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. Usual regression model untenable if applied to discrete choicesNeed to think of the ingredients that give rise to choices.. The Forecast problem. Suppose we want to forecast demand for a new good and we observe consumption data on old goods, x1
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1. Classical Discrete Choice Theory ECON 721
Petra Todd
2. Usual regression model untenable if applied to discrete choices
Need to think of the ingredients that give rise to choices.
3. The Forecast problem Suppose we want to forecast demand for a new good and we observe consumption data on old goods, x1…xI.
Each good might represent a type of car or a mode of transportation, for example.
McFadden wanted to forecast demand for San Fransisco BART subway
Need to find way of putting new good on a basis with the old
4. Two predominant modeling approaches Luce (1953)-McFadden conditional logit model
Widely used in economics
Easy to compute
Identifiability of parameters well understood
Restrictive substitution possibilities among goods
5. Thurstone (1929) -Quandt multivariate probit model
Very general substitution possibilities
Allows for general forms of heterogeneity
More difficult to compute
Identifiability less easily established
6. Luce/McFadden Conditional Logit Model References: Manski and McFadden, Chapter 5; Greene, Chapter 21; Amemiya, Chapter 9
Notation
7.
We might assume there is a distribution of choice rules, because
In observation we lost some information governing chocies
There can be random variaion in choices due to unmeasured psychological factors
Define the probability that an individual drawn randomly from the population with attributes x and alternatives set B chooses x
Luce Axioms – maintain some restrictions on P(x|s,B) and derive implications for functional form of P
13. Can now derive multinomial logit (MNL) form
14. Random Utility Models (RUMs) First proposed by Thurstone (1920’s,1930’s). Link between Luce model and RUM established by Marshak (1959)
18. Marshak (1959) result Assuming weibull errors, get Luce logit model from RUM framework
19. Weibull is sufficient, but not necessary
Yellot (1977) showed that if we require invariance of choice probabilities under uniform expansions of the choice set, then weibull is the only distribution that yields logistic form
{coffee, tea, milk}
{coffee,tea,milk,coffee,tea,milk}
20. The forecast problem Suppose want to forecast demand for a new good
22. Debreu Red-Bus-Blue-Bus Critique
23. Criteria for a good probability choice system (PCS) Flexible functional form
Computationally practical
Allows for flexibility in representing substitution patterns
Is consistent with a RUM
24. How do you know if a PCS is consistent with a RUM?
25. Daly-Zachary-Williams Theorem Provide a set of conditions that make it easier to derive a PCS from an RUM for a class of models called generalized extreme value models
McFadden shows that under certain assumptions about the form of V, the DZW result can be seen as a form of Roy’s identity
34. Example: Choice of Transportation mode Neighborhood m, transportation mode t