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Probit

Identification in Probit Model. In most probit models, the index function is linear in its parameters, so that and cannot be separately identified. Typically normalize . Logit Model. The standard logit model results if the errors are iid extreme value variates, with . This in turn yields .

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Probit

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    1. Probit

    2. Identification in Probit Model

    3. Logit Model

    4. Identification in the Logit Model

    5. Estimation: Nonlinear Least Squares

    6. Maximum Likelihood

    7. Maximum Likelihood (cont’d)

    8. Maximum Likelihood (cont’d)

    9. Maximum Likelihood (cont’d)

    10. Maximum Likelihood (cont’d)

    11. Maximum Likelihood Given that you have specified your model correctly, the resulting parameter estimates will be consistent estimates of the true parameters efficient (i.e., lowest variance) asymptotically normal

    12. Inference in Discrete Choice Models

    13. Fitted Choice Probabilities Issues What are the statistical properties of fitted choice probabilities? How does one aggregate over decision makers to make inference for the population?

    14. Taylor Series Expansion

    15. Taylor Series Expansion (cont’d)

    16. Simulation

    17. Example: Chicago Transit Authority Explanatory variables TW: walking time from nearest train stop to place of work (+) AIVTSS: Difference between drive time and train ride time (-) ACF: Difference between auto-parking charge and train fare (-) AW: Number of vehicles in the household (+)

    18. Estimated Parameters (MLE)

    19. Fitted Choice Probabilities

    20. Aggregation

    21. Train (2002) Example

    23. Errors in Aggregation

    24. Sample Enumeration In aggregating over individuals or projecting to the population as a whole, one needs to keep in mind Degree to which sample is representative of target population Endogeneities in sample selection Sample enumeration frequently used when sample is exogeneously determined Controlling for endogenous sampling is more difficult

    25. Sample Enumeration (cont’d)

    26. Marginal Effects

    27. Marginal Effects – Logit

    28. Marginal Effects Continued

    30. Logit and Probit Yield Similar Results

    31. Example #1: Greene

    33. For Transportation Study: Marginal Effects

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