Choice model simulation

Simulating choice models”” requires attention to issues that arise when the researcher attempts to simulate some choice models with an actual data set.

Defining 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:

1. 0 , 1, More than 1 laptop
2. 0 , 1 , 2 , More than 2 laptops
3. Less than 2 , 2 , 3 , 4 , More than 4 laptops

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:

$U_{i}=\alpha P_{i}+\beta D_{i}+\varepsilon _{i}\,$

$U_{a}=\alpha P_{a}+\beta D_{a}+\varepsilon _{a}\,$

where $\varepsilon$ 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

$P_{i}=Prob(U_{i}>U_{a})=Prob(\alpha P_{i}+\beta D_{i}+\varepsilon _{i}>\alpha P_{a}+\beta D_{a}+\varepsilon _{a})=Prob(\varepsilon _{i}-\varepsilon _{a}>\alpha P_{a}+\beta D_{a}-\alpha P_{i}-\beta D_{i})$

Denoting the observed portion of the utility function as V,

$P_{i}=Prob(\varepsilon _{i}-\varepsilon _{a}>V_{a}-V_{i})$

In the end, discrete choice modeling comes down to specifying the distribution of $\varepsilon$ (or \varepsilon_i - \varepsilon_a) and solving the integral over the range of $\varepsilon$ to calculate $P_{i}$. Extending this to more general situations with

1. N consumers (n = 1, 2, …, N),
2. J choices of consumption (j = 1, 2, … , J),

The choice probability of consumer n choosing j can be written as

$P_{nj}=Prob(U_{nj}>U_{mni})$ for all i other than j

General Models

1. Logit
2. GEV (Generalized Extreme-Value
3. Probit
4. Mixed Logit

References

A Nevo (2000). "Practitioners Guide to Estimation of Random Coefficients Logit Models of Demand,” Journal of Economics & Management Strategy, 9(4), 513-548

Kenneth E. Train, " Discrete Choice Methods with Simulation", Massachusetts: Cambridge University Press, 2003.