In order to compute the probabilities in the previous example, we have
arbitrarily decided to constraint 
to 1. Alternatively, we could
have decided to constraint 
to 1. It is easy to show that,
in this case, we have
and
which is equivalent to (51) and (52),
replacing 
by .
A model where the scale parameter is arbitrarily constrained to 1 is said to be
``normalized from the top''. A model where one of the parameters 
is
constrained to 1 is said to be ``normalized from the bottom''.
The latter may produce a simpler formulation of the model. We
illustrate it using the
example of Figure 11.
Figure 11: A mode choice example
We have
and
If we impose 
, we can define 
, 
, 
and 
to obtain the following expressions.
and
This formulation, proposed by Daly (1987), simplifies the estimation process. For this reason, it has been adopted in estimation packages like ALOGIT (Daly, 1987) or HieLoW (Bierlaire, 1995, Bierlaire and Vandevyvere, 1995).
We emphasize here that this formulation should be used with caution when the same parameters are present in more than one nest. In this case, specific techniques, inspired from artificial trees proposed by Bradley and Daly (1991) must be used to obtain a correct specification of the model. The description of these techniques is out of the scope of this paper.
A direct extension of the nested logit model consists in partionning some or all nests into sub-nests, which can, in turn, be divided into sub-nests. Because of the complexity of these models, their structure is usually represented as a tree, as suggested by Daly (1987). Clearly, the number of potential structures, reflecting the correlation among alternatives, can be very large. No technique has been proposed thus far to identify the most appropriate correlation structure directly from the data.
We conclude our introduction of nested logit models by mentioning their limitations. These models are designed to capture choice problems where alternatives within each nest are correlated. No correlation across nests can be captured by the Nested Logit Model. When alternatives cannot be partitioned into well separated nests to reflect their correlation, Nested Logit Models are not applicable. This is the case for most route choice problems. Several models within the ``logit family'' have been designed to capture specific correlation structures. For example, Cascetta (1996) captures overlapping paths in a route choice context using commonality factors, Koppelman and Wen (1997) capture correlation between pair of alternatives, and Vovsha (1997) proposes a cross-nested model allowing alternatives to belong to more than one nest. The two last models are derived from the Generalized Extreme Value model, presented in the next section.