Finding choice probabilities by using utility with logit and probit models

Question

I am using a formula to calculate the utility, which is as follows:

v_{ij} = 1 - x*beta + delta_i + e_{ij}

delta_i ~ N(0,phi^2)
e_ij ~ N(0,sigma^2)

v_{ij} is the utility of state ij, x is the vector of dummy variables, beta is the vector of regression coefficients, delta is the i-th random error and e_{ij} is the usual error term.

So, let's say there are two alternatives to make a choice from and their utility functions are

    v_{i1} = 1 - x_{i1}*beta + delta_i + e_{i1} 

and

    v_{i2} = 1 - x_{i2}*beta + delta_i + e_{i2} .

I am wondering how I could calculate the probability of choosing the first option.

I know I need to use this

P(Choosing the first) = P( v_{i1} >  v_{i2}) 
= ( 1 -x_{i1}*beta + delta_i + e_{i1}  > 1 - x_{i2}*beta+ delta_i + e_{i2} )

Can anyone please help me find this probability by using both logit and probit models.

Answer 1

A good reference is Train's text which is free online.

http://elsa.berkeley.edu/books/choice2.html

Start by rearranging your terms.

$P($Choosing the first) = $P(v_{i1} > v_{i2}) = P(1-x \beta + \delta_i + e_{i1} > 1 - x \beta + \delta_i + e_{i2}) = P(e_{i1} - e_{i2} > 0)$

To evaluate this expression you need to consider the distribution of $e_{i}^{*} = e_{i1} - e_{i2}$.

While you've already assumed that $e_{i}$ is normally distributed, this assumption gives you the probit. The logit model comes out of assuming a type I extreme value distribution for the individual error terms (since then $e_{i}^{*}$ follows a Logistic distribution.)