Choice model simulation

Although the concept choice models is widely understood and practiced these days, it is often difficult to acquire hands-on knowledge in simulating choice models. While many stat packages provide useful tools to simulate, researchers attempting to test and simulate new choice models with data often encounter problems from as simple as scaling parameter to misspecification. This article goes beyond simply defining discrete choice models. Rather, it aims at providing a comprehensive overview of how to simulate such models in computer.
Defining choice set.
When a researcher has some consumer choice data in his/her hand and tries to construct a choice model and simulate it against the data, he/she needs to first define a choice set. A Choice Set in discrete choice models is defined to be finite, exhaustive, and mutually exclusive.
For instance, consider households' choice of how many laptops to own. The researcher can define the choice set depending on the nature of the data and the interpretation they wish to draw, as long as it satisfies three properties mentioned above. Some examples of choice sets that meet the categories are the following:
Defining consumer utility.
Suppose a student is trying to decide which pub he/she should go for a beer after his/her last final exam. Suppose there are two pubs in the town of the college: an Irish pub and an American pub. The researcher wishes to predict which pub he/she will choose based on the price (P) of beer and the distance (D) to each pub, assuming they are known to the researcher. Then, the consumer utilities for choosing the Irish pub and the American pub can be defined:
where formula_3 captures unobserved variables that affect consumer utilities.
Defining choice probabilities.
Once the consumer utilities have been specified, the researcher can derive choice probabilities. Namely, the probability of the student choosing the Irish pub over the American pub is
Denoting the observed portion of the utility function as V,
In the end, discrete choice modeling comes down to specifying the distribution of formula_3 (or formula_7) and solving the integral over the range of formula_3 to calculate formula_9. Extending this to more general situations with
The choice probability of consumer "n" choosing "j" can be written as
for all "i" other than "j"
Identification.
1. What's irrelevant
From equation (4), it's obvious that formula_11 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 formula_12 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 formula_16 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 formula_17, 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 formula_20 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:
2. GEV (Generalized extreme value distribution)
3. Probit
4. Mixed logit