Table 1515 Values Of Cumulative Probability Functions

Cumulative normal

Cumulative logistic

V2n J-c e-s2'zds o.ooia 0.0228 0.0ee8

0.i587 0.a085 0.5000 0.e9i5 0.84ia 0.9aa2 0.9772 0.9987

0.0474 0.ii92 0.i824 0.2e89 0.a775 0.5000 0.e225 0.73ii 0.8i7e 0.8808 0.952e

Though the models are similar, one has to be careful in interpreting the coefficients estimated by the two models. For example, for our grade example, the coefficient of GPA of 1.6258 of the probit model and 2.8261 of the logit model are not directly comparable. The reason is that, although the standard logistic (the basis of logit) and the standard normal distributions (the basis of probit) both have a mean value of zero, their variances are different; 1 for the standard normal (as we already know) and n2/3 for the logistic distribution, where n ~ 22/7. Therefore, if you multiply the probit coefficient by about 1.81 (which is approximately = n/V3), you will get approximately the logit coefficient. For our example, the probit coefficient of GPA is 1.6258. Multiplying this by 1.81, we obtain 2.94, which is close to the logit coefficient. Alternatively, if you multiply a logit coefficient by 0.55 ( = 1/1.81), you will get the probit coefficient. Amemiya, however, suggests multiplying a logit estimate by 0.625 to get a better estimate of the corresponding probit estimate.34 Conversely, multiplying a probit coefficient by 1.6(= 1/0.625) gives the corresponding logit coefficient.

Incidentally, Amemiya has also shown that the coefficients of LPM and logit models are related as follows:

and fiLPM = 0.25^iogit except for intercept

We leave it to the reader to find out if these approximations hold for our grade example.

34T. Amemiya, "Qualitative Response Model: A Survey," Journal of Economic Literature, vol. 19, 1981, pp. 481-536.

616 PARTTHREE: TOPICS IN ECONOMETRICS

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