I use features new to Stata 14.1 to estimate an average treatment effect (ATE) for a probit model with an endogenous treatment. In 14.1, we added new prediction statistics after mlexp that margins can use to estimate an ATE.

I am building on a previous post in which I demonstrated how to use mlexp to estimate the parameters of a probit model with sample selection. Our results match those obtained with biprobit; see [R] biprobit for more details. In a future post, I use these techniques to estimate treatment-effect parameters not yet available from another Stata command.

Probit model with treatment

In this section, I describe the potential-outcome framework used to define an ATE. For each treatment level, there is an outcome that we would observe if a person were to select that treatment level. When the outcome is binary and there are two treatment levels, we can specify how the potential outcomes \(y_{0i}\) and \(y_{1i}\) are generated from the regressors \({\bf x}_i\) and the error terms \(\epsilon_{0i}\) and \(\epsilon_{1i}\):

\[\begin{eqnarray*}
y_{0i} &=& {\bf 1}({\bf x}_i{\boldsymbol \beta}_0 + \epsilon_{0i} > 0) \cr
y_{1i} &=& {\bf 1}({\bf x}_i{\boldsymbol \beta}_1 + \epsilon_{1i} > 0)
\end{eqnarray*}\]

(Assuming that each error is standard normal, this gives us a bivariate probit model.) The indicator function \({\bf 1}(\cdot)\) outputs 1 when its input is true and 0 otherwise.

The probit model for potential outcomes \(y_{0i}\) and \(y_{1i}\) with treatment \(t_i\) assumes that we observe the outcome

\[\begin{equation}
y_i = (1-t_i) y_{0i} + t_i y_{1i}
\nonumber
\end{equation}\]

So we observe \(y_{1i}\) under the treatment (\(t_{i}=1\)) and \(y_{0i}\) when the treatment is withheld (\(t_{i}=0\)).

The treatment \(t_i\) is determined by regressors \({\bf z}_i\) and standard normal error \(u_i\):

\[\begin{equation}
t_i = {\bf 1}({\bf z}_i{\boldsymbol \psi} + u_i > 0)
\nonumber
\end{equation}\]

Probit model with endogenous treatment

We could estimate the parameters \({\boldsymbol \beta}_0\) and \({\boldsymbol \beta}_1\) using a probit regression on \(y_i\) if \(t_i\) was not related to the unobserved errors \(\epsilon_{0i}\) and \(\epsilon_{1i}\). This may not always be the case. Suppose we modeled whether parents send their children to private school and used private tutoring for the child as a treatment. Unobserved factors that influence private school enrollment may be correlated with the unobserved factors that influence whether private tutoring is given. The treatment would be correlated with the unobserved errors of the outcome.

We can treat \(t_i\) as endogenous by allowing \(\epsilon_{0i}\) and \(\epsilon_{1i}\) to be correlated with \(u_i\). In this post, we will assume that these correlations are the same. Formally, \(\epsilon_{0i}\), \(\epsilon_{1i}\), and \(u_i\) are trivariate normal with covariance:

\[\begin{equation}
\left[\begin{matrix}
1 & \rho_{01} & \rho_{t} \cr
\rho_{01} & 1 & \rho_{t} \cr
\rho_{t} & \rho_{t} & 1
\end{matrix}\right]
\nonumber
\end{equation}\]

The correlation \(\rho_{01}\) cannot be identified because we never observe both \(y_{0i}\) and \(y_{1i}\). However, identification of \(\rho_{01}\) is not necessary to estimate the other parameters, because we will observe the covariates and outcome in observations from each treatment group.

The log-likelihood for observation \(i\) is

\[\begin{eqnarray*}
\ln L_i = & & {\bf 1}(y_i =1 \mbox{ and } t_i = 1) \ln \Phi_2({\bf x}_i{\boldsymbol \beta}_1, {\bf z}_i{\boldsymbol \gamma},\rho_t) + \cr
& & {\bf 1}(y_i=0 \mbox{ and } t_i=1)\ln \Phi_2(-{\bf x}_i{\boldsymbol \beta}_1, {\bf z}_i{\boldsymbol \gamma},-\rho_t) + \cr
& & {\bf 1}(y_i=1 \mbox{ and } t_i=0) \ln \Phi_2({\bf x}_i{\boldsymbol \beta}_0, -{\bf z}_i{\boldsymbol \gamma},-\rho_t) + \cr
& & {\bf 1}(y_i=0 \mbox{ and } t_i = 0)\ln \Phi_2(-{\bf x}_i{\boldsymbol \beta}_0, -{\bf z}_i{\boldsymbol \gamma},\rho_t)
\end{eqnarray*}\]

where \(\Phi_2\) is the bivariate normal cumulative distribution function.

This model is a variation of the bivariate probit model. For a good introduction to the bivariate probit model, see Pindyck and Rubinfeld (1998).

The data

We will simulate data from a probit model with an endogenous treatment and then estimate the parameters of the model using mlexp. Then, we will use margins to estimate the ATE. We simulate a random sample of 10,000 observations.

. set seed 3211

. set obs 10000
number of observations (_N) was 0, now 10,000

. gen x = rnormal() + 4

. gen b = rpoisson(1)

. gen z = rnormal()

First, we generate the regressors. The variable \(x\) has a normal distribution with a mean of 4 and variance of 1. It is used as a regressor for the outcome and treatment. The variable \(b\) has a Poisson distribution with a mean of 1 and will be used as a treatment regressor. A standard normal variable \(z\) is also used as a treatment regressor.

. matrix cm = (1, .3,.7 \ .3, 1, .7 \ .7, .7, 1)

. drawnorm ey0 ey1 et, corr(cm)

. gen t = .5*x - .1*b + .4*z - 2.4 + et > 0

. gen y0 = .6*x - .8 + ey0 > 0

. gen y1 = .3*x - 1.2 + ey1 > 0

. gen y = (1-t)*y0 + t*y1

Next, we draw the unobserved errors. The potential outcome and treatment errors will have correlation \(.7\). We generate the errors using the drawnorm command. Finally, the outcome and treatment indicators are created.

Estimating the model parameters

Now, we will use mlexp to estimate the parameters of the probit model with an endogenous treatment. As in the previous post, we use the cond() function to calculate different values of the likelihood based on the different values of \(y\) and \(t\). We use the factor variable operator ibn on \(t\) in equation y to allow for a different intercept at each level of \(t\). An interaction between \(t\) and \(x\) is also specified in equation y. This allows for a different coefficient on \(x\) at each level of \(t\). We also specify vce(robust) so that we can use vce(unconditional) when we use margins later.

. mlexp (ln(cond(t,cond(y,binormal({y: i.t#c.x ibn.t},            ///
>                                  {t: x b z _cons}, {rho}),      /// 
>                         binormal(-{y:},{t:}, -{rho})),          ///
>                  cond(y,binormal({y:},-{t:},-{rho}),            ///
>                         binormal(-{y:},-{t:},{rho})))))         ///
>         , vce(robust)

initial:       log pseudolikelihood = -13862.944
alternative:   log pseudolikelihood = -15511.071
rescale:       log pseudolikelihood = -13818.369
rescale eq:    log pseudolikelihood = -10510.488
Iteration 0:   log pseudolikelihood = -10510.488  (not concave)
Iteration 1:   log pseudolikelihood = -10004.946  
Iteration 2:   log pseudolikelihood = -9487.4032  
Iteration 3:   log pseudolikelihood = -9286.0118  
Iteration 4:   log pseudolikelihood =  -9183.901  
Iteration 5:   log pseudolikelihood = -9181.9207  
Iteration 6:   log pseudolikelihood = -9172.0256  
Iteration 7:   log pseudolikelihood = -9170.8198  
Iteration 8:   log pseudolikelihood = -9170.7994  
Iteration 9:   log pseudolikelihood = -9170.7994  

Maximum likelihood estimation

Log pseudolikelihood = -9170.7994               Number of obs     =     10,000

------------------------------------------------------------------------------
             |               Robust
             |      Coef.   Std. Err.      z    P>|z|     [95% Conf. Interval]
-------------+----------------------------------------------------------------
y            |
       t#c.x |
          0  |   .5829362   .0223326    26.10   0.000     .5391651    .6267073
          1  |   .2745585   .0259477    10.58   0.000     .2237021     .325415
             |
           t |
          0  |  -.7423227   .0788659    -9.41   0.000     -.896897   -.5877483
          1  |  -1.088765   .1488922    -7.31   0.000    -1.380589   -.7969419
-------------+----------------------------------------------------------------
t            |
           x |   .4900691   .0148391    33.03   0.000     .4609851    .5191532
           b |  -.1086717   .0132481    -8.20   0.000    -.1346375   -.0827059
           z |   .4135792   .0150112    27.55   0.000     .3841579    .4430006
       _cons |  -2.354418   .0640056   -36.78   0.000    -2.479867   -2.228969
-------------+----------------------------------------------------------------
        /rho |   .7146737   .0377255    18.94   0.000     .6407331    .7886143
------------------------------------------------------------------------------

Our parameter estimates are close to their true values.

Estimating the ATE

The ATE of \(t\) is the expected value of the difference between \(y_{1i}\) and \(y_{0i}\), the average difference between the potential outcomes. Using the law of iterated expectations, we have

\[\begin{eqnarray*}
E(y_{1i}-y_{0i}) &=& E\{E(y_{1i}-y_{0i}|{\bf x}_i)\} \cr
&=& E\{\Phi({\bf x}_i{\boldsymbol \beta}_1)-
\Phi({\bf x}_i{\boldsymbol \beta}_0)\}
\end{eqnarray*}\]

This can be estimated as a predictive margin.

Now, we estimate the ATE using margins. We specify the normal probability expression in the expression() option. The xb() term refers to the linear prediction of the first equation, which we can now predict in Stata 14.1. We specify r.t so that margins will take the difference of the expression under \(t=1\) and \(t=0\). We specify vce(unconditional) to obtain standard errors for the population ATE rather than the sample ATE. The contrast(nowald) option is specified to omit the Wald test for the difference.

. margins r.t, expression(normal(xb())) vce(unconditional) contrast(nowald)

Contrasts of predictive margins

Expression   : normal(xb())

--------------------------------------------------------------
             |            Unconditional
             |   Contrast   Std. Err.     [95% Conf. Interval]
-------------+------------------------------------------------
           t |
   (1 vs 0)  |  -.4112345   .0248909     -.4600197   -.3624493
--------------------------------------------------------------

We estimate that the ATE of \(t\) on \(y\) is \(-.41\). So taking the treatment decreases the probability of a positive outcome by \(.41\) on average over the population.

We will compare this estimate to the sample difference of \(y_{1}\) and \(y_{0}\).

. gen diff = y1 - y0

. sum diff

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
        diff |     10,000      -.4132    .5303715         -1          1

In our sample, the average difference of \(y_{1}\) and \(y_{0}\) is also \(-.41\).

Conclusion

I have demonstrated how to estimate the parameters of a model with a complex likelihood function: the probit model with an endogenous treatment using mlexp. See [R] mlexp for more details about mlexp. I have also demonstrated how to use margins to estimate the ATE for the probit model with an endogenous treatment. See [R] margins for more details about margins.

Reference

Pindyck, R. S., and D. L. Rubinfeld. 1998. Econometric Models and Economic Forecasts. 4th ed. New York: McGraw-Hill.