Choice Model Simulation - Identification

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 C_i and C_a 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