26For an accessible discussion, see J. Scott Long, Regression Models for Categorical and Limited Dependent Variables, Sage Publications, Newbury Park, California, 1997, pp. 102-113.
606 PART THREE: TOPICS IN ECONOMETRICS
value in our example is 0.3740.27 Like R2, RMcF also ranges between 0 and 1. Another comparatively simple measure of goodness of fit is the count R2, which is defined as:
2 number of correct predictions Count R2 =-:-:--:--(15.8.2)
total number of observations
Since the regressand in the logit model takes a value of 1 or zero, if the predicted probability is greater than 0.5, we classify that as 1, but if it is less than 0.5, we classify that as 0. We then count the number of correct predictions and compute the R2 as given in (15.7.2). We will illustrate this shortly.
It should be noted, however, that in binary regressand models, goodness of fit is of secondary importance. What matters is the expected signs of the regression coefficients and their statistical and/or practical significance.
4. To test the null hypothesis that all the slope coefficients are simultaneously equal to zero, the equivalent of the F test in the linear regression model is the likelihood ratio (LR) statistic. Given the null hypothesis, the LR statistic follows the x2 distribution with df equal to the number of explanatory variables, three in the present example. (Note: Exclude the intercept term in computing the df).
Now let us interpret the regression results given in (15.8.1). Each slope coefficient in this equation is a partial slope coefficient and measures the change in the estimated logit for a unit change in the value of the given re-gressor (holding other regressors constant). Thus, the GPA coefficient of 2.8261 means, with other variables held constant, that if GPA increases by a unit, on average the estimated logit increases by about 2.83 units, suggesting a positive relationship between the two. As you can see, all the other re-gressors have a positive effect on the logit, although statistically the effect of TUCE is not significant. However, together all the regressors have a significant impact on the final grade, as the LR statistic is 15.40, whose p value is about 0.0015, which is very small.
As noted previously, a more meaningful interpretation is in terms of odds, which are obtained by taking the antilog of the various slope coefficients. Thus, if you take the antilog of the PSI coefficient of 2.3786 you will get 10.7897 (~ e23786). This suggests that students who are exposed to the new method of teaching are more than 10 times likely to get an A than students who are not exposed to it, other things remaining the same.
Suppose we want to compute the actual probability of a student getting an A grade. Consider student number 10 in Table 15.7. Putting the actual data for this student in the estimated logit model given in Table 15.8, the reader can check that the estimated logit value for this student is 0.8178.
"Technically, this is defined as: 1 — (LLFur/LLFr), where LLFur is the unrestricted log likelihood function where all regressors are included in the model and LLFr is the restricted log likelihood function where only the intercept is included in the model. Conceptually, LLFur is equivalent to RSS and LLFr is the equivalent to TSS of the linear regression model.
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Using Eq. (15.5.2), the reader can easily check that the estimated probability is 0.69351. Since this student's actual final grade was an A, and since our logit model assigns a probability of 1 to a student who gets an A, the estimated probability of 0.69351 is not exactly 1 but close to it.
Recall the count R2 defined earlier. Table 15.9 gives you the actual and predicted values of the regressand for our illustrative example. From this table you can observe that, out of 32 observations, there were 6 incorrect predictions (students 14, 19, 24, 26, 31, and 32). Hence the count R2 value is 26/32 = 0.8125, whereas the McFadden R2 value is 0.3740. Although these two values are not directly comparable, they give you some idea about the orders of magnitude. Besides, one should not overplay the importance of goodness of fit in models where the regressand is dichotomous.
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