qualitative and limited dependent variable models prepared by vera tabakova, east carolina...

Chapter 16

Qualitative and Limited Dependent Variable Models

Prepared by Vera Tabakova, East Carolina University

Chapter 16: Qualitative and Limited Dependent Variable Models

16.1 Models with Binary Dependent Variables

16.2 The Logit Model for Binary Choice

16.3 Multinomial Logit

16.4 Conditional Logit

16.5 Ordered Choice Models

16.6 Models for Count Data

16.7 Limited Dependent Variables

Slide 16-2Principles of Econometrics, 3rd Edition

16.1 Models with Binary Dependent Variables Examples:

An economic model explaining why some states in the United States have ratified the Equal Rights Amendment, and others have not.

An economic model explaining why some individuals take a second, or third, job and engage in “moonlighting.”

An economic model of why some legislators in the U. S. House of Representatives vote for a particular bill and others do not.

An economic model of why the federal government awards development grants to some large cities and not others.

Slide16-3Principles of Econometrics, 3rd Edition


An economic model explaining why some loan applications are accepted and others not at a large metropolitan bank.

An economic model explaining why some individuals vote “yes” for increased spending in a school board election and others vote “no.”

An economic model explaining why some female college students decide to study engineering and others do not.



If the probability that an individual drives to work is p, then

It follows that the probability that a person uses public

transportation is .


(16.1)

(16.2)

1 individual drives to work

0 individual takes bus to worky

1 .P y p

0 1P y p

1( ) (1 ) , 0,1y yf y p p y

; var 1E y p y p p

16.1.1 The Linear Probability Model


(16.3)

(16.5)

(16.4)

( )y E y e p e

1 2( )E y p x

1 2( )y E y e x e


One problem with the linear probability model is that the error term is

heteroskedastic; the variance of the error term e varies from one

observation to another.


y value e value Probability

1

0

1 21 x

1 2x

1 2p x

1 21 1p x


Using generalized least squares, the estimated variance is:


(16.6)

1 2 1 2var 1e x x

21 2 1 2ˆ var 1i i i ie b b x b b x

*

*

* 1 * *1 2

ˆ

ˆ

ˆ

i i i

i i i

i i i i

y y

x x

y x e


Problems:

We can easily obtain values of that are less than 0 or greater than 1.

Some of the estimated variances in (16.6) may be negative.


(16.7)

(16.8)

1 2p̂ b b x

2

dp

dx

p̂

16.1.2 The Probit Model

Figure 16.1 (a) Standard normal cumulative distribution function (b) Standard normal probability density function


16.1.2 The Probit Model


(16.9)

p̂

2.51( )

2zz e

2.51( ) [ ]

2uz

z P Z z e du

(16.10)1 2 1 2[ ] ( )p P Z x x

16.1.3 Interpretation of the Probit Model

where and is the standard normal probability

density function evaluated at


(16.11)1 2 2

( )( )

dp d t dtx

dx dt dx

1 2t x 1 2( )x

1 2 .x


Equation (16.11) has the following implications:

1. Since is a probability density function its value is always

positive. Consequently the sign of dp/dx is determined by the sign of

2. In the transportation problem we expect 2 to be positive so that

dp/dx > 0; as x increases we expect p to increase.


1 2( )x


2. As x changes the value of the function Φ(β1 + β2x) changes. The

standard normal probability density function reaches its maximum

when z = 0, or when β1 + β2x = 0. In this case p = Φ(0) = .5 and an

individual is equally likely to choose car or bus transportation.

The slope of the probit function p = Φ(z) is at its maximum when

z = 0, the borderline case.



3. On the other hand, if β1 + β2x is large, say near 3, then the

probability that the individual chooses to drive is very large and

close to 1. In this case a change in x will have relatively little effect

since Φ(β1 + β2x) will be nearly 0. The same is true if β1 + β2x is a

large negative value, say near 3. These results are consistent with

the notion that if an individual is “set” in their ways, with p near 0 or

1, the effect of a small change in commuting time will be negligible.



Predicting the probability that an individual chooses the alternative

y = 1:


(16.12)1 2ˆ ( )p x

ˆ1 0.5ˆ

ˆ0 0.5

py

p

16.1.4 Maximum Likelihood Estimation of the Probit Model

Suppose that y1 = 1, y2 = 1 and y3 = 0.

Suppose that the values of x, in minutes, are x1 = 15, x2 = 20 and x3 = 5.


(16.13)11 2 1 2( ) [ ( )] [1 ( )] , 0,1i iy y

i i i if y x x y

1 2 3 1 2 3( , , ) ( ) ( ) ( )f y y y f y f y f y

16.1.4 Maximum Likelihood Estimation of the Probit Model

In large samples the maximum likelihood estimator is normally

distributed, consistent and best, in the sense that no competing

estimator has smaller variance.


(16.14)

1 2 3[ 1, 1, 0] (1,1,0) (1) (1) (0)P y y y f f f f

1 2 3

1 2 1 2 1 2

[ 1, 1, 0]

[ (15)] [ (20)] 1 [ (5)]

P y y y

16.1.5 An Example


16.1.5 An Example


(16.15)1 2 .0644 .0299

(se) (.3992) (.0103) i iDTIME DTIME

1 2 2( ) ( 0.0644 0.0299 20)(0.0299)

(.5355)(0.0299) 0.3456 0.0299 0.0104

dpDTIME

dDTIME

16.1.5 An Example

If an individual is faced with the situation that it takes 30 minutes

longer to take public transportation than to drive to work, then the

estimated probability that auto transportation will be selected is

Since the estimated probability that the individual will choose to

drive to work is 0.798, which is greater than 0.5, we “predict” that

when public transportation takes 30 minutes longer than driving to

work, the individual will choose to drive.Slide16-21Principles of Econometrics, 3rd Edition

1 2ˆ ( ) ( 0.0644 0.0299 30) .798p DTIME



(16.16) 2( ) ,1

l

l

el l

e

(16.18)

(16.17) 1[ ]

1 ll p L l

e

1 21 2 1 2

1

1 xp P L x x

e



1 2

1 2

1 2

exp1

1 exp1 x

xp

xe

1 2

11

1 expp

x

16.3 Multinomial Logit

Examples of multinomial choice situations:1. Choice of a laundry detergent: Tide, Cheer, Arm & Hammer, Wisk,

etc. 2. Choice of a major: economics, marketing, management, finance or

accounting.3. Choices after graduating from high school: not going to college,

going to a private 4-year college, a public 4 year-college, or a 2-year college.

The explanatory variable xi is individual specific, but does not change across alternatives.


16.3.1 Multinomial Logit Choice Probabilities


(16.19a)

(16.19c)

(16.19b)

112 22 13 23

1, 1

1 exp expii i

p jx x

12 222

12 22 13 23

exp, 2

1 exp expi

ii i

xp j

x x

13 233

12 22 13 23

exp, 3

1 exp expi

ii i

xp j

x x

individual chooses alternative ijp P i j

16.3.2 Maximum Likelihood Estimation


11 22 33 11 22 33

12 22 1 13 23 1

12 22 2

12 22 2 13 23 2

13 23 3

12 22 3 13 23 3

12 22 13 23

1, 1, 1

1

1 exp exp

exp

1 exp exp

exp

1 exp exp

, , ,

P y y y p p p

x x

x

x x

x

x x

L

16.3.3 Post-Estimation Analysis


01

12 22 0 13 23 0

1

1 exp expp

x x

(16.20)3

2 21all else constant

im imim m j ij

ji i

p pp p

x x

1 1 1

12 22 13 23 12 22 13 23

1 1

1 exp exp 1 exp exp

b a

b b a a

p p p

x x x x


An interesting feature of the odds ratio (16.21) is that the odds of choosing

alternative j rather than alternative 1 does not depend on how many alternatives

there are in total. There is the implicit assumption in logit models that the odds

between any pair of alternatives is independent of irrelevant alternatives

(IIA). Slide16-28Principles of Econometrics, 3rd Edition

(16.21)

(16.22)

1 2

1

exp 2,31

ijij j i

i i

pP y jx j

P y p

1

2 1 2exp 2,3ij i

j j j ii

p px j

x

16.3.4 An Example


16.4 Conditional Logit

Example: choice between three types (J = 3) of soft drinks, say Pepsi,

7-Up and Coke Classic.

Let yi1, yi2 and yi3 be dummy variables that indicate the choice made

by individual i. The price facing individual i for brand j is PRICEij.

Variables like price are to be individual and alternative specific, because they vary from individual to individual and are different for each choice the consumer might make


16.4.1 Conditional Logit Choice Probabilities


(16.23)

individual chooses alternative ijp P i j

1 2

11 2 1 12 2 2 13 2 3

exp

exp exp expj ij

iji i i

PRICEp

PRICE PRICE PRICE

16.4.1 Conditional Logit Choice Probabilities


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


The own price effect is:

The cross price effect is:


(16.24)

(16.25)

21ijij ij

ij

pp p

PRICE

2ij

ij ikik

pp p

PRICE


The odds ratio depends on the difference in prices, but not on the prices

themselves. As in the multinomial logit model this ratio does not depend on

the total number of alternatives, and there is the implicit assumption of the

independence of irrelevant alternatives (IIA).


1 2

1 1 21 2

expexp

expj ijij

j k ij ikik k ik

PRICEpPRICE PRICE

p PRICE

16.4.3 An Example


16.4.3 An Example

The predicted probability of a Pepsi purchase, given that the price of

Pepsi is $1, the price of 7-Up is $1.25 and the price of Coke is $1.10

is:


11 2

1

11 2 12 2 2

exp 1.00ˆ .4832

exp 1.00 exp 1.25 exp 1.10ip


The choice options in multinomial and conditional logit models have no natural ordering or arrangement. However, in some cases choices are ordered in a specific way. Examples include:

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

2. Assignment of grades or work performance ratings. Students receive grades A, B, C, D, F which are ordered on the basis of a teacher’s evaluation of their performance. Employees are often given evaluations on scales such as Outstanding, Very Good, Good, Fair and Poor which are similar in spirit.



3. Standard and Poor’s rates bonds as AAA, AA, A, BBB and so on, as a judgment about the credit worthiness of the company or country issuing a bond, and how risky the investment might be.

4. Levels of employment are unemployed, part-time, or full-time.

When modeling these types of outcomes numerical values are assigned to the outcomes, but the numerical values are ordinal, and reflect only the ranking of the outcomes.



Example:


1 strongly disagree

2 disagree

3 neutral

4 agree

5 strongly agree

y


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.


(16.26)

3 4-year college (the full college experience)

2 2-year college (a partial college experience)

1 no college

y

16.5.1 Ordinal Probit Choice Probabilities


*i i iy GRADES e

*2*

1 2*

1

3 (4-year college) if

2 (2-year college) if

1 (no college) if

i

i

i

y

y y

y


Figure 16.2 Ordinal Choices Relation to Thresholds




*1 1

1

1

1i i i i

i i

i

P y P y P GRADES e

P e GRADES

GRADES



*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



*2 2

2

2

3

1

i i i i

i i

i

P y P y P GRADES e

P e GRADES

GRADES

16.5.2 Estimation and Interpretation

The parameters are obtained by maximizing the log-likelihood

function using numerical methods. Most software includes options

for both ordered probit, which depends on the errors being standard

normal, and ordered logit, which depends on the assumption that the

random errors follow a logistic distribution.


1 2 1 2 3, , 1 2 3L P y P y P y


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

1. What is the probability that a high-school graduate with GRADES = 2.5 (on a 13 point scale, with 1 being the highest) will attend a 2-year college? The answer is obtained by plugging in the specific value of GRADES into the predicted probability based on the maximum likelihood estimates of the parameters,


2 12 | 2.5 2.5 2.5P y GRADES


2. What is the difference in probability of attending a 4-year college for two students, one with GRADES = 2.5 and another with GRADES = 4.5? The difference in the probabilities is calculated directly as


2 | 4.5 2 | 2.5P y GRADES P y GRADES


3. If we treat GRADES as a continuous variable, what is the marginal effect on the probability of each outcome, given a 1-unit change in GRADES? These derivatives are:


1

1 2

2

1

2

3

P yGRADES

GRADES

P yGRADES GRADES

GRADES

P yGRADES

GRADES

16.5.3 An Example



When the dependent variable in a regression model is a count of the number of occurrences of an event, the outcome variable is y = 0, 1, 2, 3, … These numbers are actual counts, and thus different from the ordinal numbers of the previous section. Examples include:

The number of trips to a physician a person makes during a year.

The number of fishing trips taken by a person during the previous year.

The number of children in a household.

The number of automobile accidents at a particular intersection during a month.

The number of televisions in a household.

The number of alcoholic drinks a college student takes in a week.



If Y is a Poisson random variable, then its probability function is

This choice defines the Poisson regression model for count data.


(16.27) , 0,1,2,!

yef y P Y y y

y

! 1 2 1y y y y

(16.28) 1 2expE Y x



1 2

1 2

, 0 2 2

ln , ln 0 ln 2 ln 2

L P Y P Y P Y

L P Y P Y P Y

1 2 1 2

ln ln ln ln !!

exp ln !

yeP Y y y y

y

x y x y

1 2 1 2 1 21

ln , exp ln !N

i i i ii

L x y x y

16.6.2 Interpretation in the Poisson Regression Model


0 0 1 2 0expE y x

0 0expPr , 0,1,2,

!

y

Y y yy



(16.29)

2i

ii

E y

x

2

%100 100 %i i

i i

E y E y E y

x x



1 2

1 2

1 2

1 2 1 2

1 2

exp

| 0 exp

| 1 exp

exp exp100 % 100 1 %

exp

i i i i

i i i

i i i

i i

i

E y x D

E y D x

E y D x

x xe

x

16.6.3 An Example


16.7 Limited Dependent Variables

16.7.1 Censored Data

Figure 16.3 Histogram of Wife’s Hours of Work in 1975



Having censored data means that a substantial fraction of the

observations on the dependent variable take a limit value. The

regression function is no longer given by (16.30).

The least squares estimators of the regression parameters obtained by

running a regression of y on x are biased and inconsistent—least

squares estimation fails.


(16.30) 1 2|E y x x

16.7.2 A Monte Carlo Experiment

We give the parameters the specific values and

Assume


(16.31)

1 29 and 1.

*1 2 9i i i i iy x e x e

2~ 0, 16 .ie N

*

* *

0 if 0;

if 0.

i i

i i i

y y

y y y


Create N = 200 random values of xi that are spread evenly (or

uniformly) over the interval [0, 20]. These we will keep fixed in

further simulations.

Obtain N = 200 random values ei from a normal distribution with

mean 0 and variance 16.

Create N = 200 values of the latent variable.

Obtain N = 200 values of the observed yi using


*

* *

0 if 0

if 0

i

i

i i

yy

y y


Figure 16.4 Uncensored Sample Data and Regression Function



Figure 16.5 Censored Sample Data, and Latent Regression Function and

Least Squares Fitted Line




(16.32a)ˆ 2.1477 .5161

(se) (.3706) (.0326)i iy x

(16.32b)ˆ 3.1399 .6388

(se) (1.2055) (.0827)i iy x

(16.33) ( )1

1 NSAM

MC k k mm

E b bNSAM


The maximum likelihood procedure is called Tobit in honor of James

Tobin, winner of the 1981 Nobel Prize in Economics, who first

studied this model.

The probit probability that yi = 0 is:


1 20 [ 0] 1i i iP y P y x

1

221 2 21 2 1 22

0 0

1, , 1 2 exp

2i i

ii i

y y

xL y x


The maximum likelihood estimator is consistent and asymptotically

normal, with a known covariance matrix.

Using the artificial data the fitted values are:


(16.34)10.2773 1.0487

(se) (1.0970) (.0790)i iy x

16.7.4 Tobit Model Interpretation

Because the cdf values are positive, the sign of the coefficient does

tell the direction of the marginal effect, just not its magnitude. If

β2 > 0, as x increases the cdf function approaches 1, and the slope of

the regression function approaches that of the latent variable model.


(16.35) 1 2

2

|E y x x

x

16.7.4 Tobit Model Interpretation

Figure 16.6 Censored Sample Data, and Regression Functions for Observed and Positive y values


16.7.5 An Example


(16.36)1 2 3 4 4 6HOURS EDUC EXPER AGE KIDSL e

2 73.29 .3638 26.34

E HOURS

EDUC

16.7.5 An Example


16.7.6 Sample Selection

Problem: our sample is not a random sample. The data we observe

are “selected” by a systematic process for which we do not account.

Solution: a technique called Heckit, named after its developer, Nobel

Prize winning econometrician James Heckman.


16.7.6a The Econometric Model

The econometric model describing the situation is composed of two equations. The first, is the selection equation that determines whether the variable of interest is observed.


(16.37)*1 2 1, ,i i iz w u i N

(16.38)

*1 0

0 otherwise

i

i

zz


The second equation is the linear model of interest. It is


(16.39)

(16.40)

1 2 1, ,i i iy x e i n N n

(16.41)

*1 2| 0 1, ,i i i iE y z x i n

1 2

1 2

ii

i

w

w


The estimated “Inverse Mills Ratio” is

The estimating equation is


(16.42)

1 2

1 2

ii

i

w

w

1 2 1, ,i i i iy x v i n

16.7.6b Heckit Example: Wages of Married Women


(16.43) 2ln .4002 .1095 .0157 .1484

(t-stat) ( 2.10) (7.73) (3.90)

WAGE EDUC EXPER R

1 1.1923 .0206 .0838 .3139 1.3939

(t-stat) ( 2.93) (3.61) ( 2.54) ( 2.26)

P LFP AGE EDUC KIDS MTR

1.1923 .0206 .0838 .3139 1.3939

1.1923 .0206 .0838 .3139 1.3939

AGE EDUC KIDS MTRIMR

AGE EDUC KIDS MTR

16.7.6b Heckit Example: Wages of Married Women

The maximum likelihood estimated wage equation is

The standard errors based on the full information maximum likelihood procedure are smaller than those yielded by the two-step estimation method.


(16.44)

ln .8105 .0585 .0163 .8664

(t-stat) (1.64) (2.45) (4.08) ( 2.65)

(t-stat-adj) (1.33) (1.97) (3.88) ( 2.17)

WAGE EDUC EXPER IMR

ln .6686 .0658 .0118

(t-stat) (2.84) (3.96) (2.87)

WAGE EDUC EXPER

Keywords

Slide 16-80Principles of Econometrics, 3rd Edition

binary choice models censored data conditional logit count data models feasible generalized least squares Heckit identification problem independence of irrelevant

alternatives (IIA) index models individual and alternative specific

variables individual specific variables latent variables likelihood function limited dependent variables linear probability model

logistic random variable logit log-likelihood function marginal effect maximum likelihood estimation multinomial choice models multinomial logit odds ratio ordered choice models ordered probit ordinal variables Poisson random variable Poisson regression model probit selection bias tobit model truncated data

qualitative and limited dependent variable models prepared by vera tabakova, east carolina...

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