Heteroscedastic Extreme-Value Model

### Heteroscedastic Extreme-Value Model

The heteroscedastic extreme-value (HEV) model (Bhat 1995) allows the random components of the utility function to be nonidentical.
Specifically, the HEV model assumes independent but nonidentical error terms distributed with the Type I extreme-value distribution.
The HEV model allows the variances of the random components of utility to differ across alternatives. Bhat (1995) argues that
the HEV model does not have the IIA property. The HEV model contains the conditional logit model as a special case. The probability
that an individual will choose alternative from the set of available alternatives is

where the choice set has elements and

are the cumulative distribution function and probability density function of the Type I extreme-value distribution. The variance
of the error term for the th alternative is . If the scale parameters, , of the random components of utility of all alternatives are equal, then this choice probability is the same as that of the
conditional logit model. The log-likelihood function of the HEV model can be written as

where

Since the log-likelihood function contains an improper integral function, it is computationally difficult to get a stable
estimate. With the transformation , the probability can be written

Using the Gauss-Laguerre weight function, , the integration of the log-likelihood function can be replaced with a summation as follows:

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