Nested logit choice probabilities

As discussed in section 3.1 one of the main drawbacks of the multinomial logit model is the independence from irrelevant alternatives property. The nested logit (NL) model relaxes the IIA property by dividing the choice alternatives into different subsets or nests, allowing the IIA property to hold within each nest but not across nests. In other words, the ratio of the probabilities of two alternatives in different nests may depend on the attributes of the other alternatives in these two nests. The ratio of the probabilities of two alternatives in the same nest, however, will not depend on the attributes of the other alternatives.

Figure 3.2. An example of a nested logit decision tree

Public transport

Car Bus Train

Figure 3.2 above is an example of a Nested Logit decision “tree”. The alternatives that are likely to be close substitutes (bus and train) are specified to belong to the same nest. By relaxing the IIA property, the cross elasticities with respect to bus frequency are allowed to differ between the car and train modes. Hence the nested structure in

figure 3.2 accounts for the a priori belief that an increase in the probability of n choosing bus comes more from ti ain than car.

In order to give a more formal description of the nested logit model it is conceptually helpful to divide the representative utility function into two parts: one which varies between nests but not between alternatives within a nest, W„ky and one which varies between alternatives within the nest, Y„j. The utility individual n derives from choosing alternative j belonging to nest Bi is thus given by:

^nj = ^«/ + Yy + (3.11)

If the income of an individual is thought to influence her choice between private or public transport but not the choice between bus and train, for example, the income variable would enter W„i rather than Y„j. The cost of the bus and train modes on the other hand would enter Y,y since it is relevant for the choice between the modes. It should be noted that it is not uncommon to have all explanatory variables enter T„y, since they may all be thought to influence the choice between alternatives within nests. Since Y,^j = - W„, for any Wni, however, (3.11) is a fully general specification (Train, 2003). The decomposition of representative utility is paiiicularly useful when modelling multidimensional choices such as in a joint car ownership and mode choice model (see the next section and chapter 5). In this case the variables relating to the car ownership decision would be specified to enter W„i while the variables influencing mode choice enter Y„j.

McFadden (1978a) shows that if the unobserved components of the random utility function, are assumed to be distributed according to a particular generalised extreme value (GEV) distribution, the probability that individual n chooses alternative / belonging to nest Bk is given by:

P>ù - (3.12) where. 1(4,.,.,r,, e ^ ’’“ K,\j>, = r ;r (3.14) 4-<ye% and, 4 = ln E ,.« /^ (3.15)

In words the probability of choosing alternative / in nest equals the marginal probability of choosing nest B^ multiplied by the conditional probability of choosing alternative i given that B^ is chosen. The forms of the marginal and the conditional probabilities are both multinomial logit, and the nested logit model is therefore the product of two multinomial logit models. ^

The key feature of the nested logit model is that the scale of the multinomial logit models in equations (3.13 - 3.14) are allowed to differ. If the scale factors of the conditional model, —, and the marginal models, — , all equal 1 the nested logit

fj. Xf

reduces to the multinomial logit model (hence the multinomial logit model is “nested” within the NL model). Equation (3.15), which is the log of the denominator in (3.14), is often called the “inclusive value” or “log-sum term” (Ben-Akiva, 1972). The

^ It should be pointed out that the nested logit can have more than two levels. It is straightforward to describe a model with three or more levels using the framework outlined above. For the present purposes, however, the nested logit with two levels will suffice.

product of the scale factor, and the inclusive value can be interpreted as the 4

expected utility the individual recieves from choosing nest Bk analogous to the discussion of welfare analysis above.

In the context of the nested-logit model the inverse of the scale factor, , is often called the dissimilarity parameter^ since it measures the (dis)similarity between the unobserved portions of utility for alternatives within the same nest. Ben-Akiva and Lerman (1985) show that 1 is a measure of the degree of correlation among the unobserved poitions of utility for alternatives in nest Bi. Thus, when the dissimilaiity parameter equals 1 the degree of correlation between the alternatives in a nest is zero (and if this is the case in all K nests the nested logit model reduces to the multinomial logit model as discussed above).

As in the multinomial logit model one of the scale parameters must be normalized to 1 for identification purposes. It is common to impose the normalisation on the scale parameter of the upper (marginal) model such that // = 1, and this normalisation will be used in the following discussion.^ Daly and Zachary (1978) and McFadden (1978b) show that the nested logit model is globally consistent with utility maximisation if:

0<A, <1 fbrall ZeÆ (3.16)

Borch-Supan (1990) argues that this condition is unnecessarily sti’ong given that the NL model should be viewed as a local approximation. Based on the work of Borch- Supan, Herriges and Kling (1996) and Gil-Molto and Hole (2004) derive necessary

^ Koppelmann and Wen (1997), Hunt (2000) and Hensher and Greene (2002) give an overview of alternative normalisations of the nested logit model, including the so-called non-normalised nested

conditions for local consistency with utility maximization for two-level and three- level NL models respectively.