Higher-Level Nested Logit Model
The nested logit model defined in the preceding section can be regarded as implying two levels of nesting because the responses are classified into S groups and each group is further classified into the individual elements. In this section we shall consider a three-level nested logit. A generalization to a higher level can be easily deduced from the three-level case.
Figure 9.1 shows examples of two-level and three-level nested logit models for the case of eight responses.
Following McFadden (1981), we can generalize (9.3.58) to the three-level case by defining
Део. е,,. . . ,ej
: exp (-? L?. Ls. яр ^чГТ)’
Then (9.3.59) and (9.3.60) can be generalized to
pry-,-Id – exp (ft/A)_.
as 2 exP (A//A)
2 ^ – M – —T——————- V-.
je * 2 a* 2exp &*//a)
тєс, L/eBt – I
A three-step estimation method can be defined in which, first, the parameters in (9.3.64) are estimated by maximizing the product of the terms of the form (9.3.64). Then these estimates are inserted into (9.3.65), and the product of the terms of the form (9.3.65) is maximized to yield the estimates of its parameters. Finally, (9.3.66) is maximized after inserting the estimates obtained by the first and second steps.