Classical Discrete Choice Theory

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Classical Discrete Choice Theory

• ECON 721
• Petra Todd

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• Usual regression model untenable if applied to
  discrete choices
• Need to think of the ingredients that give rise
  to choices.

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The Forecast problem

• Suppose we want to forecast demand for a new good
  and we observe consumption data on old goods,
  x1xI.
• 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

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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

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

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Luce/McFadden Conditional Logit Model

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

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• 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(xs,B) and derive implications for functional
  form of P

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Can now derive multinomial logit (MNL) form
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Random Utility Models (RUMs)

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

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Marshak (1959) result

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

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• 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

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The forecast problem

• Suppose want to forecast demand for a new good

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Debreu Red-Bus-Blue-Bus Critique
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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

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

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Example Choice of Transportation mode

• Neighborhood m, transportation mode t

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