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## Chapter 16

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### Chapter 16

Econometrics

Memorial University of Newfoundland

Qualitative and Limited Dependent Variable Models

Adapted from Vera Tabakova’s notes

Chapter 16: Qualitative and Limited Dependent Variable Models

- 16.1 Models with Binary Dependent Variables
- 16.2 The Logit Model for Binary Choice
- 16.3 Multinomial Logit
- 16.4 Conditional Logit
- 16.5 Ordered Choice Models
- 16.6 Models for Count Data
- 16.7 Limited Dependent Variables

Principles of Econometrics, 3rd Edition

16.5 Ordered Choice Models

The choice options in multinomial and conditional logit models have no natural ordering or arrangement. However, in some cases choices are ordered in a specific way. Examples include:

- Results of opinion surveys in which responses can be strongly disagree, disagree, neutral, agree or strongly agree.
- Assignment of grades or work performance ratings. Students receive grades A, B, C, D, F which are ordered on the basis of a teacher’s evaluation of their performance. Employees are often given evaluations on scales such as Outstanding, Very Good, Good, Fair and Poor which are similar in spirit.

Principles of Econometrics, 3rd Edition

16.5 Ordered Choice Models

- When modeling these types of outcomes numerical values are assigned to the outcomes, but the numerical values are ordinal, and reflect only the ranking of the outcomes
- The distance between the values is not meaningful!

Principles of Econometrics, 3rd Edition

16.5 Ordered Choice Models

The usual linear regression model is not appropriate for such data, because in regression we would treat the y values as having some numerical meaning when they do not.

Principles of Econometrics, 3rd Edition

16.5.1 Ordinal Probit Choice Probabilities

Principles of Econometrics, 3rd Edition

16.5.1 Ordinal Probit Choice Probabilities

Figure 16.2 Ordinal Choices Relation to Thresholds

Principles of Econometrics, 3rd Edition

16.5.1 Ordinal Probit Choice Probabilities

Principles of Econometrics, 3rd Edition

16.5.1 Ordinal Probit Choice Probabilities

Principles of Econometrics, 3rd Edition

16.5.1 Ordinal Probit Choice Probabilities

Principles of Econometrics, 3rd Edition

16.5.2 Estimation and Interpretation

The parameters are obtained by maximizing the log-likelihood function using numerical methods. Most software includes options for both ordered probit, which depends on the errors being standard normal, and ordered logit, which depends on the assumption that the random errors follow a logistic distribution.

Principles of Econometrics, 3rd Edition

16.5.2 Estimation and Interpretation

The types of questions we can answer with this model are:

- What is the probability that a high-school graduate with GRADES = 2.5 (on a 13 point scale, with 1 being the highest) will attend a 2-year college? The answer is obtained by plugging in the specific value of GRADES into the predicted probability based on the maximum likelihood estimates of the parameters,

Principles of Econometrics, 3rd Edition

16.5.2 Estimation and Interpretation

- What is the difference in probability of attending a 4-year college for two students, one with GRADES = 2.5 and another with GRADES = 4.5? The difference in the probabilities is calculated directly as

Principles of Econometrics, 3rd Edition

16.5.2 Estimation and Interpretation

- If we treat GRADES as a continuous variable, what is the marginal effect on the probability of each outcome, given a 1-unit change in GRADES? These derivatives are:

Principles of Econometrics, 3rd Edition

16.5.3 An Example

Principles of Econometrics, 3rd Edition

16.5.3 An Example

Slide16-18

Ordered Logit vs Ordered Probit

Slide16-19

Ordered Logit vs Ordered Probit

Why is the second case more different than the first?

Why is the second case more different than the first?

Postestimation

But remember that there is no meaningful numerical interpretation behind the values of the dependent variable in this model

There are many useful postestimations commands you should consider to understand and report your results (see, e.g. Long and Freese)

Assumption of parallel regressions

Ordered Logit is known as the proportional-odds model because the odds ratio of the event is independent of the category j. The odds ratio is assumed to be constant for all categories

These models assume that the effect of the slope coefficients on the switch from every category to the next is about the same

Assumption of parallel regressions

You should test if the assumption is tenable

This test is sensitive to the number of cases. Samples with larger numbers of cases are more likely to show a statistically significant test

Assumption of parallel regressions

You should test if the assumption is tenable

Approximate likelihood-ratio test of proportionality of odds

across response categories:

chi2(1) = 0.18

Prob > chi2 = 0.6679

In standard STATA 9 for our example, too big for student version

Assumption of parallel regressions

If the assumption fails, you will have to consider other methods

Multinomial Logit

Stereotype model (mclest in STATA)

Generalized ordered logit model (gologit)

Continuation ratio model

Keywords

- binary choice models
- censored data
- conditional logit
- count data models
- feasible generalized least squares
- Heckit
- identification problem
- independence of irrelevant alternatives (IIA)
- index models
- individual and alternative specific variables
- individual specific variables
- latent variables
- likelihood function
- limited dependent variables
- linear probability model
- logistic random variable
- logit
- log-likelihood function
- marginal effect
- maximum likelihood estimation
- multinomial choice models
- multinomial logit
- odds ratio
- ordered choice models
- ordered probit
- ordinal variables
- Poisson random variable
- Poisson regression model
- probit
- selection bias
- tobit model
- truncated data

Principles of Econometrics, 3rd Edition

References

Hoffmann, 2004 for all topics

Long, S. and J. Freese for all topics

Cameron and Trivedi’s book for count data

Agresti, A. (2001) Categorical Data Analysis (2nd ed). New York: Wiley.

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