Qualitative%20and%20Limited%20Dependent%20Variable%20Models

1
Chapter 16
ECON 6002 Econometrics Memorial University of
Newfoundland

• Qualitative and Limited Dependent Variable Models

Adapted from Vera Tabakovas notes
2
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

3
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 teachers 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.

4
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!

5
16.5 Ordered Choice Models

• Example

6
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.

(16.26)
7
16.5.1 Ordinal Probit Choice Probabilities

8
16.5.1 Ordinal Probit Choice Probabilities

• Figure 16.2 Ordinal Choices Relation to Thresholds

9
16.5.1 Ordinal Probit Choice Probabilities
10
16.5.1 Ordinal Probit Choice Probabilities
11
16.5.1 Ordinal Probit Choice Probabilities
12
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.

13
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,

14
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

15
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

16
16.5.3 An Example

17
16.5.3 An Example

Slide16-17
Principles of Econometrics, 3rd Edition
18
16.5.3 An Example

Slide16-18
19
Ordered Logit vs Ordered Probit

Slide16-19
20
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?
21
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)

22
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 slop
  coefficients on he switch from every category to
  the next is about the same

23
Assumption of parallel regressions
24
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

25
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 gt chi2 0.6679
In standard STATA 9 for our example, too big for
student version
26
Assumption of parallel regressions
A Wald test, that can identify the Problem
variables
27
Assumption of parallel regressions
28
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

29
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

30
Further models

• Survival analysis (time-to-event data analysis)
• Multivariate probit (biprobit, triprobit,
  mvprobit)

31
References

• Hoffmann, 2004 for all topics
• Long, S. and J. Freese for all topics
• Cameron and Trivedis book for count data
• Agresti, A. (2001) Categorical Data Analysis (2nd
  ed). New York Wiley.