ECON 6002
Econometrics
Memorial University of Newfoundland
Qualitative and Limited Dependent Variable
Models
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
Principles of Econometrics, 3rd Edition
Slide 16-2
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.
Principles of Econometrics, 3rd Edition
Slide16-3

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
Slide16-4
Example:
1
2

y  3
4

5
strongly disagree
disagree
neutral
agree
strongly agree
Principles of Econometrics, 3rd Edition
Slide16-5
3 4-year college (the full college experience)

y  2 2-year college (a partial college experience)
1 no college

(16.26)
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
Slide16-6
yi*  GRADESi  ei
3 (4-year college) if

y  2 (2-year college) if
1 (no college)
if

Principles of Econometrics, 3rd Edition
yi*   2
1  yi*   2
yi*  1
Slide16-7
Figure 16.2 Ordinal Choices Relation to Thresholds
Principles of Econometrics, 3rd Edition
Slide16-8
P  yi  1  P  yi*  1   P GRADESi  ei  1 
 P  ei  1  GRADESi 
   1   GRADESi 
Principles of Econometrics, 3rd Edition
Slide16-9
P  yi  2  P 1  yi*   2   P 1   GRADESi  ei   2 
 P 1  GRADESi  ei   2   GRADESi 
    2   GRADESi     1   GRADESi 
Principles of Econometrics, 3rd Edition
Slide16-10
P  yi  3  P  yi*   2   P GRADESi  ei   2 
 P  ei   2   GRADESi 
 1     2   GRADESi 
Principles of Econometrics, 3rd Edition
Slide16-11
L  , 1 , 2   P  y1  1  P  y2  2  P  y3  3
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
Slide16-12
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 2year 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,



P  y  2 | GRADES  2.5   2  2.5   1  2.5
Principles of Econometrics, 3rd Edition

Slide16-13
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
P  y  2 | GRADES  4.5  P  y  2 | GRADES  2.5
Principles of Econometrics, 3rd Edition
Slide16-14
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:
P  y  1
   1   GRADES  
GRADES
P  y  2
 
 1  GRADES      2  GRADES  

GRADES
P  y  3
    2   GRADES  
GRADES
Principles of Econometrics, 3rd Edition
Slide16-15
. tab psechoice
no college
= 1, 2 =
2-year
college, 3
= 4-year
college
Freq.
Percent
Cum.
1
2
3
222
251
527
22.20
25.10
52.70
22.20
47.30
100.00
Total
1,000
100.00
Principles of Econometrics, 3rd Edition
Slide16-16
. oprobit
Iteration
Iteration
Iteration
Iteration
psechoice grades
0:
1:
2:
3:
log
log
log
log
likelihood
likelihood
likelihood
likelihood
=
=
=
=
-1018.6575
-876.21962
-875.82172
-875.82172
Ordered probit regression
Number of obs
LR chi2(1)
Prob > chi2
Pseudo R2
Log likelihood = -875.82172
psechoice
Coef.
Std. Err.
grades
-.3066252
.0191735
/cut1
/cut2
-2.9456
-2.089993
.1468283
.1357681
Principles of Econometrics, 3rd Edition
z
-15.99
P>|z|
0.000
=
=
=
=
1000
285.67
0.0000
0.1402
[95% Conf. Interval]
-.3442045
-.2690459
-3.233378
-2.356094
-2.657822
-1.823893
Slide16-17
. oprobit
psechoice grades, nolog
Ordered probit regression
Number of obs
LR chi2(1)
Prob > chi2
Pseudo R2
Log likelihood = -875.82172
psechoice
Coef.
Std. Err.
grades
-.3066252
.0191735
/cut1
/cut2
-2.9456
-2.089993
.1468283
.1357681
z
-15.99
P>|z|
=
=
=
=
1000
285.67
0.0000
0.1402
[95% Conf. Interval]
0.000
-.3442045
-.2690459
-3.233378
-2.356094
-2.657822
-1.823893
. mfx, at(grades=6.64) predict(outcome(3))
Marginal effects after oprobit
y = Pr(psechoice==3) (predict, outcome(3))
= .52153319
variable
dy/dx
grades
-.1221475
Std. Err.
.00763
z
-16.00
P>|z|
[
95% C.I.
]
X
0.000
-.137108 -.107187
6.64
P>|z|
[
X
0.000
-.060813 -.046745
. mfx, at(grades=2.635) predict(outcome(3))
Marginal effects after oprobit
y = Pr(psechoice==3) (predict, outcome(3))
= .90008498
variable
dy/dx
grades
-.0537788
Std. Err.
.00359
z
-14.99
95% C.I.
]
2.635
Slide16-18
Ordered Logit vs Ordered Probit
. estimates table oprobit ologit, b(%7.2g) style(oneline)
Variable
oprobit
ologit
psechoice
grades
-.31
-.51
_cons
-2.9
-4.9
_cons
-2.1
-3.5
cut1
cut2
Slide16-19
Ordered Logit vs Ordered Probit
. mfx, at(grades=6.64) predict(outcome(3))
Marginal effects after ologit
y = Pr(psechoice==3) (predict, outcome(3))
= .52243824
variable
dy/dx
grades
-.1272801
Std. Err.
.00844
z
-15.08
P>|z|
[
95% C.I.
]
0.000
-.143827 -.110734
X
6.64
Why is the second case more different than the first?
. mfx, at(grades=2.635) predict(outcome(3))
Marginal effects after ologit
y = Pr(psechoice==3) (predict, outcome(3))
= .89406527
variable
dy/dx
grades
-.0483174
Std. Err.
.00348
z
-13.89
P>|z|
[
95% C.I.
0.000
-.055136 -.041498
Why is the second case more different than the first?
]
X
2.635


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)


Ordered Logit is known as the proportionalodds 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
. ologit
psechoice grades, nolog or
Ordered logistic regression
Number of obs
LR chi2(1)
Prob > chi2
Pseudo R2
Log likelihood = -877.29561
psechoice
Odds Ratio
Std. Err.
grades
.6004068
.0203699
/cut1
/cut2
-4.916916
-3.477195
.2672764
.2416765
z
-15.04
=
=
=
=
1000
282.72
0.0000
0.1388
P>|z|
[95% Conf. Interval]
0.000
.5617809
.6416884
-5.440768
-3.950872
-4.393064
-3.003518


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

You should test if the assumption is tenable
In standard STATA 9 for our example, too big for student version
Approximate likelihood-ratio test of proportionality of odds
across response categories:
chi2(1) =
0.18
Prob > chi2 = 0.6679
. ologit
psechoice grades, nolog
Ordered logistic regression
Number of obs
LR chi2(1)
Prob > chi2
Pseudo R2
Log likelihood = -877.29561
psechoice
Coef.
Std. Err.
grades
-.5101479
.0339269
/cut1
/cut2
-4.916916
-3.477195
.2672764
.2416765
z
-15.04
P>|z|
0.000
=
=
=
=
1000
282.72
0.0000
0.1388
[95% Conf. Interval]
-.5766433
-.4436525
-5.440768
-3.950872
-4.393064
-3.003518
. brant
Brant Test of Parallel Regression Assumption
Variable
chi2
p>chi2
df
All
0.20
0.654
1
grades
0.20
0.654
1
A Wald test, that can identify the
Problem variables
A significant test statistic provides evidence that the parallel
regression assumption has been violated.
. brant, detail
Estimated coefficients from j-1 binary regressions
grades
_cons
y>1
-.53665426
5.11967
y>2
-.51624365
3.5222116
Brant Test of Parallel Regression Assumption
Variable
chi2
p>chi2
df
All
0.20
0.654
1
grades
0.20
0.654
1
A significant test statistic provides evidence that the parallel
regression assumption has been violated.





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















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
Principles of Econometrics, 3rd Edition

















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
Slide 16-29


Survival analysis (time-to-event data
analysis)
Multivariate probit (biprobit, triprobit,
mvprobit)




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.