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Principles of Econometrics, 4th Edition
Page 1Chapter 16: Qualitative and Limited Dependent
Variable Models
Chapter 16Qualitative and Limited
Dependent Variable Models
Walter R. Paczkowski Rutgers University
Principles of Econometrics, 4th Edition
Page 2Chapter 16: Qualitative and Limited Dependent
Variable Models
16.1 Models with Binary Dependent Variables16.2 The Logit Model for Binary Choice16.3 Multinomial Logit16.4 Conditional Logit16.5 Ordered Choice Models16.6 Models for Count Data16.7 Limited Dependent Variables
Chapter Contents
Principles of Econometrics, 4th Edition
Page 3Chapter 16: Qualitative and Limited Dependent
Variable Models
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
Page 4Chapter 16: Qualitative and Limited Dependent
Variable Models
16.1
Models with Binary Dependent Variables
Principles of Econometrics, 4th Edition
Page 5Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.1Models with Binary Dependent Variables
Principles of Econometrics, 4th Edition
Page 6Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.1Models with Binary Dependent Variables
Principles of Econometrics, 4th Edition
Page 7Chapter 16: Qualitative and Limited Dependent
Variable Models
We represent an individual’s choice by the indicator variable:
16.1Models with Binary Dependent Variables
1 individual drives to work
0 individual takes bus to worky
Eq. 16.1
Principles of Econometrics, 4th Edition
Page 8Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
with
16.1Models with Binary Dependent Variables
Eq. 16.21( ) (1 ) , 0,1y yf y p p y
, var 1E y p y p p
Principles of Econometrics, 4th Edition
Page 9Chapter 16: Qualitative and Limited Dependent
Variable Models
For our analysis, define the explanatory variable as:
x = (commuting time by bus - commuting time by car)
16.1Models with Binary Dependent Variables
Principles of Econometrics, 4th Edition
Page 10Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.1Models with Binary Dependent Variables
16.1.1The Linear
Probability Model
Principles of Econometrics, 4th Edition
Page 11Chapter 16: Qualitative and Limited Dependent
Variable Models
In regression analysis we break the dependent variable into fixed and random parts– If we do this for the indicator variable y, we
have:
– Assuming that the relationship is linear:
16.1Models with Binary Dependent Variables
16.1.1The Linear
Probability Model
( )y E y e p e Eq. 16.3
1 2( )E y p x Eq. 16.4
Principles of Econometrics, 4th Edition
Page 12Chapter 16: Qualitative and Limited Dependent
Variable Models
The linear regression model for explaining the choice variable y is called the linear probability model:
16.1Models with Binary Dependent Variables
16.1.1The Linear
Probability Model
Eq. 16.5 1 2( )y E y e x e
Principles of Econometrics, 4th Edition
Page 13Chapter 16: Qualitative and Limited Dependent
Variable Models
The probability density functions for y and e are:
16.1Models with Binary Dependent Variables
16.1.1The Linear
Probability Model
Principles of Econometrics, 4th Edition
Page 14Chapter 16: Qualitative and Limited Dependent
Variable Models
Using these values it can be shown that the variance of the error term e is:
– The estimated variance of the error term is:
16.1Models with Binary Dependent Variables
16.1.1The Linear
Probability Model
1 2 1 2var 1e x x
21 2 1 2ˆ var 1i i i ie b b x b b x Eq. 16.6
Principles of Econometrics, 4th Edition
Page 15Chapter 16: Qualitative and Limited Dependent
Variable Models
We can transform the data as:
– And estimate the model:
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
16.1Models with Binary Dependent Variables
16.1.1The Linear
Probability Model
*
*
ˆ
ˆi i i
i i i
y y
x x
* 1 * *1 2ˆi i i iy x e
Principles of Econometrics, 4th Edition
Page 16Chapter 16: Qualitative and Limited Dependent
Variable Models
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
– By substituting alternative values of x,we can easily obtain values that are less than zero or greater than one
16.1Models with Binary Dependent Variables
16.1.1The Linear
Probability Model
1 2p̂ b b x Eq. 16.7
Principles of Econometrics, 4th Edition
Page 17Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.1Models with Binary Dependent Variables
16.1.1The Linear
Probability Model
Eq. 16.8 2
dp
dx
Principles of Econometrics, 4th Edition
Page 18Chapter 16: Qualitative and Limited Dependent
Variable Models
To keep the choice probability p within the interval [0, 1], a nonlinear S-shaped relationship between x and p can be used
16.1Models with Binary Dependent Variables
16.1.2The Probit Model
Principles of Econometrics, 4th Edition
Page 19Chapter 16: Qualitative and Limited Dependent
Variable Models
16.1Models with Binary Dependent Variables
16.1.2The Probit Model
FIGURE 16.1 (a) Standard normal cumulative distribution function (b) Standard normal probability density function
Principles of Econometrics, 4th Edition
Page 20Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
– The probit function is:
16.1Models with Binary Dependent Variables
16.1.2The Probit Model
2.51( )
2zz e
2.51( ) [ ]
2uz
z P Z z e du
Eq. 16.9
Principles of Econometrics, 4th Edition
Page 21Chapter 16: Qualitative and Limited Dependent
Variable Models
The probit statistical model expresses the probability p that y takes the value 1 to be:
– The probit model is said to be nonlinear
16.1Models with Binary Dependent Variables
16.1.2The Probit Model
Eq. 16.10 1 2 1 2[ ] ( )p P Z x x
Principles of Econometrics, 4th Edition
Page 22Chapter 16: Qualitative and Limited Dependent
Variable Models
We can examine the marginal effect of a one-unit change in x on the probability that y = 1 by considering the derivative:
16.1Models with Binary Dependent Variables
16.1.3Interpretation of the
Probit Model
Eq. 16.11 1 2 2
( )( )
dp d t dtx
dx dt dx
Principles of Econometrics, 4th Edition
Page 23Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.1Models with Binary Dependent Variables
16.1.3Interpretation of the
Probit Model
Principles of Econometrics, 4th Edition
Page 24Chapter 16: Qualitative and Limited Dependent
Variable Models
We estimate the probability p to be:
– By comparing to a threshold value, like 0.5, we can predict choice using the rule:
16.1Models with Binary Dependent Variables
16.1.3Interpretation of the
Probit Model
1 2ˆ ( )p x Eq. 16.12
ˆ1 0.5ˆ
ˆ0 0.5
py
p
Principles of Econometrics, 4th Edition
Page 25Chapter 16: Qualitative and Limited Dependent
Variable Models
The probability function for y is combined with the probit model to obtain:
16.1Models with Binary Dependent Variables
16.1.4Maximum Likelihood
Estimation of the Probit Model
Eq. 16.131
1 2 1 2( ) [ ( )] [1 ( )] , 0,1i iy yi i i if y x x y
Principles of Econometrics, 4th Edition
Page 26Chapter 16: Qualitative and Limited Dependent
Variable Models
If the three individuals are independently drawn, then:
– The probability of observing y1 = 1, y2 = 1, and y3 = 0 is:
16.1Models with Binary Dependent Variables
16.1.4Maximum Likelihood
Estimation of the Probit Model
1 2 3 1 2 3( , , ) ( ) ( ) ( )f y y y f y f y f y
1 2 3[ 1, 1, 0] (1, 1, 0) (1) (1) (0)P y y y f f f f
Principles of Econometrics, 4th Edition
Page 27Chapter 16: Qualitative and Limited Dependent
Variable Models
We now have:
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
16.1Models with Binary Dependent Variables
16.1.4Maximum Likelihood
Estimation of the Probit Model
Eq. 16.14
1 2 3 1 2 1 2
1 2
1 2
[ 1, 1, 0] (15) (6)
1 (7)
β ,β
P y y y
L
Principles of Econometrics, 4th Edition
Page 28Chapter 16: Qualitative and Limited Dependent
Variable Models
In practice, instead of maximizing Eq. 16.14, we maximize the logarithm of Eq. 16.14, which is called the log-likelihood function:
– 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
16.1Models with Binary Dependent Variables
16.1.4Maximum Likelihood
Estimation of the Probit Model
Eq. 16.15
1 2 1 2 1 2 1 2
1 2 1 2 1 2
ln β ,β ln (15) (6) 1 (7)
ln (15) ln (6) ln 1 (7)
L
Principles of Econometrics, 4th Edition
Page 29Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.1Models with Binary Dependent Variables
16.1.4Maximum Likelihood
Estimation of the Probit Model
Principles of Econometrics, 4th Edition
Page 30Chapter 16: Qualitative and Limited Dependent
Variable Models
Let DTIME = (BUSTIME-AUTOTIME)÷10, which is the commuting time differential in 10-minute increments– The probit model is:
P(AUTO = 1) = Φ(β1+ β2DTIME)
– The maximum likelihood estimates of the parameters are:
16.1Models with Binary Dependent Variables
16.1.5A Transportation
Example
1 2 0.0644 0.3000
(se) (0.3992) (0.1029) iDTIME DTIME
Principles of Econometrics, 4th Edition
Page 31Chapter 16: Qualitative and Limited Dependent
Variable Models
The marginal effect of increasing public transportation time, given that travel via public transportation currently takes 20 minutes longer than auto travel is:
16.1Models with Binary Dependent Variables
16.1.5A Transportation
Example
1 2 2( ) ( 0.0644 0.3000 2)(0.3000)
(0.5355)(0.3000) 0.3456 0.3000 0.1037
dpDTIME
dDTIME
Principles of Econometrics, 4th Edition
Page 32Chapter 16: Qualitative and Limited Dependent
Variable Models
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 0.7983 > 0.5, we “predict” the individual will choose to drive
16.1Models with Binary Dependent Variables
16.1.5A Transportation
Example
1 2ˆ ( ) ( 0.0644 0.3000 3) 0.7983p DTIME
Principles of Econometrics, 4th Edition
Page 33Chapter 16: Qualitative and Limited Dependent
Variable Models
Rather than evaluate the marginal effect at a specific value, or the mean value, the average marginal effect (AME) is often considered:
– For our problem:– The sample standard deviation is: 0.0365– Its minimum and maximum values are 0.0025
and 0.1153
16.1Models with Binary Dependent Variables
16.1.6Further Post-
Estimation Analysis
1 2 21
1β β β
N
iiAME DTIME
N
0.0484AME
Principles of Econometrics, 4th Edition
Page 34Chapter 16: Qualitative and Limited Dependent
Variable Models
Consider the marginal effect:
– The marginal effect function is nonlinear
16.1Models with Binary Dependent Variables
16.1.6Further Post-
Estimation Analysis
1 2 2 1 2( ) ,
dpDTIME g
dDTIME
Principles of Econometrics, 4th Edition
Page 35Chapter 16: Qualitative and Limited Dependent
Variable Models
16.2
The Logit Model for Binary Choice
Principles of Econometrics, 4th Edition
Page 36Chapter 16: Qualitative and Limited Dependent
Variable Models
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 S-shaped curve used to constrain probabilities to the [0, 1] interval
16.2The Logit Model for
Binary Choice
Principles of Econometrics, 4th Edition
Page 37Chapter 16: Qualitative and Limited Dependent
Variable Models
If L is a logistic random variable, then its probability density function is:
– The cumulative distribution function for a logistic random variable is:
16.2The Logit Model for
Binary Choice
2( ) ,
1
l
l
el l
e
Eq. 16.16
1[ ]
1 ll p L l
e
Eq. 16.17
Principles of Econometrics, 4th Edition
Page 38Chapter 16: Qualitative and Limited Dependent
Variable Models
The probability p that the observed value y takes the value 1 is:
16.2The Logit Model for
Binary Choice
Eq. 16.18 1 21 2 1 2
1
1 xp P L x x
e
Principles of Econometrics, 4th Edition
Page 39Chapter 16: Qualitative and Limited Dependent
Variable Models
The probability that y = 1 is:
The probability that y = 0 is:
16.2The Logit Model for
Binary Choice
1 2
1 2
1 2
exp1
1 exp1 x
xp
xe
1 2
11
1 expp
x
Principles of Econometrics, 4th Edition
Page 40Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.2The Logit Model for
Binary Choice
Principles of Econometrics, 4th Edition
Page 41Chapter 16: Qualitative and Limited Dependent
Variable Models
Consider the Coke example with:
16.2The Logit Model for
Binary Choice
16.2.1An Empirical Example from
Marketing
1 if Coke is chosen
0 if Pepsi is chosenCOKE
Principles of Econometrics, 4th Edition
Page 42Chapter 16: Qualitative and Limited Dependent
Variable Models
Based on ‘‘scanner’’ data on 1,140 individuals who purchased Coke or Pepsi, the probit and logit models for the choice are:
16.2The Logit Model for
Binary Choice
16.2.1An Empirical Example from
Marketing
1 2 3 4
1 2 3 4
β β β _ β _
γ γ γ _ γ _
COKE
COKE
p E COKE PRATIO DISP COKE DISP PEPSI
p E COKE PRATIO DISP COKE DISP PEPSI
Principles of Econometrics, 4th Edition
Page 43Chapter 16: Qualitative and Limited Dependent
Variable Models
16.2The Logit Model for
Binary Choice
16.2.1An Empirical Example from
Marketing
Table 16.1 Coke-Pepsi Choice Models
Principles of Econometrics, 4th Edition
Page 44Chapter 16: Qualitative and Limited Dependent
Variable Models
The parameters and their estimates vary across the models and no direct comparison is very useful, but some rules of thumb exist– Roughly:
16.2The Logit Model for
Binary Choice
16.2.1An Empirical Example from
Marketing
Logit LPM
Probit LPM
Logit Probit
ˆγ 4β
ˆβ 2.5β
γ 1.6β
Principles of Econometrics, 4th Edition
Page 45Chapter 16: Qualitative and Limited Dependent
Variable Models
If the null hypothesis is H0: βk = c, then the test statistic using the probit model is:
– The t-test is based on the Wald principle
16.2The Logit Model for
Binary Choice
16.2.2Wald Hypothesis
Tests
β
se β
ak
N K
k
ct t
Principles of Econometrics, 4th Edition
Page 46Chapter 16: Qualitative and Limited Dependent
Variable Models
Using the probit model, consider the two hypotheses:
16.2The Logit Model for
Binary Choice
16.2.2Wald Hypothesis
Tests
0 3 4 1 3 4
0 3 4 1 3 4
Hypothesis (1) :β β , :β β
Hypothesis (2) :β 0, β 0, : either β or β is not zero
H H
H H
Principles of Econometrics, 4th Edition
Page 47Chapter 16: Qualitative and Limited Dependent
Variable Models
To test hypothesis (1) in a linear model, we would compute:
– 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
16.2The Logit Model for
Binary Choice
16.2.2Wald Hypothesis
Tests
_ _
1140 4 1136
_ _
β β
se β β
aDISP COKE DISP PEPSI
DISP COKE DISP PEPSI
t t
Principles of Econometrics, 4th Edition
Page 48Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.2The Logit Model for
Binary Choice
16.2.2Wald Hypothesis
Tests
Principles of Econometrics, 4th Edition
Page 49Chapter 16: Qualitative and Limited Dependent
Variable Models
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)
16.2The Logit Model for
Binary Choice
16.2.3Likelihood Ratio Hypothesis Tests
Principles of Econometrics, 4th Edition
Page 50Chapter 16: Qualitative and Limited Dependent
Variable Models
The restricted probit model is obtained by imposing the condition β3 = -β4:
–We have LR = -713.6595– The likelihood ratio test statistic value is:
16.2The Logit Model for
Binary Choice
16.2.3Likelihood Ratio Hypothesis Tests
1 2 3 4
1 2 4 4
1 2 4
β β β _ β _
β β β _ β _
β β _β _ _
COKEp E COKE PRATIO DISP COKE DISP PEPSI
PRATIO DISP COKE DISP PEPSI
PRATIO DISP PEPSI DISP COKE
2 ln ln
2 710.9486 713.6595
5.4218
U RLR L L
Principles of Econometrics, 4th Edition
Page 51Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.2The Logit Model for
Binary Choice
16.2.3Likelihood Ratio Hypothesis Tests
Principles of Econometrics, 4th Edition
Page 52Chapter 16: Qualitative and Limited Dependent
Variable Models
16.3
Multinomial Logit
Principles of Econometrics, 4th Edition
Page 53Chapter 16: Qualitative and Limited Dependent
Variable Models
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?
16.3Multinomial Logit
Principles of Econometrics, 4th Edition
Page 54Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.3Multinomial Logit
Principles of Econometrics, 4th Edition
Page 55Chapter 16: Qualitative and Limited Dependent
Variable Models
As in the logit and probit models, we will try to explain the probability that the ith person will choose alternative j
– Assume J = 3
16.3Multinomial Logit
16.3.1Multinomial Logit
Choice Probabilities
ij = individual chooses alternative p P i j
Principles of Econometrics, 4th Edition
Page 56Chapter 16: Qualitative and Limited Dependent
Variable Models
For a single explanatory factor, the choice probabilities are:
16.3Multinomial Logit
16.3.1Multinomial Logit
Choice Probabilities
Eq. 16.19a 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
Eq. 16.19b
Eq. 16.19c
Principles of Econometrics, 4th Edition
Page 57Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.3Multinomial Logit
16.3.1Multinomial Logit
Choice Probabilities
Principles of Econometrics, 4th Edition
Page 58Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
16.3Multinomial Logit
16.3.2Maximum Likelihood Estimation
11 22 33 11 22 331, 1, 1P y y y p p p
Principles of Econometrics, 4th Edition
Page 59Chapter 16: Qualitative and Limited Dependent
Variable Models
Or
16.3Multinomial Logit
16.3.2Maximum Likelihood Estimation
11 22 3312 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
11, 1, 1
1 exp exp
exp
1 exp exp
exp
1 exp exp
, , ,
P y y yx x
x
x x
x
x x
L
Principles of Econometrics, 4th Edition
Page 60Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.3Multinomial Logit
16.3.2Maximum Likelihood Estimation
Principles of Econometrics, 4th Edition
Page 61Chapter 16: Qualitative and Limited Dependent
Variable Models
For the value of the explanatory variable x0, we can calculate the predicted probabilities of each outcome being selected– For alternative 1:
– Similarly for alternatives 2 and 3
16.3Multinomial Logit
16.3.3Post-Estimation
Analysis
01
12 22 0 13 23 0
1
1 exp expp
x x
Principles of Econometrics, 4th Edition
Page 62Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
16.3Multinomial Logit
16.3.3Post-Estimation
Analysis
3
2 21all else constant
im imim m j ij
ji i
p pp p
x x
Eq. 16.20
Principles of Econometrics, 4th Edition
Page 63Chapter 16: Qualitative and Limited Dependent
Variable Models
Alternatively, and somewhat more simply, the difference in probabilities can be calculated for two specific values of xi
16.3Multinomial Logit
16.3.3Post-Estimation
Analysis
1 1 1
12 22 13 23
12 22 13 23
1
1 exp exp
1
1 exp exp
b a
b b
a a
p p p
x x
x x
Principles of Econometrics, 4th Edition
Page 64Chapter 16: Qualitative and Limited Dependent
Variable Models
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
– The effect on the probability ratio of changing the value of xi is given by the derivative:
16.3Multinomial Logit
16.3.3Post-Estimation
Analysis
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
Eq. 16.21
Eq. 16.22
Principles of Econometrics, 4th Edition
Page 65Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.3Multinomial Logit
16.3.3Post-Estimation
Analysis
Principles of Econometrics, 4th Edition
Page 66Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.3Multinomial Logit
16.3.3Post-Estimation
Analysis
Principles of Econometrics, 4th Edition
Page 67Chapter 16: Qualitative and Limited Dependent
Variable Models
16.3Multinomial Logit
16.3.4An Example
Table 16.2 Maximum Likelihood Estimates of PSE Choice
Principles of Econometrics, 4th Edition
Page 68Chapter 16: Qualitative and Limited Dependent
Variable Models
16.3Multinomial Logit
16.3.4An Example
Table 16.3 Effects of Grades on Probability of PSE Choice
Principles of Econometrics, 4th Edition
Page 69Chapter 16: Qualitative and Limited Dependent
Variable Models
16.4
Conditional Logit
Principles of Econometrics, 4th Edition
Page 70Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.4Conditional Logit
Principles of Econometrics, 4th Edition
Page 71Chapter 16: Qualitative and Limited Dependent
Variable Models
Consider a model for the probability that individual i chooses alternative j:
The conditional logit model specifies these probabilities as:
16.4Conditional Logit
16.4.1Conditional Logit
Choice Probabilities
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
Eq. 16.23
Principles of Econometrics, 4th Edition
Page 72Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.4Conditional Logit
16.4.1Conditional Logit
Choice Probabilities
Principles of Econometrics, 4th Edition
Page 73Chapter 16: Qualitative and Limited Dependent
Variable Models
We have:
16.4Conditional Logit
16.4.1Conditional Logit
Choice Probabilities
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
Principles of Econometrics, 4th Edition
Page 74Chapter 16: Qualitative and Limited Dependent
Variable Models
The own price effect is:
The change in probability of alternative j being selected if the price of alternative k changes (k ≠ j) is:
16.4Conditional Logit
16.4.2Post-Estimation
Analysis
21ijij ij
ij
pp p
PRICE
2ij
ij ikik
pp p
PRICE
Eq. 16.24
Eq. 16.25
Principles of Econometrics, 4th Edition
Page 75Chapter 16: Qualitative and Limited Dependent
Variable Models
An important feature of the conditional logit model is that the probability ratio between alternatives j and k is:
– The probability ratio depends on the difference in prices, but not on the prices themselves
16.4Conditional Logit
16.4.2Post-Estimation
Analysis
1 2
1 1 21 2
expexp
expj ijij
j k ij ikik k ik
PRICEpPRICE PRICE
p PRICE
Principles of Econometrics, 4th Edition
Page 76Chapter 16: Qualitative and Limited Dependent
Variable Models
16.4Conditional Logit
16.4.2Post-Estimation
Analysis
Table 16.4a Conditional Logit Parameter Estimates
Principles of Econometrics, 4th Edition
Page 77Chapter 16: Qualitative and Limited Dependent
Variable Models
16.4Conditional Logit
16.4.2Post-Estimation
Analysis
Table 16.4b Marginal Effect of Price on Probability of Pepsi Choice
Principles of Econometrics, 4th Edition
Page 78Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.4Conditional Logit
16.4.2Post-Estimation
Analysis
Principles of Econometrics, 4th Edition
Page 79Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
16.4Conditional Logit
16.4.3An Example
11 2
1
11 2 12 2 2
exp 1.00ˆ 0.4832
exp 1.00 exp 1.25 exp 1.10ip
Principles of Econometrics, 4th Edition
Page 80Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.4Conditional Logit
16.4.3An Example
Principles of Econometrics, 4th Edition
Page 81Chapter 16: Qualitative and Limited Dependent
Variable Models
16.5
Ordered Choice Models
Principles of Econometrics, 4th Edition
Page 82Chapter 16: Qualitative and Limited Dependent
Variable 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
16.5Ordered Choice
Models
Principles of Econometrics, 4th Edition
Page 83Chapter 16: Qualitative and Limited Dependent
Variable 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, part-time, or full-time
16.5Ordered Choice
Models
Principles of Econometrics, 4th Edition
Page 84Chapter 16: Qualitative and Limited Dependent
Variable 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:
16.5Ordered Choice
Models
1 strongly disagree
2 disagree
3 neutral
4 agree
5 strongly agree
y
Principles of Econometrics, 4th Edition
Page 85Chapter 16: Qualitative and Limited Dependent
Variable Models
There may be a natural ordering to college choice–We might rank the possibilities as:
– The usual linear regression model is not appropriate for such data, because in regression we would treat the y values as having some numerical meaning when they do not
16.5Ordered Choice
Models
3 4-year college (the full college experience)
2 2-year college (a partial college experience)
1 no college
y
Eq. 16.26
Principles of Econometrics, 4th Edition
Page 86Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.5Ordered Choice
Models
16.5.1Ordered Probit
Choice Probabilities
Principles of Econometrics, 4th Edition
Page 87Chapter 16: Qualitative and Limited Dependent
Variable Models
For college choice, a latent variable may be grades:
– This model is not a regression model, because the dependent variable is unobservable• Consequently it is sometimes called an
index model
16.5Ordered Choice
Models
16.5.1Ordered Probit
Choice Probabilities
*i i iy GRADES e
Principles of Econometrics, 4th Edition
Page 88Chapter 16: Qualitative and Limited Dependent
Variable Models
16.5Ordered Choice
Models
16.5.1Ordered Probit
Choice Probabilities
FIGURE 16.2 Ordinal choices relative to thresholds
Principles of Econometrics, 4th Edition
Page 89Chapter 16: Qualitative and Limited Dependent
Variable Models
We can now specify:
16.5Ordered Choice
Models
16.5.1Ordered Probit
Choice Probabilities
*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
Principles of Econometrics, 4th Edition
Page 90Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
16.5Ordered Choice
Models
16.5.1Ordered Probit
Choice Probabilities
*1 1
1
1
1i i i i
i i
i
P y P y P GRADES e
P e GRADES
GRADES
Principles of Econometrics, 4th Edition
Page 91Chapter 16: Qualitative and Limited Dependent
Variable Models
Also:
16.5Ordered Choice
Models
16.5.1Ordered Probit
Choice Probabilities
*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
Principles of Econometrics, 4th Edition
Page 92Chapter 16: Qualitative and Limited Dependent
Variable Models
Finally:
16.5Ordered Choice
Models
16.5.1Ordered Probit
Choice Probabilities
*2 2
2
2
3
1
i i i i
i i
i
P y P y P GRADES e
P e GRADES
GRADES
Principles of Econometrics, 4th Edition
Page 93Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
16.5Ordered Choice
Models
16.5.2Estimation and Interpretation
1 2 1 2 3, , 1 2 3L P y P y P y
Principles of Econometrics, 4th Edition
Page 94Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.5Ordered Choice
Models
16.5.2Estimation and Interpretation
Principles of Econometrics, 4th Edition
Page 95Chapter 16: Qualitative and Limited Dependent
Variable Models
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?
16.5Ordered Choice
Models
16.5.2Estimation and Interpretation
2 1ˆ 2 | 2.5 2.5 2.5P y GRADES
Principles of Econometrics, 4th Edition
Page 96Chapter 16: Qualitative and Limited Dependent
Variable Models
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?
16.5Ordered Choice
Models
16.5.2Estimation and Interpretation
ˆ ˆ2 | 4.5 2 | 2.5P y GRADES P y GRADES
Principles of Econometrics, 4th Edition
Page 97Chapter 16: Qualitative and Limited Dependent
Variable Models
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?
16.5Ordered Choice
Models
16.5.2Estimation and Interpretation
1
1 2
2
1
2
3
P yGRADES
GRADES
P yGRADES GRADES
GRADES
P yGRADES
GRADES
Principles of Econometrics, 4th Edition
Page 98Chapter 16: Qualitative and Limited Dependent
Variable Models
16.5Ordered Choice
Models
16.5.3An Example
Table 16.5 Ordered Probit Parameter Estimates
Principles of Econometrics, 4th Edition
Page 99Chapter 16: Qualitative and Limited Dependent
Variable Models
16.6
Models for Count Data
Principles of Econometrics, 4th Edition
Page 100Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.6Models for Count
Data
Principles of Econometrics, 4th Edition
Page 101Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.6Models for Count
Data
Principles of Econometrics, 4th Edition
Page 102Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
where
16.6Models for Count
Data
, 0,1,2,!
yef y P Y y y
y
Eq. 16.27
! 1 2 1y y y y
Principles of Econometrics, 4th Edition
Page 103Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
– This choice defines the Poisson regression model for count data
16.6Models for Count
Data
Eq. 16.28 1 2expE Y x
Principles of Econometrics, 4th Edition
Page 104Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
16.6Models for Count
Data
16.6.1Maximum Likelihood Estimation
1 2, 0 2 2L P Y P Y P Y
Principles of Econometrics, 4th Edition
Page 105Chapter 16: Qualitative and Limited Dependent
Variable Models
The log-likelihood function is:
– Using Eq. 16.28 for λ, the log of the probability function is:
16.6Models for Count
Data
16.6.1Maximum Likelihood Estimation
1 2ln , ln 0 ln 2 ln 2L 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
Principles of Econometrics, 4th Edition
Page 106Chapter 16: Qualitative and Limited Dependent
Variable Models
Then the log-likelihood function, given a sample of N observations, becomes:
16.6Models for Count
Data
16.6.1Maximum Likelihood Estimation
1 2 1 2 1 21
ln , exp ln !N
i i i ii
L x y x y
Principles of Econometrics, 4th Edition
Page 107Chapter 16: Qualitative and Limited Dependent
Variable Models
Prediction of the conditional mean of y is straightforward:
The probability of a particular number of occurrences can be estimated by inserting the estimated conditional mean into the probability function, as:
16.6Models for Count
Data
16.6.2Interpretation of the Poisson Regression
Model
0 0 1 2 0expE y x
0 0expPr , 0,1,2,
!
y
Y y yy
Principles of Econometrics, 4th Edition
Page 108Chapter 16: Qualitative and Limited Dependent
Variable Models
The marginal effect is:
– This can be expressed as a percentage, which can be useful:
16.6Models for Count
Data
16.6.2Interpretation of the Poisson Regression
Model
2
ii
i
E y
x
Eq. 16.29
2
%100 100 %i i
i i
E y E y E y
x x
Principles of Econometrics, 4th Edition
Page 109Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.6Models for Count
Data
16.6.2Interpretation of the Poisson Regression
Model
1 2
1 2
| 0 exp
| 1 exp
i i i
i i i
E y D x
E y D x
Principles of Econometrics, 4th Edition
Page 110Chapter 16: Qualitative and Limited Dependent
Variable Models
The percentage change in the conditional mean is:
16.6Models for Count
Data
16.6.2Interpretation of the Poisson Regression
Model
1 2 1 2
1 2
exp exp100 % 100 1 %
expi i
i
x xe
x
Principles of Econometrics, 4th Edition
Page 111Chapter 16: Qualitative and Limited Dependent
Variable Models
16.6Models for Count
Data
16.6.3An Example
Table 16.6 Poisson Regression Estimates
Principles of Econometrics, 4th Edition
Page 112Chapter 16: Qualitative and Limited Dependent
Variable Models
16.7
Limited Dependent Variables
Principles of Econometrics, 4th Edition
Page 113Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.7Limited Dependent
Variables
Principles of Econometrics, 4th Edition
Page 114Chapter 16: Qualitative and Limited Dependent
Variable Models
16.7Limited Dependent
Variables
16.7.1Censored Data
FIGURE 16.3 Histogram of wife’s hours of work in 1975
Principles of Econometrics, 4th Edition
Page 115Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.7Limited Dependent
Variables
16.7.1Censored Data
Principles of Econometrics, 4th Edition
Page 116Chapter 16: Qualitative and Limited Dependent
Variable Models
We previously showed the probability density functions for the dependent variable y, at different x-values, centered on the regression function
– 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
16.7Limited Dependent
Variables
16.7.1Censored Data
1 2|E y x x Eq. 16.30
Principles of Econometrics, 4th Edition
Page 117Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.7Limited Dependent
Variables
16.7.1Censored Data
Principles of Econometrics, 4th Edition
Page 118Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
–We assume:
16.7Limited Dependent
Variables
16.7.2A Monte Carlo
Experiment
*1 2 9i i i i iy x e x e Eq. 16.31
2~ 0, 16ie N
*
* *
0 if 0
if 0
i i
i i i
y y
y y y
Principles of Econometrics, 4th Edition
Page 119Chapter 16: Qualitative and Limited Dependent
Variable Models
16.7Limited Dependent
Variables
16.7.2A Monte Carlo
Experiment
FIGURE 16.4 Uncensored sample data and regression function
Principles of Econometrics, 4th Edition
Page 120Chapter 16: Qualitative and Limited Dependent
Variable Models
In Figure 16.5 we show the estimated regression function for the 200 observed y-values, which is given by:
– If we restrict our sample to include only the 100 positive y-values, the fitted regression is:
16.7Limited Dependent
Variables
16.7.2A Monte Carlo
Experiment
ˆ 2.1477 0.5161
(se) (.3706) (0.0326)
y x Eq. 16.32a
ˆ 3.1399 0.6388
(se) (1.2055) (0.0827)
y x Eq. 16.32b
Principles of Econometrics, 4th Edition
Page 121Chapter 16: Qualitative and Limited Dependent
Variable Models
16.7Limited Dependent
Variables
16.7.2A Monte Carlo
Experiment
FIGURE 16.5 Censored sample data, and latent regression function and least squares fitted line
Principles of Econometrics, 4th Edition
Page 122Chapter 16: Qualitative and Limited Dependent
Variable Models
We can compute the average values of the estimates, which is the Monte Carlo ‘‘expected value’’:
where bk(m) is the estimate of βk in the mth Monte Carlo sample
16.7Limited Dependent
Variables
16.7.2A Monte Carlo
Experiment
Eq. 16.33 ( )1
1 NSAM
MC k k mm
E b bNSAM
Principles of Econometrics, 4th Edition
Page 123Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.7Limited Dependent
Variables
16.7.3Maximum Likelihood Estimation
Principles of Econometrics, 4th Edition
Page 124Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.7Limited Dependent
Variables
16.7.3Maximum Likelihood Estimation
Principles of Econometrics, 4th Edition
Page 125Chapter 16: Qualitative and Limited Dependent
Variable Models
The (probit) probability that y = 0 is:
16.7Limited Dependent
Variables
16.7.3Maximum Likelihood Estimation
1 20 [ 0] 1i i iP y P y x
Principles of Econometrics, 4th Edition
Page 126Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
– The maximum likelihood estimator is consistent and asymptotically normal, with a known covariance matrix.
16.7Limited Dependent
Variables
16.7.3Maximum Likelihood Estimation
1
221 2 21 2 1 22
0 0
1, , 1 2 exp
2i i
ii i
y y
xL y x
Principles of Econometrics, 4th Edition
Page 127Chapter 16: Qualitative and Limited Dependent
Variable Models
For artificial data, we estimate:
16.7Limited Dependent
Variables
16.7.3Maximum Likelihood Estimation
10.2773 1.0487
(se) (1.0970) (0.0790)i iy x
Eq. 16.34
Principles of Econometrics, 4th Edition
Page 128Chapter 16: Qualitative and Limited Dependent
Variable Models
16.7Limited Dependent
Variables
16.7.3Maximum Likelihood Estimation
Table 16.7 Censored Data Monte Carlo Results
Principles of Econometrics, 4th Edition
Page 129Chapter 16: Qualitative and Limited Dependent
Variable Models
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)
16.7Limited Dependent
Variables
16.7.4Tobit Model
Interpretation
Principles of Econometrics, 4th Edition
Page 130Chapter 16: Qualitative and Limited Dependent
Variable Models
The slope of E(y|x) is:
16.7Limited Dependent
Variables
16.7.4Tobit Model
Interpretation
1 22
|E y x x
x
Eq. 16.35
Principles of Econometrics, 4th Edition
Page 131Chapter 16: Qualitative and Limited Dependent
Variable Models
The marginal effect can be decomposed into two factors called the ‘‘McDonald-Moffit’’ decomposition:
– 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
16.7Limited Dependent
Variables
16.7.4Tobit Model
Interpretation
| | , 0 Prob 0Prob 0 | , 0
E y x E y x y yy E y x y
x x x
Principles of Econometrics, 4th Edition
Page 132Chapter 16: Qualitative and Limited Dependent
Variable Models
16.7Limited Dependent
Variables
16.7.4Tobit Model
Interpretation
FIGURE 16.6 Censored sample data, and regression functions for observed and positive y-values
Principles of Econometrics, 4th Edition
Page 133Chapter 16: Qualitative and Limited Dependent
Variable Models
Consider the regression model:
16.7Limited Dependent
Variables
16.7.5An Example
1 2 3 4 5 6HOURS EDUC EXPER AGE KIDSL e Eq. 16.36
Principles of Econometrics, 4th Edition
Page 134Chapter 16: Qualitative and Limited Dependent
Variable Models
16.7Limited Dependent
Variables
16.7.5An Example
Table 16.8 Estimates of Labor Supply Function
Principles of Econometrics, 4th Edition
Page 135Chapter 16: Qualitative and Limited Dependent
Variable Models
The calculated scale factor is– The marginal effect on observed hours of work
of another year of education is:
• 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
16.7Limited Dependent
Variables
16.7.5An Example
0.3638
2 73.29 0.3638 26.34
E HOURS
EDUC
Principles of Econometrics, 4th Edition
Page 136Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.7Limited Dependent
Variables
16.7.6Sample Selection
Principles of Econometrics, 4th Edition
Page 137Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.7Limited Dependent
Variables
16.7.6Sample Selection
Principles of Econometrics, 4th Edition
Page 138Chapter 16: Qualitative and Limited Dependent
Variable Models
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
16.7Limited Dependent
Variables
16.7.6Sample Selection
Principles of Econometrics, 4th Edition
Page 139Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
– The latent variable is not observed, but we do observe the indicator variable:
16.7Limited Dependent
Variables
16.7.6aThe Econometric
Model
*1 2 1, ,i i iz w u i N Eq. 16.37
*1 0
0 otherwise
i
i
zz
Eq. 16.38
Principles of Econometrics, 4th Edition
Page 140Chapter 16: Qualitative and Limited Dependent
Variable Models
The second equation is the linear model of interest:
– 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
16.7Limited Dependent
Variables
16.7.6aThe Econometric
Model
Eq. 16.39 1 2 1, , ,i i iy x e i n N n
Principles of Econometrics, 4th Edition
Page 141Chapter 16: Qualitative and Limited Dependent
Variable Models
Consistent estimators are based on the conditional regression function:
where the additional variable λi is the ‘‘inverse Mills ratio”:
16.7Limited Dependent
Variables
16.7.6aThe Econometric
Model
Eq. 16.40 *1 2| 0 , 1, ,i i i iE y z x i n
1 2
1 2
ii
i
w
w
Eq. 16.41
Principles of Econometrics, 4th Edition
Page 142Chapter 16: Qualitative and Limited Dependent
Variable Models
Consistent estimators are based on the conditional regression function:
where the additional variable λi is the ‘‘inverse Mills ratio”:
16.7Limited Dependent
Variables
16.7.6aThe Econometric
Model
Eq. 16.40 *1 2| 0 , 1, ,i i i iE y z x i n
1 2
1 2
ii
i
w
w
Eq. 16.41
Principles of Econometrics, 4th Edition
Page 143Chapter 16: Qualitative and Limited Dependent
Variable Models
The parameters γ1 and γ2 can be estimated using a probit model, based on the observed binary outcome zi so that the estimated IMR:
– Therefore:
16.7Limited Dependent
Variables
16.7.6aThe Econometric
Model
1 2
1 2
ii
i
w
w
1 2 , 1, ,i i i iy x v i n Eq. 16.42
Principles of Econometrics, 4th Edition
Page 144Chapter 16: Qualitative and Limited Dependent
Variable Models
An estimated model is:
The Heckit procedure starts by estimating a probit model:
The inverse Mills ratio is:
16.7Limited Dependent
Variables
16.7.6bHeckit Example: Wages of Married
Women
Eq. 16.43 2ln 0.4002 0.1095 0.0157 0.1484
(t) ( 2.10) (7.73) (3.90)
WAGE EDUC EXPER R
1 1.1923 0.0206 0.0838 0.3139 1.3939
( ) ( 2.93) (3.61) ( 2.54) ( 2.26)
P LFP AGE EDUC KIDS MTR
t
1.1923 0.0206 0.0838 0.3139 1.3939
1.1923 0.0206 0.0838 0.3139 1.3939
AGE EDUC KIDS MTRIMR
AGE EDUC KIDS MTR
Principles of Econometrics, 4th Edition
Page 145Chapter 16: Qualitative and Limited Dependent
Variable Models
The final combined model is:
16.7Limited Dependent
Variables
16.7.6bHeckit Example: Wages of Married
Women
Eq. 16.44
ln 0.8105 0.0585 0.0163 0.8664
( ) (1.64) (2.45) (4.08) ( 2.65)
( -adj) (1.33) (1.97) (3.88)
WAGE EDUC EXPER IMR
t
t
( 2.17)
Principles of Econometrics, 4th Edition
Page 146Chapter 16: Qualitative and Limited Dependent
Variable Models
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:
– The standard errors based on the full information maximum likelihood procedure are smaller than those yielded by the two-step estimation method
16.7Limited Dependent
Variables
16.7.6bHeckit Example: Wages of Married
Women
ln 0.6686 0.0658 0.0118
( ) (2.84) (3.96) (2.87)
WAGE EDUC EXPER
t
Principles of Econometrics, 4th Edition
Page 147Chapter 16: Qualitative and Limited Dependent
Variable Models
Key Words
Principles of Econometrics, 4th Edition
Page 148Chapter 16: Qualitative and Limited Dependent
Variable Models
Keywords
binary choice models
censored data
conditional logit
count data models
feasible generalized least squares
Heckit
identification problem
independence of irrelevant alternatives (IIA)
index models
individual and alternative specific variables
individual specific variables
latent variables
likelihood function
limited dependent variables
linear probability model
logistic random variable
logit
log-likelihood function
marginal effect
maximum likelihood estimation
multinomial choice models
multinomial logit
Principles of Econometrics, 4th Edition
Page 149Chapter 16: Qualitative and Limited Dependent
Variable Models
Keywords
odds ratio
ordered choice models
ordered probit
ordinal variables
Poisson random variable
Poisson regression model
probit
selection bias
Tobit model
truncated data
Principles of Econometrics, 4th Edition
Page 150Chapter 16: Qualitative and Limited Dependent
Variable Models
Appendices
Principles of Econometrics, 4th Edition
Page 151Chapter 16: Qualitative and Limited Dependent
Variable Models
Consider the probit model p = Φ(β1 + β2x)
– The marginal effect at x = x0 is:
– The estimator of the marginal effect, based on maximum likelihood, is:
16AProbit Marginal Effects: Details
0
1 2 0 2 1 2β β β β ,βx x
dpx g
dx
1 2β ,βg
16A.1Standard Error
of Marginal Effect at a Given
Point
Principles of Econometrics, 4th Edition
Page 152Chapter 16: Qualitative and Limited Dependent
Variable Models
The variance is:
16AProbit Marginal Effects: Details
2 2
1 2 1 21 2 1 2
1 2
1 2 1 21 2
1 2
β ,β β ,βvar β ,β var β var β
β β
β ,β β ,β 2 cov β ,β
β β
g gg
g g
Eq. 16A.1
16A.1Standard Error
of Marginal Effect at a Given
Point
Principles of Econometrics, 4th Edition
Page 153Chapter 16: Qualitative and Limited Dependent
Variable Models
To implement the delta method we require the derivative:
16AProbit Marginal Effects: Details
1 2 0 21 2
1 1
1 2 0 22 1 2 0
1 1
1 2 0 1 2 0 2
β β ββ ,β
β β
β β ββ β β
β β
β β β β β
xg
xx
x x
16A.1Standard Error
of Marginal Effect at a Given
Point
Principles of Econometrics, 4th Edition
Page 154Chapter 16: Qualitative and Limited Dependent
Variable Models
To obtain the final result, we used and:
We then obtain the key derivative:
16AProbit Marginal Effects: Details
21 2 0
21 2 0
1β β1 2 0 2
1 1
1β β
21 2 0
1 2 0 1 2 0
β β 1
β β 2
1 12 β β
22
β β β β
x
x
xe
e x
x x
2 1β β 0
1 21 2 0 1 2 0 2 0
2
β ,ββ β 1 β β β
β
gx x x
16A.1Standard Error
of Marginal Effect at a Given
Point
Principles of Econometrics, 4th Edition
Page 155Chapter 16: Qualitative and Limited Dependent
Variable Models
Using the transportation data, we get:
16AProbit Marginal Effects: Details
1 1 2
1 2 2
var β cov β ,β 0.1593956 0.0003261
0.0003261 0.0105817cov β ,β var β
16A.1Standard Error
of Marginal Effect at a Given
Point
Principles of Econometrics, 4th Edition
Page 156Chapter 16: Qualitative and Limited Dependent
Variable Models
For DTIME = 2 (x0 = 2), the calculated values of the derivatives are:
The estimated variance and standard error of the marginal effect are:
16AProbit Marginal Effects: Details
1 2 1 2
1 2
β ,β β ,β0.055531 and 0.2345835
β β
g g
1 2 1 2var β ,β 0.0010653 and se β ,β 0.0326394g g
16A.1Standard Error
of Marginal Effect at a Given
Point
Principles of Econometrics, 4th Edition
Page 157Chapter 16: Qualitative and Limited Dependent
Variable Models
The average marginal effect of this continuous variable is:
We require the derivatives:
16AProbit Marginal Effects: Details
1 2 2 2 1 21
1β β β β ,βN
iiAME DTIME g
N
16A.2Standard Error
of Average Marginal Effect
2 1 21 2 21
1 1
1 2 211
1 2
11
β ,β 1β β β
β β
1β β β
β
β ,β1
β
N
ii
N
ii
N
i
gDTIME
N
DTIMEN
g
N
Principles of Econometrics, 4th Edition
Page 158Chapter 16: Qualitative and Limited Dependent
Variable Models
Similarly:
16AProbit Marginal Effects: Details
16A.2Standard Error
of Average Marginal Effect
2 1 21 2 21
2 2
1 2 212
1 2
12
β ,β 1β β β
β β
1β β β
β
β ,β1
β
N
ii
N
ii
N
i
gDTIME
N
DTIMEN
g
N
Principles of Econometrics, 4th Edition
Page 159Chapter 16: Qualitative and Limited Dependent
Variable Models
For the transportation data:
The estimated variance and standard error of the average marginal effect are:
16AProbit Marginal Effects: Details
16A.2Standard Error
of Average Marginal Effect
2 1 2 2 1 2
1 2
β ,β β ,β0.00185 and 0.032366
β β
g g
2 1 2 2 1 2var β ,β 0.0000117 and β ,β 0.003416g se g