Chapter 16
Qualitative and Limited Dependent Variable Models
Prepared by Vera Tabakova, East Carolina University
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
Slide 16-2Principles of Econometrics, 3rd Edition
16.1 Models with Binary Dependent Variables Examples:
An economic model explaining why some states in the United States have ratified the Equal Rights Amendment, and others have not.
An economic model explaining why some individuals take a second, or third, job and engage in “moonlighting.”
An economic model of why some legislators in the U. S. House of Representatives vote for a particular bill and others do not.
An economic model of why the federal government awards development grants to some large cities and not others.
Slide16-3Principles of Econometrics, 3rd Edition
16.1 Models with Binary Dependent Variables
An economic model explaining why some loan applications are accepted and others not at a large metropolitan bank.
An economic model explaining why some individuals vote “yes” for increased spending in a school board election and others vote “no.”
An economic model explaining why some female college students decide to study engineering and others do not.
Slide16-4Principles of Econometrics, 3rd Edition
16.1 Models with Binary Dependent Variables
If the probability that an individual drives to work is p, then
It follows that the probability that a person uses public
transportation is .
Slide16-5Principles of Econometrics, 3rd Edition
(16.1)
(16.2)
1 individual drives to work
0 individual takes bus to worky
1 .P y p
0 1P y p
1( ) (1 ) , 0,1y yf y p p y
; var 1E y p y p p
16.1.1 The Linear Probability Model
Slide16-6Principles of Econometrics, 3rd Edition
(16.3)
(16.5)
(16.4)
( )y E y e p e
1 2( )E y p x
1 2( )y E y e x e
16.1.1 The Linear Probability Model
One problem with the linear probability model is that the error term is
heteroskedastic; the variance of the error term e varies from one
observation to another.
Slide16-7Principles of Econometrics, 3rd Edition
y value e value Probability
1
0
1 21 x
1 2x
1 2p x
1 21 1p x
16.1.1 The Linear Probability Model
Using generalized least squares, the estimated variance is:
Slide16-8Principles of Econometrics, 3rd Edition
(16.6)
1 2 1 2var 1e x x
21 2 1 2ˆ var 1i i i ie b b x b b x
*
*
* 1 * *1 2
ˆ
ˆ
ˆ
i i i
i i i
i i i i
y y
x x
y x e
16.1.1 The Linear Probability Model
Problems:
We can easily obtain values of that are less than 0 or greater than 1.
Some of the estimated variances in (16.6) may be negative.
Slide16-9Principles of Econometrics, 3rd Edition
(16.7)
(16.8)
1 2p̂ b b x
2
dp
dx
p̂
16.1.2 The Probit Model
Figure 16.1 (a) Standard normal cumulative distribution function (b) Standard normal probability density function
Slide16-10Principles of Econometrics, 3rd Edition
16.1.2 The Probit Model
Slide16-11Principles of Econometrics, 3rd Edition
(16.9)
p̂
2.51( )
2zz e
2.51( ) [ ]
2uz
z P Z z e du
(16.10)1 2 1 2[ ] ( )p P Z x x
16.1.3 Interpretation of the Probit Model
where and is the standard normal probability
density function evaluated at
Slide16-12Principles of Econometrics, 3rd Edition
(16.11)1 2 2
( )( )
dp d t dtx
dx dt dx
1 2t x 1 2( )x
1 2 .x
16.1.3 Interpretation of the Probit Model
Equation (16.11) has the following implications:
1. Since is a probability density function its value is always
positive. Consequently the sign of dp/dx is determined by the sign of
2. In the transportation problem we expect 2 to be positive so that
dp/dx > 0; as x increases we expect p to increase.
Slide16-13Principles of Econometrics, 3rd Edition
1 2( )x
16.1.3 Interpretation of the Probit Model
2. As x changes the value of the function Φ(β1 + β2x) changes. The
standard normal probability density function reaches its maximum
when z = 0, or when β1 + β2x = 0. In this case p = Φ(0) = .5 and an
individual is equally likely to choose car or bus transportation.
The slope of the probit function p = Φ(z) is at its maximum when
z = 0, the borderline case.
Slide16-14Principles of Econometrics, 3rd Edition
16.1.3 Interpretation of the Probit Model
3. On the other hand, if β1 + β2x is large, say near 3, then the
probability that the individual chooses to drive is very large and
close to 1. In this case a change in x will have relatively little effect
since Φ(β1 + β2x) will be nearly 0. The same is true if β1 + β2x is a
large negative value, say near 3. These results are consistent with
the notion that if an individual is “set” in their ways, with p near 0 or
1, the effect of a small change in commuting time will be negligible.
Slide16-15Principles of Econometrics, 3rd Edition
16.1.3 Interpretation of the Probit Model
Predicting the probability that an individual chooses the alternative
y = 1:
Slide16-16Principles of Econometrics, 3rd Edition
(16.12)1 2ˆ ( )p x
ˆ1 0.5ˆ
ˆ0 0.5
py
p
16.1.4 Maximum Likelihood Estimation of the Probit Model
Suppose that y1 = 1, y2 = 1 and y3 = 0.
Suppose that the values of x, in minutes, are x1 = 15, x2 = 20 and x3 = 5.
Slide16-17Principles of Econometrics, 3rd Edition
(16.13)11 2 1 2( ) [ ( )] [1 ( )] , 0,1i iy y
i i i if y x x y
1 2 3 1 2 3( , , ) ( ) ( ) ( )f y y y f y f y f y
16.1.4 Maximum Likelihood Estimation of the Probit Model
In large samples the maximum likelihood estimator is normally
distributed, consistent and best, in the sense that no competing
estimator has smaller variance.
Slide16-18Principles of Econometrics, 3rd Edition
(16.14)
1 2 3[ 1, 1, 0] (1,1,0) (1) (1) (0)P y y y f f f f
1 2 3
1 2 1 2 1 2
[ 1, 1, 0]
[ (15)] [ (20)] 1 [ (5)]
P y y y
16.1.5 An Example
Slide16-19Principles of Econometrics, 3rd Edition
16.1.5 An Example
Slide16-20Principles of Econometrics, 3rd Edition
(16.15)1 2 .0644 .0299
(se) (.3992) (.0103) i iDTIME DTIME
1 2 2( ) ( 0.0644 0.0299 20)(0.0299)
(.5355)(0.0299) 0.3456 0.0299 0.0104
dpDTIME
dDTIME
16.1.5 An Example
If an individual is faced with the situation that it takes 30 minutes
longer to take public transportation than to drive to work, then the
estimated probability that auto transportation will be selected is
Since the estimated probability that the individual will choose to
drive to work is 0.798, which is greater than 0.5, we “predict” that
when public transportation takes 30 minutes longer than driving to
work, the individual will choose to drive.Slide16-21Principles of Econometrics, 3rd Edition
1 2ˆ ( ) ( 0.0644 0.0299 30) .798p DTIME
16.2 The Logit Model for Binary Choice
Slide16-22Principles of Econometrics, 3rd Edition
(16.16) 2( ) ,1
l
l
el l
e
(16.18)
(16.17) 1[ ]
1 ll p L l
e
1 21 2 1 2
1
1 xp P L x x
e
16.2 The Logit Model for Binary Choice
Slide16-23Principles of Econometrics, 3rd Edition
1 2
1 2
1 2
exp1
1 exp1 x
xp
xe
1 2
11
1 expp
x
16.3 Multinomial Logit
Examples of multinomial choice situations:1. Choice of a laundry detergent: Tide, Cheer, Arm & Hammer, Wisk,
etc. 2. Choice of a major: economics, marketing, management, finance or
accounting.3. Choices after graduating from high school: not going to college,
going to a private 4-year college, a public 4 year-college, or a 2-year college.
The explanatory variable xi is individual specific, but does not change across alternatives.
Slide16-24Principles of Econometrics, 3rd Edition
16.3.1 Multinomial Logit Choice Probabilities
Slide16-25Principles of Econometrics, 3rd Edition
(16.19a)
(16.19c)
(16.19b)
112 22 13 23
1, 1
1 exp expii i
p jx x
12 222
12 22 13 23
exp, 2
1 exp expi
ii i
xp j
x x
13 233
12 22 13 23
exp, 3
1 exp expi
ii i
xp j
x x
individual chooses alternative ijp P i j
16.3.2 Maximum Likelihood Estimation
Slide16-26Principles of Econometrics, 3rd Edition
11 22 33 11 22 33
12 22 1 13 23 1
12 22 2
12 22 2 13 23 2
13 23 3
12 22 3 13 23 3
12 22 13 23
1, 1, 1
1
1 exp exp
exp
1 exp exp
exp
1 exp exp
, , ,
P y y y p p p
x x
x
x x
x
x x
L
16.3.3 Post-Estimation Analysis
Slide16-27Principles of Econometrics, 3rd Edition
01
12 22 0 13 23 0
1
1 exp expp
x x
(16.20)3
2 21all else constant
im imim m j ij
ji i
p pp p
x x
1 1 1
12 22 13 23 12 22 13 23
1 1
1 exp exp 1 exp exp
b a
b b a a
p p p
x x x x
16.3.3 Post-Estimation Analysis
An interesting feature of the odds ratio (16.21) is that the odds of choosing
alternative j rather than alternative 1 does not depend on how many alternatives
there are in total. There is the implicit assumption in logit models that the odds
between any pair of alternatives is independent of irrelevant alternatives
(IIA). Slide16-28Principles of Econometrics, 3rd Edition
(16.21)
(16.22)
1 2
1
exp 2,31
ijij j i
i i
pP y jx j
P y p
1
2 1 2exp 2,3ij i
j j j ii
p px j
x
16.3.4 An Example
Slide16-29Principles of Econometrics, 3rd Edition
16.3.4 An Example
Slide16-30Principles of Econometrics, 3rd Edition
16.4 Conditional Logit
Example: choice between three types (J = 3) of soft drinks, say Pepsi,
7-Up and Coke Classic.
Let yi1, yi2 and yi3 be dummy variables that indicate the choice made
by individual i. The price facing individual i for brand j is PRICEij.
Variables like price are to be individual and alternative specific, because they vary from individual to individual and are different for each choice the consumer might make
Slide16-31Principles of Econometrics, 3rd Edition
16.4.1 Conditional Logit Choice Probabilities
Slide16-32Principles of Econometrics, 3rd Edition
(16.23)
individual chooses alternative ijp P i j
1 2
11 2 1 12 2 2 13 2 3
exp
exp exp expj ij
iji i i
PRICEp
PRICE PRICE PRICE
16.4.1 Conditional Logit Choice Probabilities
Slide16-33Principles of Econometrics, 3rd Edition
11 22 33 11 22 33
11 2 11
11 2 11 12 2 12 2 13
12 2 22
11 2 21 12 2 22 2 23
2 33
11 2 31 12 2
1, 1, 1
exp
exp exp exp
exp
exp exp exp
exp
exp exp
P y y y p p p
PRICE
PRICE PRICE PRICE
PRICE
PRICE PRICE PRICE
PRICE
PRICE PRICE
32 2 33
12 22 2
exp
, ,
PRICE
L
16.4.2 Post-Estimation Analysis
The own price effect is:
The cross price effect is:
Slide16-34Principles of Econometrics, 3rd Edition
(16.24)
(16.25)
21ijij ij
ij
pp p
PRICE
2ij
ij ikik
pp p
PRICE
16.4.2 Post-Estimation Analysis
The odds ratio depends on the difference in prices, but not on the prices
themselves. As in the multinomial logit model this ratio does not depend on
the total number of alternatives, and there is the implicit assumption of the
independence of irrelevant alternatives (IIA).
Slide16-35Principles of Econometrics, 3rd Edition
1 2
1 1 21 2
expexp
expj ijij
j k ij ikik k ik
PRICEpPRICE PRICE
p PRICE
16.4.3 An Example
Slide16-36Principles of Econometrics, 3rd Edition
16.4.3 An Example
The predicted probability of a Pepsi purchase, given that the price of
Pepsi is $1, the price of 7-Up is $1.25 and the price of Coke is $1.10
is:
Slide16-37Principles of Econometrics, 3rd Edition
11 2
1
11 2 12 2 2
exp 1.00ˆ .4832
exp 1.00 exp 1.25 exp 1.10ip
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:
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.
Slide16-38Principles of Econometrics, 3rd Edition
16.5 Ordered Choice Models
3. Standard and Poor’s rates bonds as AAA, AA, A, BBB and so on, as a judgment about the credit worthiness of the company or country issuing a bond, and how risky the investment might be.
4. Levels of employment are unemployed, part-time, or full-time.
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.
Slide16-39Principles of Econometrics, 3rd Edition
16.5 Ordered Choice Models
Example:
Slide16-40Principles of Econometrics, 3rd Edition
1 strongly disagree
2 disagree
3 neutral
4 agree
5 strongly agree
y
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.
Slide16-41Principles of Econometrics, 3rd Edition
(16.26)
3 4-year college (the full college experience)
2 2-year college (a partial college experience)
1 no college
y
16.5.1 Ordinal Probit Choice Probabilities
Slide16-42Principles of Econometrics, 3rd Edition
*i i iy GRADES e
*2*
1 2*
1
3 (4-year college) if
2 (2-year college) if
1 (no college) if
i
i
i
y
y y
y
16.5.1 Ordinal Probit Choice Probabilities
Figure 16.2 Ordinal Choices Relation to Thresholds
Slide16-43Principles of Econometrics, 3rd Edition
16.5.1 Ordinal Probit Choice Probabilities
Slide16-44Principles of Econometrics, 3rd Edition
*1 1
1
1
1i i i i
i i
i
P y P y P GRADES e
P e GRADES
GRADES
16.5.1 Ordinal Probit Choice Probabilities
Slide16-45Principles of Econometrics, 3rd Edition
*1 2 1 2
1 2
2 1
2i i i i
i i i
i i
P y P y P GRADES e
P GRADES e GRADES
GRADES GRADES
16.5.1 Ordinal Probit Choice Probabilities
Slide16-46Principles of Econometrics, 3rd Edition
*2 2
2
2
3
1
i i i i
i i
i
P y P y P GRADES e
P e GRADES
GRADES
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.
Slide16-47Principles of Econometrics, 3rd Edition
1 2 1 2 3, , 1 2 3L P y P y P y
16.5.2 Estimation and Interpretation
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,
Slide16-48Principles of Econometrics, 3rd Edition
2 12 | 2.5 2.5 2.5P y GRADES
16.5.2 Estimation and Interpretation
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
Slide16-49Principles of Econometrics, 3rd Edition
2 | 4.5 2 | 2.5P y GRADES P y GRADES
16.5.2 Estimation and Interpretation
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:
Slide16-50Principles of Econometrics, 3rd Edition
1
1 2
2
1
2
3
P yGRADES
GRADES
P yGRADES GRADES
GRADES
P yGRADES
GRADES
16.5.3 An Example
Slide16-51Principles of Econometrics, 3rd Edition
16.6 Models for Count Data
When the dependent variable in a regression model is a count of the number of occurrences of an event, the outcome variable is y = 0, 1, 2, 3, … These numbers are actual counts, and thus different from the ordinal numbers of the previous section. Examples include:
The number of trips to a physician a person makes during a year.
The number of fishing trips taken by a person during the previous year.
The number of children in a household.
The number of automobile accidents at a particular intersection during a month.
The number of televisions in a household.
The number of alcoholic drinks a college student takes in a week.
Slide16-52Principles of Econometrics, 3rd Edition
16.6 Models for Count Data
If Y is a Poisson random variable, then its probability function is
This choice defines the Poisson regression model for count data.
Slide16-53Principles of Econometrics, 3rd Edition
(16.27) , 0,1,2,!
yef y P Y y y
y
! 1 2 1y y y y
(16.28) 1 2expE Y x
16.6.1 Maximum Likelihood Estimation
Slide16-54Principles of Econometrics, 3rd Edition
1 2
1 2
, 0 2 2
ln , ln 0 ln 2 ln 2
L P Y P Y P Y
L P Y P Y P Y
1 2 1 2
ln ln ln ln !!
exp ln !
yeP Y y y y
y
x y x y
1 2 1 2 1 21
ln , exp ln !N
i i i ii
L x y x y
16.6.2 Interpretation in the Poisson Regression Model
Slide16-55Principles of Econometrics, 3rd Edition
0 0 1 2 0expE y x
0 0expPr , 0,1,2,
!
y
Y y yy
16.6.2 Interpretation in the Poisson Regression Model
Slide16-56Principles of Econometrics, 3rd Edition
(16.29)
2i
ii
E y
x
2
%100 100 %i i
i i
E y E y E y
x x
16.6.2 Interpretation in the Poisson Regression Model
Slide16-57Principles of Econometrics, 3rd Edition
1 2
1 2
1 2
1 2 1 2
1 2
exp
| 0 exp
| 1 exp
exp exp100 % 100 1 %
exp
i i i i
i i i
i i i
i i
i
E y x D
E y D x
E y D x
x xe
x
16.6.3 An Example
Slide16-58Principles of Econometrics, 3rd Edition
16.7 Limited Dependent Variables
16.7.1 Censored Data
Figure 16.3 Histogram of Wife’s Hours of Work in 1975
Slide16-59Principles of Econometrics, 3rd Edition
16.7.1 Censored Data
Having censored data means that a substantial fraction of the
observations on the dependent variable take a limit value. The
regression function is no longer given by (16.30).
The least squares estimators of the regression parameters obtained by
running a regression of y on x are biased and inconsistent—least
squares estimation fails.
Slide16-60Principles of Econometrics, 3rd Edition
(16.30) 1 2|E y x x
16.7.1 Censored Data
Having censored data means that a substantial fraction of the
observations on the dependent variable take a limit value. The
regression function is no longer given by (16.30).
The least squares estimators of the regression parameters obtained by
running a regression of y on x are biased and inconsistent—least
squares estimation fails.
Slide16-61Principles of Econometrics, 3rd Edition
(16.30) 1 2|E y x x
16.7.2 A Monte Carlo Experiment
We give the parameters the specific values and
Assume
Slide16-62Principles of Econometrics, 3rd Edition
(16.31)
1 29 and 1.
*1 2 9i i i i iy x e x e
2~ 0, 16 .ie N
*
* *
0 if 0;
if 0.
i i
i i i
y y
y y y
16.7.2 A Monte Carlo Experiment
Create N = 200 random values of xi that are spread evenly (or
uniformly) over the interval [0, 20]. These we will keep fixed in
further simulations.
Obtain N = 200 random values ei from a normal distribution with
mean 0 and variance 16.
Create N = 200 values of the latent variable.
Obtain N = 200 values of the observed yi using
Slide16-63Principles of Econometrics, 3rd Edition
*
* *
0 if 0
if 0
i
i
i i
yy
y y
16.7.2 A Monte Carlo Experiment
Figure 16.4 Uncensored Sample Data and Regression Function
Slide16-64Principles of Econometrics, 3rd Edition
16.7.2 A Monte Carlo Experiment
Figure 16.5 Censored Sample Data, and Latent Regression Function and
Least Squares Fitted Line
Slide16-65Principles of Econometrics, 3rd Edition
16.7.2 A Monte Carlo Experiment
Slide16-66Principles of Econometrics, 3rd Edition
(16.32a)ˆ 2.1477 .5161
(se) (.3706) (.0326)i iy x
(16.32b)ˆ 3.1399 .6388
(se) (1.2055) (.0827)i iy x
(16.33) ( )1
1 NSAM
MC k k mm
E b bNSAM
16.7.3 Maximum Likelihood Estimation
The maximum likelihood procedure is called Tobit in honor of James
Tobin, winner of the 1981 Nobel Prize in Economics, who first
studied this model.
The probit probability that yi = 0 is:
Slide16-67Principles of Econometrics, 3rd Edition
1 20 [ 0] 1i i iP y P y x
1
221 2 21 2 1 22
0 0
1, , 1 2 exp
2i i
ii i
y y
xL y x
16.7.3 Maximum Likelihood Estimation
The maximum likelihood estimator is consistent and asymptotically
normal, with a known covariance matrix.
Using the artificial data the fitted values are:
Slide16-68Principles of Econometrics, 3rd Edition
(16.34)10.2773 1.0487
(se) (1.0970) (.0790)i iy x
16.7.3 Maximum Likelihood Estimation
Slide16-69Principles of Econometrics, 3rd Edition
16.7.4 Tobit Model Interpretation
Because the cdf values are positive, the sign of the coefficient does
tell the direction of the marginal effect, just not its magnitude. If
β2 > 0, as x increases the cdf function approaches 1, and the slope of
the regression function approaches that of the latent variable model.
Slide16-70Principles of Econometrics, 3rd Edition
(16.35) 1 2
2
|E y x x
x
16.7.4 Tobit Model Interpretation
Figure 16.6 Censored Sample Data, and Regression Functions for Observed and Positive y values
Slide16-71Principles of Econometrics, 3rd Edition
16.7.5 An Example
Slide16-72Principles of Econometrics, 3rd Edition
(16.36)1 2 3 4 4 6HOURS EDUC EXPER AGE KIDSL e
2 73.29 .3638 26.34
E HOURS
EDUC
16.7.5 An Example
Slide16-73Principles of Econometrics, 3rd Edition
16.7.6 Sample Selection
Problem: our sample is not a random sample. The data we observe
are “selected” by a systematic process for which we do not account.
Solution: a technique called Heckit, named after its developer, Nobel
Prize winning econometrician James Heckman.
Slide16-74Principles of Econometrics, 3rd Edition
16.7.6a The Econometric Model
The econometric model describing the situation is composed of two equations. The first, is the selection equation that determines whether the variable of interest is observed.
Slide16-75Principles of Econometrics, 3rd Edition
(16.37)*1 2 1, ,i i iz w u i N
(16.38)
*1 0
0 otherwise
i
i
zz
16.7.6a The Econometric Model
The second equation is the linear model of interest. It is
Slide16-76Principles of Econometrics, 3rd Edition
(16.39)
(16.40)
1 2 1, ,i i iy x e i n N n
(16.41)
*1 2| 0 1, ,i i i iE y z x i n
1 2
1 2
ii
i
w
w
16.7.6a The Econometric Model
The estimated “Inverse Mills Ratio” is
The estimating equation is
Slide16-77Principles of Econometrics, 3rd Edition
(16.42)
1 2
1 2
ii
i
w
w
1 2 1, ,i i i iy x v i n
16.7.6b Heckit Example: Wages of Married Women
Slide16-78Principles of Econometrics, 3rd Edition
(16.43) 2ln .4002 .1095 .0157 .1484
(t-stat) ( 2.10) (7.73) (3.90)
WAGE EDUC EXPER R
1 1.1923 .0206 .0838 .3139 1.3939
(t-stat) ( 2.93) (3.61) ( 2.54) ( 2.26)
P LFP AGE EDUC KIDS MTR
1.1923 .0206 .0838 .3139 1.3939
1.1923 .0206 .0838 .3139 1.3939
AGE EDUC KIDS MTRIMR
AGE EDUC KIDS MTR
16.7.6b Heckit Example: Wages of Married Women
The maximum likelihood estimated wage equation is
The standard errors based on the full information maximum likelihood procedure are smaller than those yielded by the two-step estimation method.
Slide16-79Principles of Econometrics, 3rd Edition
(16.44)
ln .8105 .0585 .0163 .8664
(t-stat) (1.64) (2.45) (4.08) ( 2.65)
(t-stat-adj) (1.33) (1.97) (3.88) ( 2.17)
WAGE EDUC EXPER IMR
ln .6686 .0658 .0118
(t-stat) (2.84) (3.96) (2.87)
WAGE EDUC EXPER
Keywords
Slide 16-80Principles 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