Qualitative and Limited Dependent Variable Models ECON 6002 Econometrics Memorial University of Newfoundland Adapted from Vera Tabakova’s notes

 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

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: 1. Results of opinion surveys in which responses can be strongly disagree, disagree, neutral, agree or strongly agree. 2. 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.

 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!

Example:

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.

Figure 16.2 Ordinal Choices Relation to Thresholds

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.

The types of questions we can answer with this model are: 1. 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,

2. 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

3. 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:

Ordered Logit vs Ordered Probit

Why is the second case more different than the first? Ordered Logit vs Ordered Probit 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 slop coefficients on he 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 = In standard STATA 9 for our example, too big for student version

Assumption of parallel regressions A Wald test, that can identify the Problem variables

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

Slide Principles of Econometrics, 3rd Edition  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

Further models  Survival analysis (time-to-event data analysis)  Multivariate probit (biprobit, triprobit, mvprobit)

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.