0206031 10.34 0.000 .1727481 .2535108 | |
| _cons | .4857752 1.077037 0.45 0.652 -1.625179 2.59673 | |
| select | ||
| married | .4451721 .0673954 6.61 0.000 .3130794 .5772647 | |
| children | .4387068 .0277828 15.79 0.000 .3842534 .4931601 | |
| education | .0557318 .0107349 5.19 0.000 .0346917 .0767718 | |
| age | .0365098 .0041533 8.79 0.000 .0283694 .0446502 | |
| _cons | -2.491015 .1893402 -13.16 0.000 -2.862115 -2.119915 | |
| /athrho | .8742086 .1014225 8.62 0.000 .6754241 1.072993 | |
| /lnsigma | 1.792559 .027598 64.95 0.000 1.738468 1.84665 | |
| rho | .7035061 .0512264 .5885365 .7905862 | |
| sigma | 6.004797 .1657202 5.68862 6.338548 | |
| lambda | 4.224412 .3992265 3.441942 5.006881 | |
To fit its Bayesian analog, we use bayes: heckman.
. bayes: heckman wage educ age, select(married children educ age)
| Model summary | ||
| Likelihood: | ||
| wage ~ heckman(xb_wage,xb_select,{athrho} {lnsigma}) | ||
| Priors: | ||
| {wage:education age _cons} ~ normal(0,10000) (1) | ||
| {select:married children education age _cons} ~ normal(0,10000) (2) | ||
| {athrho lnsigma} ~ normal(0,10000) | ||
| (1) Parameters are elements of the linear form xb_wage. (2) Parameters are elements of the linear form xb_select. | ||
| Equal-tailed | ||
| Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||
| wage | ||
| education | .9919131 .051865 .002609 .9931531 .8884407 1.090137 | |
| age | .2131372 .0209631 .001071 .2132548 .1720535 .2550835 | |
| _cons | .4696264 1.089225 .0716 .4406188 -1.612032 2.65116 | |
| select | ||
| married | .4461775 .0681721 .003045 .4456493 .3178532 .5785857 | |
| children | .4401305 .0255465 .001156 .4402145 .3911135 .4903804 | |
| education | .0559983 .0104231 .000484 .0556755 .0360289 .076662 | |
| age | .0364752 .0042497 .000248 .0362858 .0280584 .0449843 | |
| _cons | -2.494424 .18976 .011327 -2.498414 -2.861266 -2.114334 | |
| athrho | .868392 .099374 .005961 .8699977 .6785641 1.062718 | |
| lnsigma | 1.793428 .0269513 .001457 1.793226 1.740569 1.846779 | |
Unlike heckman, bayes: heckman reports the ancillary parameters only in the estimation metric. We can use bayesstats summary to obtain the parameters in the original metric.
. bayesstats summary (rho:1-2/(exp(2*{athrho})+1)) (sigma:exp({lnsigma}))
Posterior summary statistics MCMC sample size = 10,000
rho : 1-2/(exp(2*{athrho})+1)
sigma : exp({lnsigma})
| Equal-tailed | ||
| Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||
| rho | .6970522 .0510145 .003071 .701373 .5905851 .7867018 | |
| sigma | 6.012205 .1621422 .008761 6.008807 5.700587 6.339366 | |
Parameter rho is a correlation coefficient that measures the dependence between the outcome and participation models. If rho is zero, the two models are independent and can be analyzed separately. In other words, there is no sample selection, and we can model the wages using only the sample of women who work without introducing any bias in our results. In our example, rho is estimated to be between 0.59 and 0.79 with a probability of 0.95, so the decision to work is related to the wages in this example.
We can test for sample selection formally by using, for example, Bayes factors. A Bayes factor of two models is simply the ratio of their marginal likelihoods. The larger the value of the marginal likelihood, the better the model fits the data. To test for sample selection, we can compare the marginal likelihoods of the current model and of the model with rho equal to zero.
First, we store the current Bayesian estimation results from the sample-selection model.
. bayes, saving(heckman_mcmc) . estimates store heckman
Next, we fit a model that assumes no sample selection. When rho equals zero, {athrho} also equals zero. So we specify a strong prior saturated at zero for parameter {athrho}.
. bayes, prior({athrho}, normal(0,1e-4)) saving(nosel_mcmc):
heckman wage educ age, select(married children educ age)
| Model summary | ||
| Likelihood: | ||
| wage ~ heckman(xb_wage,xb_select,{athrho} {lnsigma}) | ||
| Priors: | ||
| {wage:education age _cons} ~ normal(0,10000) (1) | ||
| {select:married children education age _cons} ~ normal(0,10000) (2) | ||
| {athrho} ~ normal(0,1e-4) | ||
| {lnsigma} ~ normal(0,10000) | ||
| (1) Parameters are elements of the linear form xb_wage. (2) Parameters are elements of the linear form xb_select. | ||
| Equal-tailed | ||
| Mean Std. Dev. MCSE Median [95% Cred. Interval] | ||
| wage | ||
| education | .8981219 .0509913 .001578 .8973616 .8013416 1.000497 | |
| age | .1477784 .01854 .00066 .1477496 .1115628 .1850257 | |
| _cons | 5.994764 .890318 .030657 6.014622 4.150738 7.658942 | |
| select | ||
| married | .4351031 .0748102 .003577 .4377313 .2821176 .5752786 | |
| children | .4501657 .0285028 .001045 .4492015 .3937091 .5048498 | |
| education | .0584037 .0110582 .000524 .0579573 .0370387 .0814287 | |
| age | .034779 .0043677 .00022 .0348894 .0259916 .043139 | |
| _cons | -2.47607 .1962162 .009818 -2.467739 -2.862694 -2.10733 | |
| athrho | .0062804 .010209 .00023 .0062746 -.014139 .0261746 | |
| lnsigma | 1.69586 .019056 .000386 1.695649 1.65948 1.734115 | |
We now use bayesstats ic to obtain the Bayes factor of the two models.
. bayesstats ic heckman nosel Bayesian information criteria
| DIC log(ML) log(BF) | ||
| heckman | 10376.05 -5260.202 . | |
| nosel | 10435.29 -5283.025 -22.82221 | |
The value of the log-Bayes factor of -23 indicates a very strong preference for the sample-selection model heckman and thus for the presence of sample selection in these data.
Learn more about the general features of the bayes prefix.
Learn more about Stata's Bayesian analysis features.
Read more about the bayes prefix and Bayesian analysis in the Stata Bayesian Analysis Reference Manual.
