3 Logit Choice Probabilities By far the easiest and most widely used discrete choice model is logit. Its popularity is due to the fact that the formula for the choice probabilities takes a closed form and is readily interpretable. Originally, the logit formula was derived by Luce (1959) from assumptions about the characteristics | P1 GEM IKJ CB495-03Drv P2 GEM IKJ QC GEM ABE CB495 Train KEY BOARDED T1 GEM August 20 2002 12 14 Char Count 0 3 Logit Choice Probabilities By far the easiest and most widely used discrete choice model is logit. Its popularity is due to the fact that the formula for the choice probabilities takes a closed form and is readily interpretable. Originally the logit formula was derived by Luce 1959 from assumptions about the characteristics of choice probabilities namely the independence from irrelevant alternatives IIA property discussed in Section . Marschak 1960 showed that these axioms implied that the model is consistent with utility maximization. The relation of the logit formula to the distribution of unobserved utility as opposed to the characteristics of choice probabilities was developed by Marley as cited by Luce and Suppes 1965 who showed that the extreme value distribution leads to the logit formula. McFadden 1974 completed the analysis by showing the converse that the logit formula for the choice probabilities necessarily implies that unobserved utility is distributed extreme value. In his Nobel lecture McFadden 2001 provides a fascinating history of the development of this path-breaking model. To derive the logit model we use the general notation from Chapter 2 and add a specific distribution for unobserved utility. A decision maker labeled n faces J alternatives. The utility that the decision maker obtains from alternative j is decomposed into 1 a part labeled Vnj that is known by the researcher up to some parameters and 2 an unknown part snj that is treated by the researcher as random Unj Vnj snj V j .The logit model is obtained by assuming that each enj is independently identically distributed extreme value. The distribution is also called Gumbel and type I extreme value and sometimes mistakenly Weibull . The density for each unobserved component of utility is f 8n e-enje e-ni and the cumulative distribution is F 8nj . 38 P1 GEM IKJ P2 .
