Decision Models in Skiable Areas - EPFL

3 METHODOLOGY - DISCRETE CHOICE MODELS 6
As seen in the previous section the deterministic part includes an alternative specific
constant. The mean of the random variable can be added to the ASC’s. So that in
continuation, the means of the random variables are always supposed to be zero.
The variance can be chosen arbitrary. The argumentation is also illustrated in the
same example as above. Suppose that the random terms ε i 1 resp. ε i 2 are replaced by
αε i 1 resp. αε i 2 with α > 0. Indeed, we have
P(V1 − V2 ≥ αε2 − αε1) = P( 1
α (V1 − V2) ≥ ε2 − ε1) (7)
so that the deterministic term, in fact the coefficients βi, is modified by a factor 1
α .
Usually, two different distributions for the random term are used. They define two
different families of models.
First, the normal or Gauß distribution leads to the family of the Probit models. In
this models, it is assumed that the random terms are normally distributed, with mean
zero. The density function is
fnormal(x) = 1
σ √ 2π
1 x
e− 2 ( σ )2
with σ a positive scale parameter. Indeed, σ 2 is the variance. It is important to notice
that the different ε i a don’t have to be independent and can be correlated.
In the second case, the family of the Logit models, it is assumed that random the
term is independent and identically Gumbel distributed. The density function is
fgumbel(x) = µe −µ(x−η) e −eµ(x−η)
where η is the location parameter and µ the scale parameter. This two families are
explained precisely in the next sections
3.5 Probit Models
3.5.1 Multinomial Probit Model
As mentioned above, the Probit models assumes a normal distribution for the errors.
As consequence and advantage, it allows to capture all correlations between different
alternatives because they are not supposed to be independent. Due to the assumption
that C is finite, the elements of C can be numerated and as consequence C is of the
form C = {a1, a2, . . . , an}. The utility function U i a, ∀a ∈ C, can be written in a vector
form. We define −→ U i , −→ V i and −→ εi componentwise by −→ U i (j) = U i aj , −→ V i (j) = V i
aj and
−→
i i ε (j) = εaj . Then we have
−→
U i = −→ V i + −→ ε i
(10)
where −→ ε i follows a multivariate normal distribution of mean zero and variance covariance
matrix Σ i . So that
PC(k) = P(U i (k) ≥ U i (j)) ∀j ∈ C (11)
The disadvantage of the Probit Models is the normal distribution. Since the analytic
formula of the distribution fonction does not exist, it has to be approximated and causes
a lot of problems.
(8)
(9)