choice probability of consumer n choosing j can be written as (4) for all i other than j Identification 1. What's irrelevant From equation (4), it's obvious that does not change as long as the inequality in the probability argument on the right side stays the same. In other words, adding or multiplying by a constant to all does not change the choice probably, thus no change in interpretation. 2. Alternative-specific constants Unlike adding a constant to all the utilities, adding alternative-specific constants does alter the choice probabilities. Suppose alternative-specific constants Ci and Ca are added to (1) and (2): Then, depending on the value of the estimated alternative-specific constants, the choice probability may change. Also, if we write the choice probability in the format of (3), only the difference between affects the choice probability (i.e. our estimation can only identify the difference). So it's convenient to normalize all the alternative-specific constants to one of the alternatives. If we normalize to , then we estimate the following model: When there are more than 2 choices in the choice set, we can pick any choice i and normalize the alternative-specific constants to that choice by subtracting from all other alternative-specific constants. 3. Sociodemographic variables In deciding between the Irish pub and the American pub, if the researcher has access to additional sociodemographic variables such as income, they can enter the consumer utility equation in various ways. Denote the student's income as Y. If the researcher believes that the income affects the utility linearly, then If the researcher believes that the sociodemographic variable interacts with other variable such as price, then the utility can be written as General models As mentioned earlier, calculation and justification of choice probabilities rely on the properties of the error (i.e. the unobservables) distribution function the researcher specifies. Here is the quick overview of frequently used models that each differs in specification 1. Logit: Assumes unobserved factors have the same variance with zero correlation across alternatives. iid extreme value unobserved factors The cumulative distribution of difference in extreme values is Logistics function Logistics function has a closed form solution => No simulation necessary. 2. GEV (Generalized extreme value distribution) Allows correlation in unobserved factors across alternatives. iid extreme value unobserved factors The cumulative distribution of difference in extreme values is Logistics function Logistics function has a closed form solution => No simulation necessary. 3. Probit Unobserved factors