Note: These results are not corrected for heteroscedasticity (see exercise 15.12).
Interpretation of the Probit Estimates in Table 15.11. How do we interpret the preceding results? Suppose we want to find out the effect of a unit change in X (income measured in thousands of dollars) on the probability that Y = 1, that is, a family purchases a house. To do this, look at Eq. (15.9.2). We want to take the derivative of this function with respect to X (that is, the rate of change of the probability with respect to income). It turns out that this derivative is:
31The following results are not corrected for heteroscedasticity. See exercise 15.12 for the appropriate procedure to correct heteroscedasticity.
32We use the chain rule of derivatives:
612 PARTTHREE: TOPICS IN ECONOMETRICS
where f (p1 + p2Xi) is the standard normal probability density function evaluated at p1 + p2Xi. As you will realize, this evaluation will depend on the particular value of the X variables. Let us take a value of X from Table 15.5, say, X = 6 (thousand dollars). Using the estimated values of the parameters given in Table 15.11, we thus want to find the normal density function at f[—1.0166 + 0.04846(6)] = f(—0.72548). If you refer to the normal distribution tables, you will find that for Z = —0.72548, the normal density is about 0.3066.33 Now multiplying this value by the estimated slope coefficient of 0.04846, we obtain 0.01485. This means that starting with an income level of $6000, if the income goes up by $1000, the probability of a family purchasing a house goes up by about 1.4 percent. (Compare this result with that given in Table 15.6.)
As you can see from the preceding discussion, compared with the LPM and logit models, the computation of changes in probability using the probit model is a bit tedious.
Instead of computing changes in probability, suppose you want to find the estimated probabilities from the fitted gprobit model. This can be done easily. Using the data in Table 15.11 and inserting the values of X from Table 15.5, the reader can check that the estimated n.i.d. values (to two digits) are as follows:
Estimated n.i.d. -0.72 -0.63 -0.53 -0.39 -0.29 -0.05 0.19 0.43 0.68 0.92
Now statistical packages such as Minitab can easily compute the (cumulative) probabilities associated with the various n.i.d.'s. For example, corresponding to an n.i.d. value —0.63, the estimated probability is 0.2647 and, corresponding to an n.i.d. value of 0.43, the estimated probability is 0.6691. If you compare these estimates with the actual values given in Table 15.5, you will find that the two are fairly close, suggesting that the fitted model is quite good. Graphically, what we have just done is already shown in Figure 15.4.
Let us revisit Table 15.7, which gives data on 32 individuals about their final grade in intermediate microeconomics examination in relation to the variables GPA, TUCE, and PSI. The results of the logit regression are given in Table 15.8. Let us see what the probit results look like. Notice that as in the case of the logit model for individual data, we will have to use a nonlinear estimating procedure based on the method of maximum likelihood. The regression results calculated by Eviews 4 are given in Table 15.13.
33Note that the standard normal Z can range from —to to +to, but the density function f (Z) is always positive.
CHAPTER FIFTEEN: QUALITATIVE RESPONSE REGRESSION MODELS 613
TABLE 15.13 Dependent Variable: grade Method: ML—binary probit Convergence achieved after 5 iterations
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