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Chapter 16
Qualitative and Limited Dependent
Variable Models
Walter R. Paczkowski
Rutgers University
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 1
Chapter Contents
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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, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 2
In this chapter, we:
– Examine models that are used to describe
choice behavior, and which do not have the
usual continuous dependent variable
– Introduce a class of models with dependent
variables that are limited
• They are continuous, but that their range of
values is constrained in some way, and their
values not completely observable
• Alternatives to least squares estimation are
needed since the least squares estimator is
biased and inconsistent
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 3
16.1
Models with Binary Dependent
Variables
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 4
16.1
Models with Binary
Dependent Variables
Many of the choices that individuals and firms
make are ‘‘either–or’’ in nature
– Such choices can be represented by a binary
(indicator) variable that takes the value 1 if one
outcome is chosen and the value 0 otherwise
– The binary variable describing a choice is the
dependent variable rather than an independent
variable
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 5
16.1
Models with Binary
Dependent Variables
Examples:
– Models of why some individuals take a second or
third job, and engage in ‘‘moonlighting’’
– Models of why some legislators in the U.S. House
of Representatives vote for a particular bill and
others do not
– Models explaining why some loan applications are
accepted and others are not at a large metropolitan
bank
– Models explaining why some individuals vote for
increased spending in a school board election and
others vote against
– Models explaining why some female college
students decide to study engineering and others do
not
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 6
16.1
Models with Binary
Dependent Variables
We represent an individual’s choice by the
indicator variable:
Eq. 16.1
Principles of Econometrics, 4th Edition
1 individual drives to work
y
0 individual takes bus to work
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 7
16.1
Models with Binary
Dependent Variables
If the probability that an individual drives to work
is p, then P[y = 1] = p
– The probability that a person uses public
transportation is P[y = 0] = 1 – p
– The probability function for such a binary
random variable is:
f ( y)  p y (1  p)1 y ,
Eq. 16.2
y  0,1
with
E  y   p, var  y   p 1  p 
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 8
16.1
Models with Binary
Dependent Variables
For our analysis, define the explanatory variable
as:
x = (commuting time by bus - commuting time by car)
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 9
16.1
Models with Binary
Dependent Variables
16.1.1
The Linear
Probability Model
We could model the indicator variable y using the
linear model, however, there are several problems:
– It implies marginal effects of changes in
continuous explanatory variables are constant,
which cannot be the case for a probability
model
• This feature also can result in predicted
probabilities outside the [0, 1] interval
– The linear probability model error term is
heteroskedastic, so that a better estimator is
generalized least squares
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 10
16.1
Models with Binary
Dependent Variables
16.1.1
The Linear
Probability Model
In regression analysis we break the dependent
variable into fixed and random parts
– If we do this for the indicator variable y, we
have:
y  E ( y)  e  p  e
Eq. 16.3
– Assuming that the relationship is linear:
Eq. 16.4
Principles of Econometrics, 4th Edition
E ( y )  p  1  2 x
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 11
16.1
Models with Binary
Dependent Variables
16.1.1
The Linear
Probability Model
The linear regression model for explaining the
choice variable y is called the linear probability
model:
Eq. 16.5
Principles of Econometrics, 4th Edition
y  E ( y )  e  1  2 x  e
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 12
16.1
Models with Binary
Dependent Variables
16.1.1
The Linear
Probability Model
The probability density functions for y and e are:
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 13
16.1
Models with Binary
Dependent Variables
16.1.1
The Linear
Probability Model
Using these values it can be shown that the
variance of the error term e is:
var  e   1  2 x 1  1  2 x 
– The estimated variance of the error term is:
Eq. 16.6
Principles of Econometrics, 4th Edition
ˆ i2  var  ei    b1  b2 xi 1  b1  b2 xi 
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 14
16.1
Models with Binary
Dependent Variables
16.1.1
The Linear
Probability Model
We can transform the data as:
yi*  yi ˆ i
xi*  xi ˆ i
– And estimate the model:
ˆ i1  2 xi*  ei*
yi*  1
by least squares to produce the feasible
generalized least squares estimates
– Both least squares and feasible generalized least
squares are consistent estimators of the regression
parameters
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 15
16.1
Models with Binary
Dependent Variables
16.1.1
The Linear
Probability Model
If we estimate the parameters of Eq. 16.5 by least
squares, we obtain the fitted model explaining the
systematic portion of y
– This systematic portion is p
p̂  b1  b2 x
Eq. 16.7
– By substituting alternative values of x,we can
easily obtain values that are less than zero or
greater than one
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 16
16.1
Models with Binary
Dependent Variables
16.1.1
The Linear
Probability Model
The underlying feature that causes these problems
is that the linear probability model implicitly
assumes that increases in x have a constant effect
on the probability of choosing to drive
Eq. 16.8
Principles of Econometrics, 4th Edition
dp
 2
dx
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 17
16.1
Models with Binary
Dependent Variables
16.1.2
The Probit Model
To keep the choice probability p within the
interval [0, 1], a nonlinear S-shaped relationship
between x and p can be used
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 18
16.1
Models with Binary
Dependent Variables
FIGURE 16.1 (a) Standard normal cumulative distribution function
(b) Standard normal probability density function
16.1.2
The Probit Model
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 19
16.1
Models with Binary
Dependent Variables
16.1.2
The Probit Model
A functional relationship that is used to represent
such a curve is the probit function
– The probit function is related to the standard
normal probability distribution:
1 .5 z 2
( z ) 
e
2
– The probit function is:
Eq. 16.9
Principles of Econometrics, 4th Edition
 ( z )  P[ Z  z ]  
z

1 .5u 2
e du
2
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 20
16.1
Models with Binary
Dependent Variables
16.1.2
The Probit Model
The probit statistical model expresses the
probability p that y takes the value 1 to be:
p  P[ Z  1  2 x]  (1  2 x)
Eq. 16.10
– The probit model is said to be nonlinear
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 21
16.1
Models with Binary
Dependent Variables
16.1.3
Interpretation of the
Probit Model
We can examine the marginal effect of a one-unit
change in x on the probability that y = 1 by
considering the derivative:
Eq. 16.11
Principles of Econometrics, 4th Edition
dp d (t ) dt

  (1  2 x)2
dx
dt dx
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 22
16.1
Models with Binary
Dependent Variables
16.1.3
Interpretation of the
Probit Model
Eq. 16.11 has the following implications:
1. Since Φ(β1+ β2x) is a probability density
function, its value is always positive
2. As x changes, the value of the function
Φ(β1+ β2x) changes
3. if β1+ β2x is large, then the probability that the
individual chooses to drive is very large and
close to one
• Similarly if β1+ β2x is small
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 23
16.1
Models with Binary
Dependent Variables
16.1.3
Interpretation of the
Probit Model
We estimate the probability p to be:
pˆ  (1  2 x)
Eq. 16.12
– By comparing to a threshold value, like 0.5, we
can predict choice using the rule:
1
yˆ  
0
Principles of Econometrics, 4th Edition
pˆ  0.5
pˆ  0.5
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 24
16.1
Models with Binary
Dependent Variables
16.1.4
Maximum
Likelihood
Estimation of the
Probit Model
The probability function for y is combined with
the probit model to obtain:
Eq. 16.13
f ( yi )  [(1 2 xi )]yi [1  (1 2 xi )]1 yi , yi  0,1
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 25
16.1
Models with Binary
Dependent Variables
16.1.4
Maximum
Likelihood
Estimation of the
Probit Model
If the three individuals are independently drawn,
then:
f ( y1 , y2 , y3 )  f ( y1 ) f ( y2 ) f ( y3 )
– The probability of observing y1 = 1, y2 = 1, and
y3 = 0 is:
P[ y1  1, y2  1, y3  0]  f (1, 1, 0)  f (1) f (1) f (0)
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 26
16.1
Models with Binary
Dependent Variables
16.1.4
Maximum
Likelihood
Estimation of the
Probit Model
We now have:
P[ y1  1, y2  1, y3  0]    1  2 (15)    1  2 (6) 
Eq. 16.14
 L  β1 ,β 2 
 1   1  2 (7) 
for x1 = 15, x2 = 6, and x3 = 7
– This function, which gives us the probability of
observing the sample data, is called the likelihood
function
• The notation L(β1, β2) indicates that the
likelihood function is a function of the unknown
parameters, β1and β2
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 27
16.1
Models with Binary
Dependent Variables
16.1.4
Maximum
Likelihood
Estimation of the
Probit Model
Eq. 16.15
In practice, instead of maximizing Eq. 16.14, we
maximize the logarithm of Eq. 16.14, which is called
the log-likelihood function:


ln L  β1 ,β 2   ln   1  2 (15)    1  2 (6)   1   1  2 (7) 
 ln   1  2 (15)   ln  1  2 (6)   ln 1   1  2 (7) 
– The maximization of the log-likelihood function is
easier than the maximization of Eq. 16.14
– The values that maximize the log-likelihood
function also maximize the likelihood function
• They are the maximum likelihood estimates
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 28
16.1
Models with Binary
Dependent Variables
16.1.4
Maximum
Likelihood
Estimation of the
Probit Model
A feature of the maximum likelihood estimation
procedure is that while its properties in small
samples are not known, we can show that in large
samples the maximum likelihood estimator is
normally distributed, consistent and best, in the
sense that no competing estimator has smaller
variance
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 29
16.1
Models with Binary
Dependent Variables
16.1.5
A Transportation
Example
Let DTIME = (BUSTIME-AUTOTIME)÷10,
which is the commuting time differential in 10minute increments
– The probit model is:
P(AUTO = 1) = Φ(β1+ β2DTIME)
– The maximum likelihood estimates of the
parameters are:
1  2 DTIME  0.0644  0.3000 DTIMEi
(se)
Principles of Econometrics, 4th Edition
(0.3992) (0.1029)
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 30
16.1
Models with Binary
Dependent Variables
16.1.5
A Transportation
Example
The marginal effect of increasing public
transportation time, given that travel via public
transportation currently takes 20 minutes longer
than auto travel is:
dp
 (1  2 DTIME )2  (0.0644  0.3000  2)(0.3000)
dDTIME
 (0.5355)(0.3000)  0.3456  0.3000  0.1037
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 31
16.1
Models with Binary
Dependent Variables
16.1.5
A Transportation
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:
pˆ  (1  2 DTIME )  (0.0644  0.3000  3)  0.7983
– Since 0.7983 > 0.5, we “predict” the individual
will choose to drive
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 32
16.1
Models with Binary
Dependent Variables
16.1.6
Further PostEstimation Analysis
Rather than evaluate the marginal effect at a
specific value, or the mean value, the average
marginal effect (AME) is often considered:
1
AME 
N

N
i 1


 β1  β 2 DTIMEi β 2
– For our problem: AME  0.0484
– The sample standard deviation is: 0.0365
– Its minimum and maximum values are 0.0025
and 0.1153
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 33
16.1
Models with Binary
Dependent Variables
16.1.6
Further PostEstimation Analysis
Consider the marginal effect:

dp
 (1  2 DTIME )2  g 1 , 2
dDTIME

– The marginal effect function is nonlinear
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 34
16.2
The Logit Model for Binary Choice
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 35
16.2
The Logit Model for
Binary Choice
Probit model estimation is numerically
complicated because it is based on the normal
distribution
– A frequently used alternative to the probit
model for binary choice situations is the logit
model
– These models differ only in the particular Sshaped curve used to constrain probabilities to
the [0, 1] interval
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 36
16.2
The Logit Model for
Binary Choice
Eq. 16.16
Eq. 16.17
If L is a logistic random variable, then its
probability density function is:
e l
(l ) 
,  l  
2
l
1

e


– The cumulative distribution function for a
logistic random variable is:
1
  l   p[ L  l ] 
1  el
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 37
16.2
The Logit Model for
Binary Choice
The probability p that the observed value y takes
the value 1 is:
Eq. 16.18
p  P  L  1  2 x    1  2 x  
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
1
1  e1 2 x 
Page 38
16.2
The Logit Model for
Binary Choice
The probability that y = 1 is:
p
1
1 e
1 2 x 

exp 1  2 x 
1  exp 1  2 x 
The probability that y = 0 is:
1
1 p 
1  exp  1  2 x 
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 39
16.2
The Logit Model for
Binary Choice
The shapes of the logistic and normal probability
density functions are somewhat different, and
maximum likelihood estimates of β1 and β2 will be
different
– However, the marginal probabilities and the
predicted probabilities differ very little in most
cases
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 40
16.2
The Logit Model for
Binary Choice
16.2.1
An Empirical
Example from
Marketing
Consider the Coke example with:
1 if Coke is chosen
COKE  
0 if Pepsi is chosen
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 41
16.2
The Logit Model for
Binary Choice
16.2.1
An Empirical
Example from
Marketing
Based on ‘‘scanner’’ data on 1,140 individuals
who purchased Coke or Pepsi, the probit and logit
models for the choice are:
pCOKE  E  COKE     β1  β 2 PRATIO  β3 DISP_COKE  β 4 DISP_PEPSI 
pCOKE  E  COKE     γ1  γ 2 PRATIO  γ3 DISP_COKE  γ 4 DISP_PEPSI 
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 42
16.2
The Logit Model for
Binary Choice
Table 16.1 Coke-Pepsi Choice Models
16.2.1
An Empirical
Example from
Marketing
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 43
16.2
The Logit Model for
Binary Choice
16.2.1
An Empirical
Example from
Marketing
The parameters and their estimates vary across the
models and no direct comparison is very useful,
but some rules of thumb exist
– Roughly:
ˆ
γ Logit  4β LPM
β Probit  2.5βˆ LPM
γ Logit  1.6β Probit
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 44
16.2
The Logit Model for
Binary Choice
16.2.2
Wald Hypothesis
Tests
If the null hypothesis is H0: βk = c, then the test
statistic using the probit model is:
t
βk  c
 
se β k
a
t N  K 
– The t-test is based on the Wald principle
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 45
16.2
The Logit Model for
Binary Choice
16.2.2
Wald Hypothesis
Tests
Using the probit model, consider the two
hypotheses:
Hypothesis (1) H 0 : β3  β 4 , H1 :β3  β 4
Hypothesis (2) H 0 : β3  0, β 4  0, H1 : either β3 or β4 is not zero
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 46
16.2
The Logit Model for
Binary Choice
16.2.2
Wald Hypothesis
Tests
To test hypothesis (1) in a linear model, we would
compute:
t
β DISP _ COKE  β DISP _ PEPSI

se β DISP _ COKE  β DISP _ PEPSI
a

t1140 41136
– Noting that it is a two-tail hypothesis, we reject the
null hypothesis at the α = 0.05 level if t ≥ 1.96 or
t ≤ -1.96
– The calculated t-value is t = -2.3247, so we reject
the null hypothesis
• We conclude that the effects of the Coke and
Pepsi displays are not of equal magnitude with
opposite sign
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 47
16.2
The Logit Model for
Binary Choice
16.2.2
Wald Hypothesis
Tests
A generalization of the Wald statistic is used to
test the joint null hypothesis (2) that neither the
Coke nor Pepsi display affects the probability of
choosing Coke
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 48
16.2
The Logit Model for
Binary Choice
16.2.3
Likelihood Ratio
Hypothesis Tests
When using maximum likelihood estimators, such
as probit and logit, tests based on the likelihood
ratio principle are generally preferred
– The idea is much like the F-test
• One test component is the log-likelihood
function value in the unrestricted, full model
(ln LU) evaluated at the maximum likelihood
estimates
• The second ingredient is the log-likelihood
function value from the model that is
‘‘restricted’’ by imposing the condition that
the null hypothesis is true (ln LR)
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 49
16.2
The Logit Model for
Binary Choice
16.2.3
Likelihood Ratio
Hypothesis Tests
The restricted probit model is obtained by
imposing the condition β3 = -β4:
pCOKE  E  COKE     β1  β 2 PRATIO  β3 DISP _ COKE  β 4 DISP _ PEPSI 
   β1  β 2 PRATIO  β 4 DISP _ COKE  β 4 DISP _ PEPSI 
  β1  β 2 PRATIO _ β 4  DISP _ PEPSI  DISP _ COKE  
– We have LR = -713.6595
– The likelihood ratio test statistic value is:
LR  2  ln LU  ln LR 
 2  710.9486   713.6595  
 5.4218
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 50
16.2
The Logit Model for
Binary Choice
16.2.3
Likelihood Ratio
Hypothesis Tests
To test the null hypothesis (2), use the restricted
model E(COKE) = Φ(β1 + β2PRATIO)
– The value of the likelihood ratio test statistic is
19.55
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 51
16.3
Multinomial Logit
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 52
16.3
Multinomial Logit
We are often faced with choices involving more
than two alternatives
– These are called multinomial choice situations
• If you are shopping for a laundry detergent,
which one do you choose? Tide, Cheer, Arm
& Hammer, Wisk, and so on
• If you enroll in the business school, will you
major in economics, marketing,
management, finance, or accounting?
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 53
16.3
Multinomial Logit
The estimation and interpretation of the models is,
in principle, similar to that in logit and probit
models
– The models go under the names
• multinomial logit
• conditional logit
• multinomial probit
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 54
16.3
Multinomial Logit
16.3.1
Multinomial Logit
Choice Probabilities
As in the logit and probit models, we will try to
explain the probability that the ith person will
choose alternative j
pij = P individual i chooses alternative j 
– Assume J = 3
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 55
16.3
Multinomial Logit
16.3.1
Multinomial Logit
Choice Probabilities
For a single explanatory factor, the choice
probabilities are:
1
Eq. 16.19a
pi1 
Eq. 16.19b
exp  12  22 xi 
pi 2 
, j2
1  exp  12  22 xi   exp  13  23 xi 
Eq. 16.19c
exp  13  23 xi 
pi 3 
, j 3
1  exp  12  22 xi   exp  13  23 xi 
Principles of Econometrics, 4th Edition
1  exp  12  22 xi   exp 13  23 xi 
Chapter 16: Qualitative and Limited Dependent Variable
Models
, j 1
Page 56
16.3
Multinomial Logit
16.3.1
Multinomial Logit
Choice Probabilities
A distinguishing feature of the multinomial logit
model in Eq. 16.19 is that there is a single
explanatory variable that describes the individual,
not the alternatives facing the individual
– Such variables are called individual specific
– To distinguish the alternatives, we give them
different parameter values
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 57
16.3
Multinomial Logit
16.3.2
Maximum
Likelihood
Estimation
Suppose that we observe three individuals, who
choose alternatives 1, 2, and 3, respectively
– Assuming that their choices are independent,
then the probability of observing this outcome
is:
P  y11  1, y22  1, y33  1  p11  p22  p33
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 58
16.3
Multinomial Logit
16.3.2
Maximum
Likelihood
Estimation
Or
P  y11  1, y22  1, y33  1 
1
1  exp  12  22 x1   exp  13  23 x1 
exp  12  22 x2 
1  exp  12  22 x2   exp  13  23 x2 
exp  13  23 x3 
1  exp  12  22 x3   exp  13  23 x3 
 L  12 , 22 , 13 , 23 
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 59


16.3
Multinomial Logit
16.3.2
Maximum
Likelihood
Estimation
Maximum likelihood estimation seeks those
values of the parameters that maximize the
likelihood or, more specifically, the log-likelihood
function, which is easier to work with
mathematically
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 60
16.3
Multinomial Logit
16.3.3
Post-Estimation
Analysis
For the value of the explanatory variable x0, we
can calculate the predicted probabilities of each
outcome being selected
– For alternative 1:
p01 

1


1  exp 12  22 x0  exp 13  23 x0

– Similarly for alternatives 2 and 3
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 61
16.3
Multinomial Logit
16.3.3
Post-Estimation
Analysis
The βs are not ‘‘slopes’’
– The marginal effect is the effect of a change in
x, everything else held constant, on the
probability that an individual chooses
alternative m = 1, 2, or 3:
Eq. 16.20
Principles of Econometrics, 4th Edition
pim
xi
all else constant
3


pim

 pim 2 m   2 j pij 
xi
j 1


Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 62
16.3
Multinomial Logit
16.3.3
Post-Estimation
Analysis
Alternatively, and somewhat more simply, the
difference in probabilities can be calculated for
two specific values of xi
p1  pb1  pa1






1  exp 12  22 xb  exp 13  23 xb

Principles of Econometrics, 4th Edition

1

1

1  exp 12  22 xa  exp 13  23 xa
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 63
16.3
Multinomial Logit
16.3.3
Post-Estimation
Analysis
Eq. 16.21
Another useful interpretive device is the
probability ratio
– It shows how many times more likely category
j is to be chosen relative to the first category
P  yi  j  pij

 exp  1 j  2 j xi 
P  yi  1 pi1
j  2,3
– The effect on the probability ratio of changing
the value of xi is given by the derivative:
Eq. 16.22
  pij pi1 
Principles of Econometrics, 4th Edition
xi
 2 j exp 1 j  2 j xi 
Chapter 16: Qualitative and Limited Dependent Variable
Models
j  2,3
Page 64
16.3
Multinomial Logit
16.3.3
Post-Estimation
Analysis
An interesting feature of the probability ratio Eq.
16.21 is that it does not depend on how many
alternatives there are in total
– There is the implicit assumption in logit models
that the probability ratio between any pair of
alternatives is independent of irrelevant
alternatives (IIA)
• This is a strong assumption, and if it is
violated, multinomial logit may not be a
good modeling choice
• It is especially likely to fail if several
alternatives are similar
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 65
16.3
Multinomial Logit
16.3.3
Post-Estimation
Analysis
Tests for the IIA assumption work by dropping
one or more of the available options from the
choice set and then re-estimating the multinomial
model
– If the IIA assumption holds, then the estimates
should not change very much
– A statistical comparison of the two sets of
estimates, one set from the model with a full set
of alternatives, and the other from the model
using a reduced set of alternatives, is carried
out using a Hausman contrast test proposed by
Hausman and McFadden
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 66
16.3
Multinomial Logit
Table 16.2 Maximum Likelihood Estimates of PSE Choice
16.3.4
An Example
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 67
16.3
Multinomial Logit
Table 16.3 Effects of Grades on Probability of PSE Choice
16.3.4
An Example
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 68
16.4
Conditional Logit
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 69
16.4
Conditional Logit
Variables like PRICE are individual- and
alternative-specific because they vary from
individual to individual and are different for each
choice the consumer might make
– This type of information is very different from
what we assumed was available in the
multinomial logit model, where the explanatory
variable xi was individual-specific; it did not
change across alternatives
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 70
16.4
Conditional Logit
16.4.1
Conditional Logit
Choice Probabilities
Consider a model for the probability that
individual i chooses alternative j:
pij  P individual i chooses alternative j 
The conditional logit model specifies these
probabilities as:
Eq. 16.23
pij 
exp  1 j  2 PRICEij 
exp  11  2 PRICEi1   exp  12  2 PRICEi 2   exp 13   2 PRICEi 3 
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 71
16.4
Conditional Logit
16.4.1
Conditional Logit
Choice Probabilities
Set β13 = 0
– Estimation of the unknown parameters is by
maximum likelihood
• Suppose that we observe three individuals,
who choose alternatives one, two, and three,
respectively
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 72
16.4
Conditional Logit
16.4.1
Conditional Logit
Choice Probabilities
We have:
P  y11  1, y22  1, y33  1  p11  p22  p33

exp  11  2 PRICE11 
exp  11  2 PRICE11   exp  12  2 PRICE12   exp  2 PRICE13 
exp  12  2 PRICE22 
exp  11  2 PRICE21   exp  12  2 PRICE22   exp 2 PRICE23 
exp  2 PRICE33 
exp  11  2 PRICE31   exp  12  2 PRICE32   exp  2 PRICE33 
 L  12 , 22 , 2 
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 73


16.4
Conditional Logit
16.4.2
Post-Estimation
Analysis
The own price effect is:
pij
Eq. 16.24
PRICEij
 pij 1  pij  2
The change in probability of alternative j being
selected if the price of alternative k changes (k ≠ j)
is:
Eq. 16.25
Principles of Econometrics, 4th Edition
pij
PRICEik
  pij pik 2
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 74
16.4
Conditional Logit
16.4.2
Post-Estimation
Analysis
An important feature of the conditional logit
model is that the probability ratio between
alternatives j and k is:
pij
pik

exp  1 j  2 PRICEij 
exp  1k  2 PRICEik 
 exp 1 j  1k   2  PRICEij  PRICEik  
– The probability ratio depends on the difference
in prices, but not on the prices themselves
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 75
16.4
Conditional Logit
Table 16.4a Conditional Logit Parameter Estimates
16.4.2
Post-Estimation
Analysis
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 76
16.4
Conditional Logit
Table 16.4b Marginal Effect of Price on Probability of Pepsi Choice
16.4.2
Post-Estimation
Analysis
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 77
16.4
Conditional Logit
16.4.2
Post-Estimation
Analysis
Models that do not require the IIA assumption
have been developed, but they are difficult
– These include the multinomial probit model,
which is based on the normal distribution, and
the nested logit and mixed logit models
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 78
16.4
Conditional Logit
16.4.3
An Example
The predicted probability of a Pepsi purchase,
given that the price of Pepsi is $1.00, the price of
7-Up is $1.25 and the price of Coke is $1.10 is:
pˆ i1 


exp 11  2  1.00





exp 11  2  1.00  exp 12  2 1.25  exp 2  1.10
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models

Page 79
 0.4832
16.4
Conditional Logit
16.4.3
An Example
The standard error for this predicted probability is
0.0154, which is computed via ‘‘the delta method.’’
– If we raise the price of Pepsi to $1.10, we estimate
that the probability of its purchase falls to 0.4263
(se = 0.0135)
– If the price of Pepsi stays at $1.00 but we increase
the price of Coke by 15 cents, then we estimate that
the probability of a consumer selecting Pepsi rises
by 0.0445 (se = 0.0033)
– These numbers indicate to us the responsiveness of
brand choice to changes in prices, much like
elasticities
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 80
16.5
Ordered Choice Models
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 81
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
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 82
16.5
Ordered Choice
Models
Examples:
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
3. Standard and Poor’s rates bonds as AAA, AA,
A, BBB and so on
4. Levels of employment are unemployed, parttime, or full-time
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 83
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
– In the first example, we might assign a
dependent variable y the values:
1
2

y  3
4

5
Principles of Econometrics, 4th Edition
strongly disagree
disagree
neutral
agree
strongly agree
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 84
16.5
Ordered Choice
Models
There may be a natural ordering to college choice
– We might rank the possibilities as:
Eq. 16.26
3 4-year college (the full college experience)

y  2 2-year college (a partial college experience)
1 no college

– 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, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 85
16.5
Ordered Choice
Models
16.5.1
Ordered Probit
Choice Probabilities
When faced with a ranking problem, we develop a
‘‘sentiment’’ about how we feel concerning the
alternative choices, and the higher the sentiment,
the more likely a higher-ranked alternative will be
chosen
– This sentiment is, of course, unobservable to
the econometrician
– Unobservable variables that enter decisions are
called latent variables
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 86
16.5
Ordered Choice
Models
16.5.1
Ordered Probit
Choice Probabilities
For college choice, a latent variable may be
grades:
yi*  GRADESi  ei
– This model is not a regression model, because
the dependent variable is unobservable
• Consequently it is sometimes called an
index model
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 87
16.5
Ordered Choice
Models
FIGURE 16.2 Ordinal choices relative to thresholds
16.5.1
Ordered Probit
Choice Probabilities
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 88
16.5
Ordered Choice
Models
16.5.1
Ordered Probit
Choice Probabilities
We can now specify:
3 (4-year college) if

y  2 (2-year college) if
1 (no college)
if

Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
yi*   2
1  yi*   2
yi*  1
Page 89
16.5
Ordered Choice
Models
16.5.1
Ordered Probit
Choice Probabilities
If we assume that the errors have the standard
normal distribution, N(0, 1), an assumption that
defines the ordered probit model, then we can
calculate the following:
P  yi  1  P  yi*  1   P GRADESi  ei  1 
 P  ei  1  GRADESi 
   1  GRADESi 
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 90
16.5
Ordered Choice
Models
16.5.1
Ordered Probit
Choice Probabilities
Also:
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, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 91
16.5
Ordered Choice
Models
16.5.1
Ordered Probit
Choice Probabilities
Finally:
P  yi  3  P  yi*   2   P GRADESi  ei   2 
 P ei   2  GRADESi 
 1     2  GRADESi 
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 92
16.5
Ordered Choice
Models
16.5.2
Estimation and
Interpretation
If we observe a random sample of N = 3
individuals, with the first not going to college
(y1 = 1), the second attending a two-year college
(y2 = 2), and the third attending a four-year college
(y3 = 3), then the likelihood function is:
L  , 1 , 2   P  y1  1  P  y2  2  P  y3  3
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 93
16.5
Ordered Choice
Models
16.5.2
Estimation and
Interpretation
Econometric 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
– Most economists will use the normality
assumption
– Many other social scientists use the logistic
– There is little difference between the results
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 94
16.5
Ordered Choice
Models
16.5.2
Estimation and
Interpretation
The types of questions we can answer with this
model are the following:
1. What is the probability that a high school
graduate with GRADES = 2.5 (on a 13-point
scale, with one being the highest) will attend a
two-year college?



Pˆ  y  2 | GRADES  2.5    2    2.5   1    2.5
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 95

16.5
Ordered Choice
Models
16.5.2
Estimation and
Interpretation
The types of questions (Continued):
2. What is the difference in probability of
attending a four-year college for two students,
one with GRADES = 2.5 and another with
GRADES = 4.5?
Pˆ  y  2 | GRADES  4.5  Pˆ  y  2 | GRADES  2.5
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 96
16.5
Ordered Choice
Models
16.5.2
Estimation and
Interpretation
The types of questions (Continued):
3. If we treat GRADES as a continuous variable,
what is the marginal effect on the probability of
each outcome, given a one-unit change in
GRADES?
P  y  1
GRADES
P  y  2
GRADES
P  y  3
GRADES
Principles of Econometrics, 4th Edition
   1   GRADES   
 
  1  GRADES      2   GRADES   
    2  GRADES   
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 97
16.5
Ordered Choice
Models
Table 16.5 Ordered Probit Parameter Estimates
16.5.3
An Example
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 98
16.6
Models for Count Data
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 99
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 ordinal numbers
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 100
16.6
Models for Count
Data
Examples:
– 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
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 101
16.6
Models for Count
Data
The probability distribution we use as a
foundation is the Poisson, not the normal or the
logistic
– If Y is a Poisson random variable, then its
probability function is:
Eq. 16.27
e y
f  y   P Y  y  
,
y!
y  0,1,2,
where
y !  y   y  1   y  2  
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
1
Page 102
16.6
Models for Count
Data
In a regression model, we try to explain the
behavior of E(Y) as a function of some
explanatory variables
– We do the same here, keeping the value of
E(Y) ≥ 0 by defining:
E Y     exp 1  2 x 
Eq. 16.28
– This choice defines the Poisson regression
model for count data
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 103
16.6
Models for Count
Data
16.6.1
Maximum
Likelihood
Estimation
Suppose we randomly select N = 3 individuals
from a population and observe that their counts are
y1 = 0, y2 = 2, and y3 = 2, indicating 0, 2, and 2
occurrences of the event for these three
individuals
– The likelihood function is the joint probability
function of the observed data is:
L 1 , 2   P Y  0   P Y  2   P Y  2 
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 104
16.6
Models for Count
Data
16.6.1
Maximum
Likelihood
Estimation
The log-likelihood function is:
ln L 1 , 2   ln P Y  0   ln P Y  2   ln P Y  2 
– Using Eq. 16.28 for λ, the log of the probability
function is:
 e   y 
ln  P Y  y    ln 

 y! 
   y ln     ln  y !
  exp  1  2 x   y  1  2 x   ln  y !
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 105
16.6
Models for Count
Data
16.6.1
Maximum
Likelihood
Estimation
Then the log-likelihood function, given a sample
of N observations, becomes:
ln L 1 , 2    exp 1  2 xi   yi  1  2 xi   ln  yi !
N
i 1
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 106
16.6
Models for Count
Data
16.6.2
Interpretation of the
Poisson Regression
Model
Prediction of the conditional mean of y is
straightforward:

E  y0   0  exp 1  2 x0

The probability of a particular number of
occurrences can be estimated by inserting the
estimated conditional mean into the probability
function, as:
Pr Y  y  
Principles of Econometrics, 4th Edition


exp  0  0y
y!
,
Chapter 16: Qualitative and Limited Dependent Variable
Models
y  0,1, 2,
Page 107
16.6
Models for Count
Data
16.6.2
Interpretation of the
Poisson Regression
Model
The marginal effect is:
E  yi 
  i 2
xi
Eq. 16.29
– This can be expressed as a percentage, which
can be useful:
%E  y 
xi
Principles of Econometrics, 4th Edition
 100
E  yi  E  yi 
xi
Chapter 16: Qualitative and Limited Dependent Variable
Models
 1002 %
Page 108
16.6
Models for Count
Data
16.6.2
Interpretation of the
Poisson Regression
Model
Suppose the conditional mean function contains a
indicator variable, how do we calculate its effect?
– If E(yi) = λi = exp(β1 + β2xi + δDi), we can
examine the conditional expectation when D =
0 and when D = 1
E  yi | Di  0   exp 1  2 xi 
E  yi | Di  1  exp  1  2 xi   
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 109
16.6
Models for Count
Data
16.6.2
Interpretation of the
Poisson Regression
Model
The percentage change in the conditional mean is:
 exp  1  2 xi     exp 1  2 xi  


100 
%

100
e
 1 %


exp  1  2 xi 


Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 110
16.6
Models for Count
Data
Table 16.6 Poisson Regression Estimates
16.6.3
An Example
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 111
16.7
Limited Dependent Variables
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 112
16.7
Limited Dependent
Variables
When a model has a discrete dependent variable,
the usual regression methods we have studied
must be modified
– Now we present another case in which standard
least squares estimation of a regression model
fails
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 113
16.7
Limited Dependent
Variables
FIGURE 16.3 Histogram of wife’s hours of work in 1975
16.7.1
Censored Data
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 114
16.7
Limited Dependent
Variables
16.7.1
Censored Data
This is an example of censored data, meaning
that a substantial fraction of the observations on
the dependent variable take a limit value, which is
zero in the case of market hours worked by
married women
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 115
16.7
Limited Dependent
Variables
16.7.1
Censored Data
We previously showed the probability density
functions for the dependent variable y, at different
x-values, centered on the regression function
E  y | x   1  2 x
Eq. 16.30
– This leads to sample data being scattered along
the regression function
– Least squares regression works by fitting a line
through the center of a data scatter, and in this
case such a strategy works fine, because the
true regression function also fits through the
middle of the data scatter
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 116
16.7
Limited Dependent
Variables
16.7.1
Censored Data
For our new problem when a substantial number
of observations have dependent variable values
taking the limit value of zero, the regression
function E(y|x) is no longer given by Eq. 16.30
– Instead E(y|x) is a complicated nonlinear
function of the regression parameters β1 and β2,
the error variance σ2, and x
– 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
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 117
16.7
Limited Dependent
Variables
16.7.2
A Monte Carlo
Experiment
In this example we give the parameters the
specific values β1 = -9 and β2 = 1
– The observed sample is obtained within the
framework of an index or latent variable model:
yi*  1  2 xi  ei  9  xi  ei
Eq. 16.31
– We assume:
ei ~ N  0, 2  16 
yi  0 if yi*  0
yi  yi* if yi*  0
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 118
16.7
Limited Dependent
Variables
FIGURE 16.4 Uncensored sample data and regression function
16.7.2
A Monte Carlo
Experiment
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 119
16.7
Limited Dependent
Variables
16.7.2
A Monte Carlo
Experiment
Eq. 16.32a
In Figure 16.5 we show the estimated regression
function for the 200 observed y-values, which is
given by:
yˆ  2.1477  0.5161x
(se) (.3706) (0.0326)
– If we restrict our sample to include only the 100
positive y-values, the fitted regression is:
yˆ  3.1399  0.6388 x
Eq. 16.32b
Principles of Econometrics, 4th Edition
(se) (1.2055) (0.0827)
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 120
16.7
Limited Dependent
Variables
FIGURE 16.5 Censored sample data, and latent regression function and least
squares fitted line
16.7.2
A Monte Carlo
Experiment
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 121
16.7
Limited Dependent
Variables
16.7.2
A Monte Carlo
Experiment
We can compute the average values of the
estimates, which is the Monte Carlo ‘‘expected
value’’:
1
EMC  bk  
NSAM
Eq. 16.33
NSAM

m1
bk ( m )
where bk(m) is the estimate of βk in the mth Monte
Carlo sample
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 122
16.7
Limited Dependent
Variables
16.7.3
Maximum
Likelihood
Estimation
If the dependent variable is censored, having a
lower limit and/or an upper limit, then the least
squares estimators of the regression parameters are
biased and inconsistent
– We can apply an alternative estimation
procedure, which is called Tobit
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 123
16.7
Limited Dependent
Variables
16.7.3
Maximum
Likelihood
Estimation
Tobit is a maximum likelihood procedure that
recognizes that we have data of two sorts:
1. The limit observations (y = 0)
2. The nonlimit observations (y > 0)
– The two types of observations that we observe,
the limit observations and those that are
positive, are generated by the latent variable y*
crossing the zero threshold or not crossing that
threshold
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 124
16.7
Limited Dependent
Variables
16.7.3
Maximum
Likelihood
Estimation
The (probit) probability that y = 0 is:
P  yi  0  P[ yi  0]  1   1  2 xi  
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 125
16.7
Limited Dependent
Variables
16.7.3
Maximum
Likelihood
Estimation
The full likelihood function is the product of the
probabilities that the limit observations occur
times the probability density functions for all the
positive, nonlimit, observations:
1


2 
 1  2 xi 
 1
2 2
L 1 , 2 ,     1   
    2  exp   2  yi 1 2 xi  

yi 0 
 2

   yi 0 
– The maximum likelihood estimator is
consistent and asymptotically normal, with a
known covariance matrix.
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 126
16.7
Limited Dependent
Variables
16.7.3
Maximum
Likelihood
Estimation
For artificial data, we estimate:
Eq. 16.34
Principles of Econometrics, 4th Edition
yi  10.2773  1.0487 xi
(se) (1.0970) (0.0790)
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 127
16.7
Limited Dependent
Variables
Table 16.7 Censored Data Monte Carlo Results
16.7.3
Maximum
Likelihood
Estimation
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 128
16.7
Limited Dependent
Variables
16.7.4
Tobit Model
Interpretation
In the Tobit model the parameters β1and β2 are the
intercept and slope of the latent variable model
Eq. 16.31
– In practice we are interested in the marginal
effect of a change in x on either the regression
function of the observed data E(y|x) or the
regression function conditional on y > 0,
E(y|x, y > 0)
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 129
16.7
Limited Dependent
Variables
16.7.4
Tobit Model
Interpretation
The slope of E(y|x) is:
Eq. 16.35
Principles of Econometrics, 4th Edition
E  y | x 
 1  2 x 
 2  

x



Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 130
16.7
Limited Dependent
Variables
16.7.4
Tobit Model
Interpretation
The marginal effect can be decomposed into two
factors called the ‘‘McDonald-Moffit’’ decomposition:
E  y | x 
x
 Prob  y  0 
E  y | x, y  0 
x
 E  y | x, y  0 
Prob  y  0 
x
– The first factor accounts for the marginal effect of a
change in x for the portion of the population whose
y-data is observed already
– The second factor accounts for changes in the
proportion of the population who switch from the
y-unobserved category to the y-observed category
when x changes
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 131
16.7
Limited Dependent
Variables
FIGURE 16.6 Censored sample data, and regression functions for observed
and positive y-values
16.7.4
Tobit Model
Interpretation
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 132
16.7
Limited Dependent
Variables
16.7.5
An Example
Consider the regression model:
Eq. 16.36
HOURS  1  2 EDUC  3 EXPER  4 AGE  5 KIDSL6  e
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 133
16.7
Limited Dependent
Variables
Table 16.8 Estimates of Labor Supply Function
16.7.5
An Example
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 134
16.7
Limited Dependent
Variables
16.7.5
An Example
The calculated scale factor is   0.3638
– The marginal effect on observed hours of work
of another year of education is:
E  HOURS 
EDUC
 2   73.29  0.3638  26.34
• Another year of education will increase a
wife’s hours of work by about 26 hours,
conditional upon the assumed values of the
explanatory variables
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 135
16.7
Limited Dependent
Variables
16.7.6
Sample Selection
If the data are obtained by random sampling, then
classic regression methods, such as least squares,
work well
– However, if the data are obtained by a sampling
procedure that is not random, then standard
procedures do not work well
– Economists regularly face such data problems
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 136
16.7
Limited Dependent
Variables
16.7.6
Sample Selection
If we wish to study the determinants of the wages
of married women, we face a sample selection
problem
– We only observe data on market wages when
the woman chooses to enter the workforce
– If we observe only the working women, then
our sample is not a random sample
• The data we observe are ‘‘selected’’ by a
systematic process for which we do not
account
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 137
16.7
Limited Dependent
Variables
16.7.6
Sample Selection
A solution to this problem is a technique called Heckit
– This procedure uses two estimation steps:
1. A probit model is first estimated explaining
why a woman is in the labor force or not
– The selection equation
2. A least squares regression is estimated relating
the wage of a working woman to education,
experience, and so on, and a variable called
the ‘‘inverse Mills ratio,’’ or IMR
– The IMR is created from the first step probit
estimation and accounts for the fact that the
observed sample of working women is not
random
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 138
16.7
Limited Dependent
Variables
16.7.6a
The Econometric
Model
The selection equation
– It is expressed in terms of a latent variable z*I
that depends on one or more explanatory
variables wi, and is given by:
zi*  1   2 wi  ui
Eq. 16.37
i  1, , N
– The latent variable is not observed, but we do
observe the indicator variable:
Eq. 16.38
Principles of Econometrics, 4th Edition
*

1
z
i 0

zi  

0 otherwise
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 139
16.7
Limited Dependent
Variables
16.7.6a
The Econometric
Model
The second equation is the linear model of
interest:
yi  1  2 xi  ei
Eq. 16.39
i  1,
, n,
N n
– A selectivity problem arises when yi is
observed only when zi = 1 and if the errors of
the two equations are correlated
• In such a situation the usual least squares
estimators of β1and β2 are biased and
inconsistent
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 140
16.7
Limited Dependent
Variables
16.7.6a
The Econometric
Model
Consistent estimators are based on the conditional
regression function:
Eq. 16.40
E  yi | zi*  0  1  2 xi   i , i  1,
,n
where the additional variable λi is the ‘‘inverse
Mills ratio”:
Eq. 16.41
Principles of Econometrics, 4th Edition
  1   2 wi 
i 
  1   2 wi 
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 141
16.7
Limited Dependent
Variables
16.7.6a
The Econometric
Model
Consistent estimators are based on the conditional
regression function:
Eq. 16.40
E  yi | zi*  0  1  2 xi   i , i  1,
,n
where the additional variable λi is the ‘‘inverse
Mills ratio”:
Eq. 16.41
Principles of Econometrics, 4th Edition
  1   2 wi 
i 
  1   2 wi 
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 142
16.7
Limited Dependent
Variables
16.7.6a
The Econometric
Model
The parameters γ1 and γ2 can be estimated using a
probit model, based on the observed binary
outcome zi so that the estimated IMR:
i 
  1   2 wi 
  1   2 wi 
– Therefore:
Eq. 16.42
Principles of Econometrics, 4th Edition
yi  1  2 xi   i  vi , i  1,
Chapter 16: Qualitative and Limited Dependent Variable
Models
,n
Page 143
16.7
Limited Dependent
Variables
16.7.6b
Heckit Example:
Wages of Married
Women
Eq. 16.43
An estimated model is:
ln WAGE   0.4002  0.1095EDUC  0.0157 EXPER
(t)
(  2.10)
(7.73)
R2  0.1484
(3.90)
The Heckit procedure starts by estimating a probit
model:
P  LFP  1   1.1923  0.0206 AGE  0.0838 EDUC  0.3139 KIDS  1.3939 MTR 
(t )
(  2.93)
(3.61)
(  2.54)
(  2.26)
The inverse Mills ratio is:
  IMR 
Principles of Econometrics, 4th Edition
 1.1923  0.0206 AGE  0.0838EDUC  0.3139KIDS  1.3939MTR 
 1.1923  0.0206 AGE  0.0838EDUC  0.3139KIDS  1.3939MTR 
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 144
16.7
Limited Dependent
Variables
16.7.6b
Heckit Example:
Wages of Married
Women
The final combined model is:
ln WAGE   0.8105  0.0585EDUC  0.0163EXPER  0.8664 IMR
Eq. 16.44
(t )
(t -adj)
Principles of Econometrics, 4th Edition
(1.64)
(1.33)
(2.45)
(1.97)
(4.08)
(3.88)
Chapter 16: Qualitative and Limited Dependent Variable
Models
(  2.65)
(  2.17)
Page 145
16.7
Limited Dependent
Variables
16.7.6b
Heckit Example:
Wages of Married
Women
In most instances it is preferable to estimate the
full model, both the selection equation and the
equation of interest, jointly by maximum
likelihood
– The maximum likelihood estimated wage
equation is:
ln WAGE   0.6686  0.0658 EDUC  0.0118 EXPER
(t )
(2.84)
(3.96)
(2.87)
– The standard errors based on the full
information maximum likelihood procedure are
smaller than those yielded by the two-step
estimation method
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 146
Key Words
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 147
Keywords
binary choice
models
censored data
conditional logit
count data models
feasible
generalized least
squares
Heckit
identification
problem
independence of
irrelevant
alternatives (IIA)
Principles of Econometrics, 4th Edition
index models
individual and
alternative
specific variables
individual
specific variables
latent variables
likelihood
function
limited
dependent
variables
linear probability
model
Chapter 16: Qualitative and Limited Dependent Variable
Models
logistic random
variable
logit
log-likelihood
function
marginal effect
maximum
likelihood
estimation
multinomial
choice models
multinomial logit
Page 148
Keywords
odds ratio
ordered choice
models
ordered probit
ordinal variables
Principles of Econometrics, 4th Edition
Poisson random
variable
Poisson
regression model
probit
Chapter 16: Qualitative and Limited Dependent Variable
Models
selection bias
Tobit model
truncated data
Page 149
Appendices
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 150
16A
Probit Marginal
Effects: Details
16A.1
Standard Error of
Marginal Effect at
a Given Point
Consider the probit model p = Φ(β1 + β2x)
– The marginal effect at x = x0 is:
dp
dx
   β1  β 2 x0  β 2  g  β1 ,β 2 
x  x0
– The estimator of the marginal effect, based on
maximum likelihood, is: g  β1 ,β 2 
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 151
16A
Probit Marginal
Effects: Details
16A.1
Standard Error of
Marginal Effect at
a Given Point
The variance is:
g  β1 ,β 2  
g β1 ,β 2  


var g β1 ,β 2  
 var β1  
 var β 2


 β1 
 β 2 
g  β1 ,β 2   g β1 ,β 2  
 2

 cov β1 ,β 2
 β1   β 2 

Eq. 16A.1

2
2
 

Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
 

Page 152
16A
Probit Marginal
Effects: Details
16A.1
Standard Error of
Marginal Effect at
a Given Point
To implement the delta method we require the
derivative:
g  β1 ,β 2 
β1

 
  β1  β 2 x0  β 2 
β1
  β1  β 2 x0 

β 2

 β 2     β1  β 2 x0  
β1
β1


   β1  β 2 x0    β1  β 2 x0   β 2
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 153
16A
Probit Marginal
Effects: Details
16A.1
Standard Error of
Marginal Effect at
a Given Point
To obtain the final result, we used β 2 β1  0 and:
  β1  β 2 x0 
β1
  1  12 β1 β2 x0 2 

e


β1  2

1  12 β1 β2 x0 2 
1


e
2



β

β
x
 1 2 0 

2
2


   β1  β 2 x0    β1  β 2 x0 
We then obtain the key derivative:
g  β1 ,β 2 
β 2
Principles of Econometrics, 4th Edition
   β1  β 2 x0  1   β1  β 2 x0   β 2 x0 
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 154
16A
Probit Marginal
Effects: Details
16A.1
Standard Error of
Marginal Effect at
a Given Point
Using the transportation data, we get:
 
 var β
1


cov β1 ,β 2

Principles of Econometrics, 4th Edition



cov β1 ,β 2  0.1593956 0.0003261

   0.0003261 0.0105817 

var β 2  

 
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 155
16A
Probit Marginal
Effects: Details
16A.1
Standard Error of
Marginal Effect at
a Given Point
For DTIME = 2 (x0 = 2), the calculated values of
the derivatives are:
g  β1 ,β 2 
β1
 0.055531 and
g  β1 ,β 2 
β 2
 0.2345835
The estimated variance and standard error of the
marginal effect are:




var  g β1 ,β 2   0.0010653 and se  g β1 ,β 2   0.0326394




Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 156
16A
Probit Marginal
Effects: Details
16A.2
Standard Error of
Average Marginal
Effect
The average marginal effect of this continuous
variable is:
1 N
AME   i 1  β1  β2 DTIMEi  β2  g2 β1 ,β2 
N
We require the derivatives:
g 2  β1 ,β 2 
β1
 1 N



β

β
DTIME
β

2
i 2
i 1  1

β1  N

1 N 
  i 1
  β1  β 2 DTIMEi  β 2 

N
β1
1 N g  β1 ,β 2 
  i 1
N
β1
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 157
16A
Probit Marginal
Effects: Details
16A.2
Standard Error of
Average Marginal
Effect
Similarly:
g 2  β1 ,β 2 
β 2


β 2
1 N


β

β
DTIME
β
i 2
 N  i 1  1 2


1 N 
  i 1

β1  β 2 DTIMEi  β 2 

N
β 2
1 N g  β1 ,β 2 
  i 1
N
β 2
Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 158
16A
Probit Marginal
Effects: Details
16A.2
Standard Error of
Average Marginal
Effect
For the transportation data:
g 2  β1 ,β 2 
β1
 0.00185 and
g 2  β1 ,β 2 
β 2
 0.032366
The estimated variance and standard error of the
average marginal effect are:




var  g 2 β1 ,β 2   0.0000117 and se  g 2 β1 ,β 2   0.003416




Principles of Econometrics, 4th Edition
Chapter 16: Qualitative and Limited Dependent Variable
Models
Page 159