2.3 Sample selectivity bias  (Page 5/6)

The interpretation of the numbers in Table 5 is straightforward. Consider individual 1. The z-value predicted for this individual is -0.68. Using the standard normal tables reported in Table 11 it is easy to see:

The difference between this number and the value reported for phat in Table 5 is due to rounding error.

A little later we will want to calculate the inverse Mills ratio. As noted in (8), the formula for the inverse Mills ratio is:

The variable phat is equal to $\Phi \left(z_i{}^{\prime }\widehat{\nu }\right).$ Stata offers an easy way to calculate $\varphi \left(z_i{}^{\prime }\widehat{\nu }\right)$ with the function “normden(zbhat)” as follows:

.generate imratio = normden(zbhat)/phat

Table 6 repeats Table 5 with the estimate of the inverse Mills ratio for the first 10 observations.

The two heckman estimates

One of the great advantages of using an econometrics program like Stata is that the authors quite often have created a command that does all of the work for the user. In our case, the commands we need to run to generate the maximum likelihood estimate of the Heckman model are:

. global wage_eqn wage educ age

. global seleqn married children age education

. heckman $wage_eqn, select($seleqn)

Notice that we have used the global command to create a shortcut for referring to each of the two equations in the estimation. The command for the Heckman two-stage estimate is:

.heckman $wage_eqn, select($seleqn) twostage

.predict mymills, mills

The second command reports the estimates of the inverse Mills ratio; we have retrieved these values in order to check our earlier calculations. Table 7 reports the results of these two estimations. Column 2 reports the maximum-likelihood estimates; Column 3 reports the Heckman two-step estimates; and Column 3 reports the probit estimate of selection equation as reported in Table 4. The estimates for the two methods are very similar. Of course, the probit estimates in Column 4 exactly match the results reported for the selection equation in Column 3. As a final check, Table 8 reports the values of the inverse Mills ratio reported in Table 6 with the values of the inverse Mills ratio calculated in the Heckman two-step method. The two estimates are identical except for some rounding errors.