By Carlos Daganzo
Multinomial Probit
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Extra info for Multinomial Probit. The Theory and its Application to Demand Forecasting
Example text
This approach is particularly useful for problems with three alternatives, since in that case Eq. 23) is a line integral that can be solved with just a few evaluations of the integrand by using, say, Simpson's rule. Andrews and Langdon (1976) suggested this approach for modal-split problems involving three transportation modes but did not develop a calibration method. Hausman and Wise (1978) seem to have been the first researchers to calibrate a MNP model; they used a numerical-integration technique similar to this one.
Ofk. 16b) 44 2 Maximum-Likelihood Estimation: Computational Aspects where V-3 is the yth element of V and o]k is the (j, fe)th element of Σ^. The choice probability can be found by calculating Ρί(Υ,Σξ) = ΡΓ(1/ £ > l/7;Y/ # i| ν , Σ ξ } = Pr{Z,· < 0;ν/|Ζ,Σ ζ }, which, if Σ ζ has full rank, can be expressed as 1 Γ Jzi=-oo · · · f° J S / _ I = - G O [ ( 2 π ) / - 1 | Σ ζ | ] - 1 ' 2 β χ ρ [ - Κ 2 ! 17) which is an / — 1 dimensional integral giving the value of the cumulative distribution function of Z, at z = 0.
In the car-purchasing example mentioned at the beginning of this section, it would be reasonable to postulate that if M is large, the price of automobiles would increase with the number of cars purchased by the population according to the standard principle of economic supply. Since the price of automobiles is an attribute of any reasonable car-purchasing model and the price is not known beforehand, one cannot obtain a prediction directly. However, one could obtain, at least conceptually, a function relating the figures of merit previously discussed (y,S, T) to price π [we write y(n\ 5(π), etc.