## Info

LR statistic (3 df) = 15.5458 Probability (LR stat) =0.0014

Variable Coefficient Std. error t statistic Probability

GPA 0.4638 0.1619 2.8640 0.0078

TUCE 0.0104 0.0194 0.5386 0.5943

PSI 0.3785 0.1391 2.7200 0.0110

R2 = 0.4159 Durbin-Watson d = 2.3464 F statistic = 6.6456

"Qualitatively," the results of the probit model are comparable with those obtained from the logit model in that GPA and PSI are individually statistically significant. Collectively, all the coefficients are statistically significant, since the value of the LR statistic is 15.5458 with a p value of 0.0014. For reasons discussed in the next sections, we cannot directly compare the logit and probit regression coefficients.

For comparative purposes, we present the results based on the linear probability model (LPM) for the grade data in Table 15.14. Again, qualitatively, the LPM results are similar to the logit and probit models in that GPA and PSI are individually statistically significant but TUCE is not. Also, together the explanatory variables have a significant impact on grade, as the F value of 6.6456 is statistically significant because itsp value is only 0.0015.

The Marginal Effect of a Unit Change in the Value of a Regressor in the Various Regression Models

In the linear regression model, the slope coefficient measures the change in the average value of the regressand for a unit change in the value of a regressor, with all other variables held constant.

In the LPM, the slope coefficient measures directly the change in the probability of an event occurring as the result of a unit change in the value of a regressor, with the effect of all other variables held constant.

614 PARTTHREE: TOPICS IN ECONOMETRICS

In the logit model the slope coefficient of a variable gives the change in the log of the odds associated with a unit change in that variable, again holding all other variables constant. But as noted previously, for the logit model the rate of change in the probability of an event happening is given by PjPi (1 — Pi), where Pj is the (partial regression) coefficient of the jth re-gressor. But in evaluating Pi, all the variables included in the analysis are involved.

In the probit model, as we saw earlier, the rate of change in the probability is somewhat complicated and is given by Pj f (Zi), where f (Zi) is the density function of the standard normal variable and Zi = p1 + p2 X2i + ••• + pkXki, that is, the regression model used in the analysis.

Thus, in both the logit and probit models all the regressors are involved in computing the changes in probability, whereas in the LPM only the jth regressor is involved. This difference may be one reason for the early popularity of the LPM model.