Advanced Econometrics Takeshi Amemiya
Generalized Extreme-Value Model
McFadden (1978) introduced the generalized extreme-value (GEV) distribution defined by
F(€ue2,. . . ,ej (9.3.67)
= exp {—(/[exp (-eO, exp (-e2), . . . , exp ( Cm)]), where G satisfies the conditions,
|
(i) |
G(ux, u2,. . . |
., мт) ё 0, |
«і, «2. • • • > мтё0. |
|
(ІІ) |
G(olu1,olu2, |
. . ., aum) |
= otG(ux, u2,. . ., uj. |
|
(iii) |
> A |
if к is odd if к is even, k= 1, |
|
|
duhduh. . . |
_ != U dulk SO |
If Uj — fij + €j and the alternative with the highest utility is chosen as before, (9.3.67) implies the GEV model
„ exp {/ii)G,[ep (fij), exp (fi2. . ., exp (jMm)]
/-УТ / / / | > (У. З.Оо)
3 (/[exp (fj. il exp (fj2),. . ., exp (yUm)]
where Gj is the derivative of G with respect to its yth argument.
Both the nested logit model and the higher-level nested logit model discussed in the preceding sections are special cases of the GEV model. The only known application of the GEV model that is not a nested logit model is in a study by Small (1981).
The multinomial models presented in the subsequent subsections do not belong to the class of GEV models.
