Functional Form: Logit and Probit
Having pointed out the problems with considering the functional form F as linear, we turn to two popular functional forms of F, the logit and the probit. These two c. d.f.’s differ only in the tails, and the logit resembles the c. d.f. of a t-distribution with 7 degrees of freedom, whereas the probit is the normal c. d.f., or that of a t with ж degrees of freedom. Therefore, these two forms will give similar predictions unless there are an extreme number of observations in the tails.
We will use the conventional notation T(z) = /1^ ф(п)йп, where ф(г) = e-Z/2^J2n for —ж < z < ж, for the probit. Also, Л(г) = ez/(1 + ez) = 1/(1 + e-z) for —ж < z < +ж, for the logit. Some results that we will use quite often in our derivations are the following: dT/dz = ф, and йЛ/dz = Л(1 — Л). The p. d.f. of the logistic distribution is the product of its c. d.f. and one minus this c. d.f. Therefore, the marginal effects considered above for a general F are respectively,
дФ(х’ів )/dxk = фівк
дЛ(х’ів)/дхк = Лі(1 — Лі)вк
where фі = ф(хів) and Лі = Л(хів).
One has to be careful with the computation of partial derivatives in case there is a dummy variable among the explanatory variables. For such models, one should compute the marginal effects of a change in one unit of a continuous variable хк for both values of the dummy variable.
Illustrative Example: Using the probit model, suppose that the probability of joining a union is estimated as follows: 7ті = Ф(2.5 — 0.06 WKSі + 0.95 OCCі) where WKS is the number of weeks worked and OCC = 1, if the individual is in a blue-collar occupation, and zero otherwise. Weeks worked in this sample range from 20 to 50. From (13.7), the marginal effect of one extra week of work on the probability of joining the union is given by:
For Blue-Collar Workers: —0.06 ф[2.5 — 0.06 WKS + 0.95]
= —0.06 ф[2.25] = —0.002 at WKS = 20 = —0.06 ф[1.35] = —0.010 at WKS = 35 = —0.06 ф[0.45] = —0.022 at WKS = 50
For Non Blue-Collar Workers: —0.06 ф[2.5 — 0.06 WKS]
= —0.06 ф[1.3] = —0.010 at WKS = 20 = —0.06 ф[0.4] = —0.022 at WKS = 35 = —0.06 ф[—0.5] = —0.021 at WKS = 50
Note how different these marginal effects are for blue-collar versus non blue-collar workers even for the same weeks worked. Increasing weeks worked from 20 to 21 reduces the probability of joining the union by 0.002 for a Blue-Collar worker. This is compared to five times that amount for a Non Blue-Collar worker.