Conditional Logic

Abstract

Multinomial Models: Multinomial Logit, Conditional Logit, Nested Logit, Multinomial Probit, and Random Coe¢ cients/Parameters (" Mixed " 1) Logit Discrete (qualitative) response models deal with discrete dependent variables. It is strongly recommended that you read and study Train (2009): ordered: self-rated heath status (excellent/good/bad)) latent index models categorical (mutually exclusive): transportation mode) random utility models 1 Multinomial Logit (MNL) These are the models for discrete choice with more than two alternatives which are not ordered. Suppose that the discrete response variable y 2 f0; 1; 2; ; Jg. Examples are travel modes (bus/train/car), employment status (employed/unemployed/out-of-labor-force), marital status (single/married/divorced/widowed) and many others. We wish to model the distribution of y in terms of covariates. In some cases we need to distinguish between covariates that vary by units i (individuals or …rms), denoted by x i (i = 1; 2; ; N), and covariates that vary by alternative j (and possibly individual i), denoted by x ij (i = 1; 2; ; N ; j = 0; 1; ; J). Examples of the …rst type— case speci…c covariates— include individual characteristics such as age, gender, and income. An example of the second type— alternative speci…c covariates— is the cost (or price) associated with each alternative, such as the cost of commuting by bus or train or car. McFadden developed the interpretation of these models through utility maximizing choice behavior. In that case we may be willing to impose restrictions on how covariates a¤ect alternatives— costs of a particular alternative a¤ect the utility of that alternative, but not the utility of other alternatives. The modelling strategy is to write out a likelihood function based on each individual i's conditional probability of choosing j given the covariates x ij , Pr(y i = jjx ij) = p ij (x ij ;), parameterized by : ln L() = ln N Q i=1 J Q j=0 Pr(y i = jjx ij) 1fy i =jg ! = N P i=1 J P j=0 1fy i = jg ln p ij (x ij ;). If regressors do not vary across alternatives (x ij x i), then the multinomial logit (MNL) model is used. MNL speci…es the individual response probability as: Pr(y i = jjx i) = exp(x 0 i j) P J k=0 exp(x 0 i k) , where i = 1; ; N ; j = 0; ; J; and 0 = 0.