Classical Discrete Choice Theory

ECON 721

Petra Todd

• Usual regression model untenable if applied to discrete choices

• Need 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…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

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

• Thurstone (1929) -Quandt multivariate probit model– Very general substitution possibilities– Allows for general forms of heterogeneity– More difficult to compute– Identifiability less easily established

Luce/McFadden Conditional Logit Model

• References: Manski and McFadden, Chapter 5; Greene, Chapter 21; Amemiya, Chapter 9

• Notation

• 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

Can now derive multinomial logit (MNL) form

Random Utility Models (RUMs)

• First proposed by Thurstone (1920’s,1930’s). Link between Luce model and RUM established by Marshak (1959)

Marshak (1959) result

• Assuming weibull errors, get Luce logit model from RUM framework

• 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}

The forecast problem

• Suppose want to forecast demand for a new good

Debreu Red-Bus-Blue-Bus Critique

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

How do you know if a PCS is consistent with a RUM?

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

Example: Choice of Transportation mode

• Neighborhood m, transportation mode t