2.5 Replication of econometric studies  (Page 7/10)

8. The next step in the process is to generate the term we will use in the estimation of the grade regression to account for the potential sample selection bias. To do this we will need to find a reference in the literature that offers a clear description of what we need to do. As it turns out, a reasonable explanation of the appropriate estimation technique is available in Jimenez and Kugler (1987). Since much of what follows comes directly from this article, I highly recommend you read it yourself.

The gist of the method suggests that the potential sample bias is accounted for by an inverse Mills ratio for each of the categories. What we need to do is calculate:

for the category that the individual actually is in. What we will do is calculate (6) for all of the categories and then sum the product of this number and a dummy variable indicating if a course is the highest math class completed by an individual. Since the dummy variables will equal 0 for math categories an individual is not in, the resulting sum will preserve the value of (6) that is associated with the category the individual does belong to.

It is clear from (6) that we will need to retain the 6 cutoffs. We can do this with the commands:

. generate cutoff1 = _b[_cut1]

. generate cutoff2 = _b[_cut2]

. generate cutoff3 = _b[_cut3]

. generate cutoff4 = _b[_cut4]

. generate cutoff5 = _b[_cut5]

. generate cutoff6 = _b[_cut6]

Technically, this step is not necessary since the parameter estimates are preserved until the next regression is estimated; I suggest doing this purely as a precaution.

9. Preserve the predicted values of the ordered-probit using the command:

. predict zhat, xb

. predict phat1 phat2 phat3 phat4 phat5 phat6 phat7, p

These two commands will generate for each observation the predicted mean category of math classes and the probability that this individual will fall in each category. To see what is going on we will retrieve some representative values of these variables and then graph them for one individual. Table 7 reports these values for 10 individuals in the sample. Now consider individual 2. Fitting a normal distribution with a mean of 4.25 and using the critical values from our estimation yields the probabilities that the individual is in each of the categories. For example, the probability that individual 1 will have completed no math classes is equal to 0.1223. Figure 5 illustrates the results for individual 1. The dashed vertical lines are the six cutoff values that are the same for each individual. The solid vertical line is the zhat for individual 1. The heavy blue line represents the normal probability density function for this individual. While, there is, of course, a different probability distribution for each individual, the cutoff values are the same for all members of the sample.