1. M.Sc. in Economics Econometrics Module I
Topic 6: Multiple Response Models
Carol Newman

2. Ordered Choice Models Ordered choice models
As with binary choice models we assume one underlying latent variable
Matching of latent variable to observed variable takes account of the multiple ordered choices
μ‘s are unknown parameters that are estimated with β 2

3. Ordered Choice Model
The probability that a particular outcome is chosen will be the probability that the latent variable is within the specified boundaries
Assuming F is a standard normal cumulative distribution function we have the ordered probit model, assuming a logistic distribution we have an ordered logit model 3

4. Ordered Choice Model Ordered probit model estimated using MLE
Interpretation of the coefficients:
Note: Parallel Regression Assumption required 4

5. Unordered Choice Models
Assume an individual is faced with J+1 choices with arbitrary ordering
The utility level associated with each choice is given by
Alternative j is chosen by individual i if it yields the highest utility level. i.e.
Utility levels are unobserved but assume:
The probability that category j is chosen is given by: 5

6. Unordered Choice Models
Assume eij are independent with a log Weibull distribution then the probability that option j is chosen is given by: 6

7. Unordered Choice Models
Model 1: explanatory variables are individual specific and invariant over choices
Unique parameters not identified without some normalisation. Let β0=0
This is the multinomial logit model 7

8. Unordered Choice Models
Estimation using MLE:
where dij=1 where individual i chooses option j
Interpretation: 8

9. Unordered Choice Models
Independence of irrelevant alternatives
Assuming that the error terms associated with each choice are independent implies that the utility levels associated with any two alternatives are independent of the utility levels associated with any other two alternatives
Example: difference in utility between being low skill jobs and professional jobs the same as the difference in utility between high skill jobs and professional jobs
Test using Hausman test where s are estimators from restricted subset of choices and f are estimators from model with full set of choices 9

10. Unordered Choice Models
Model 2: Data consist of choice specific attributes instead of individual specific caracteristics:
Mixed logit model:
Multinomial logit model has also been extended to allow for layers of choices with the aim of relaxing the IIA assumption among broad groups but allowing it to hold within groups of similar alternatives 10