Chapter 16

Chapter 16

Qualitative and Limited Dependent Variable Models

Adapted from Vera Tabakova’s notes

ECON 4551 Econometrics IIMemorial University of Newfoundland

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 individuals take a second, or third, job and engage in “moonlighting.”

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

An economic model explaining why someone is in the labour force or not

Slide16-3Principles of Econometrics, 3rd Edition

16.1 Models with Binary Dependent Variables

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

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

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


16.1 Models with Binary Dependent Variables

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

It follows that the probability that a person uses public

transportation is .


(16.1)

(16.2)

1 individual drives to work0 individual takes bus to work

y

1 .P y p

0 1P y p

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

; var 1E y p y p p

As long as these exhaust the possible (mutually exclusive) options

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

So the problem of heteroskedasticityis not insurmountable…



(16.7)

(16.8)

1 2p̂ b b x

2dpdx


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, so the

WLS would not work Of course, the errors are not distributed normally R2 is usually very poor and a questionable guide for goodness of fit


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( )2

zz e

2.51( ) [ ]2

uzz 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 dt x

dx dt dx

1 2t x 1 2( )x

1 2 .x

Note that this is clearly a nonlinear model: the marginal effect varies dependingon where you measure it

cumulative density


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 0.5p

yp

Although you have to be careful with thisInterpretation!

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.Slide16-19Principles of Econometrics, 3rd Edition

(16.14)

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

1 2 3

1 2 1 2 1 2

[ 1, 1, 0]

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

P y y y

16.1.5 An Example


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

dp DTIMEdDTIME

Marginal effect of DTMeasured at DTIME = 20

16.1.5 An Example

If it takes someone 30 minutes longer to take public transportation

than to drive to work, the estimated probability that auto

transportation will be selected is

Since this estimated probability is 0.798, which is greater than 0.5, we

may want to “predict” that when public transportation takes 30

minutes longer than driving to work, the individual will choose to

drive. But again use this cautiously!


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

16.1.5 An Example

In STATA:

Use transport.dta


auto 21 .4761905 .5117663 0 1 dtime 21 -1.223809 56.91037 -90.7 91 bustime 21 48.12381 34.63082 1.6 91.5 autotime 21 49.34762 32.43491 .2 99.1 Variable Obs Mean Std. Dev. Min Max

. sum

16.1.5 An Example

Slide16-24

0.2

.4.6

.81

= 1

if au

to c

hose

n

-100 -50 0 50 100bus time - auto time

Linear fit???

16.1.5 An Example


Understand but do not use this one!!!

You can choose p-values

What is the meaning of this test?

NORMAL distributionNot t distribution, because the properties of the probitare asymptotic

Principles of Econometrics, 3rd Edition 26

Evaluates at the means by default too dtime .0119068 .0041 2.90 0.004 .003871 .019942 -1.22381 variable dy/dx Std. Err. z P>|z| [ 95% C.I. ] X = .45971697 y = Pr(auto) (predict)Marginal effects after probit

. mfx compute _cons -.0644338 .3992438 -0.16 0.872 -.8469372 .7180696 dtime .029999 .0102867 2.92 0.004 .0098374 .0501606 auto Coef. Std. Err. z P>|z| [95% Conf. Interval]

Log likelihood = -6.1651585 Pseudo R2 = 0.5758 Prob > chi2 = 0.0000 LR chi2(1) = 16.73Probit regression Number of obs = 21

Iteration 4: log likelihood = -6.1651585 Iteration 3: log likelihood = -6.1651585 Iteration 2: log likelihood = -6.165583 Iteration 1: log likelihood = -6.2074806 Iteration 0: log likelihood = -14.532272

. probit auto dtime

16.1.5 An Example

Slide16-27

_cons -.0644338 .3992438 -0.16 0.872 -.8469372 .7180696 dtime .029999 .0102867 2.92 0.004 .0098374 .0501606 auto Coef. Std. Err. z P>|z| [95% Conf. Interval]

Log likelihood = -6.1651585 Pseudo R2 = 0.5758 Prob > chi2 = 0.0000 LR chi2(1) = 16.73Probit regression Number of obs = 21

Iteration 4: log likelihood = -6.1651585 Iteration 3: log likelihood = -6.1651585 Iteration 2: log likelihood = -6.165583 Iteration 1: log likelihood = -6.2074806 Iteration 0: log likelihood = -14.532272

. probit auto dtime

_cons -.0597171 .2736728 -0.22 0.827 -.596106 .4766718 auto Coef. Std. Err. z P>|z| [95% Conf. Interval]

Log likelihood = -14.532272 Pseudo R2 = -0.0000 Prob > chi2 = . LR chi2(0) = -0.00Probit regression Number of obs = 21

Iteration 1: log likelihood = -14.532272 Iteration 0: log likelihood = -14.532272

. probit auto

You can request these iterations in GRETL too

What yields cnorm(-0.0597171)???

Principles of Econometrics, 3rd Edition

This is a probability

16.1.5 An Example

Slide16-29

IN STATA

* marginal effectsmfxmfx,at (dtime=20)

* direct calculationnlcom (normalden(_b[_cons]+_b[dtime]*30)*_b[dtime] )andnlcom (normal(_b[_cons]+_b[dtime]*30) )

16.2 The Logit Model for Binary Choice


(16.16) 2( ) ,1

l

l

el le

(16.18)

(16.17) 1[ ]1 ll p L l

e

1 21 2 1 21

1 xp P L x xe



1 2

1 2

1 2

exp11 exp1 x

xp

xe

1 2

111 exp

px

P i

1 P i odds ratio exp 1 2X so



P i

1 P i odds ratio exp 1 2X so

ln P i

1 P i 1 2X

So the “logit”, the log-odds, is actually a fully linear function of X

1. As Probability goes from 0 to 1, logit goes from –infinite to +

infinite

2. The logit is linear, but the probability is not

3. The explanatory variables are individual specific, but do not

change across alternatives

4. The slope coefficient tells us by how much the log-odds changes

with a unit change in the variable

Slide16-33

1. This model can be in principle estimated with WLS (due to the

heteroskedasticity in the error term) if we have grouped data (glogit in

STATA, while blogit will run ML logit on grouped data) IN GRETL If

you want to use logit for analysis of proportions (where the dependent

variable is the proportion of cases having a certain characteristic, at each

observation, rather than a 1 or 0 variable indicating whether the

characteristic is present or not) you should not use the logit command,

but rather construct the logit variable, as in genr lgt_p = log(p/(1 - p))

2. Otherwise we use MLE on individual data

Slide16-34

Goodness of fit

McFadden’s pseudo R2 (remember that it does not have

any natural interpretation for values between 0 and 1) Count R2 (% of correct predictions) (dodgy but

common!) Etc. Measures of goodness of fit are of secondary

importance What counts is the sign of the regression

coefficients and their statistical and practical significance

Goodness of fit

Using MLE A large sample method => estimated errors are asymptotic => we use Z test statistics (based on the

normal distribution), instead of t statistics A likelihood ratio test (with a test statistic

distributed as chi-square with df= number of regressors) is equivalent to the F test

Goodness of fit: example

ho

BIC used by Stata: 18.419 AIC used by Stata: 16.330BIC: -45.516 BIC': -13.690AIC: 0.778 AIC*n: 16.330Count R2: 0.905 Adj Count R2: 0.800Variance of y*: 3.915 Variance of error: 1.000McKelvey & Zavoina's R2: 0.745 Efron's R2: 0.649ML (Cox-Snell) R2: 0.549 Cragg-Uhler(Nagelkerke) R2: 0.733McFadden's R2: 0.576 McFadden's Adj R2: 0.438 Prob > LR: 0.000D(19): 12.330 LR(1): 16.734Log-Lik Intercept Only: -14.532 Log-Lik Full Model: -6.165

Measures of Fit for probit of auto

See http://www.soziologie.uni-halle.de/langer/logitreg/books/long/stbfitstat.pdf

How do you obtain this?

Goodness of fit: example

Correctly classified 90.48% False - rate for classified - Pr( D| -) 9.09%False + rate for classified + Pr(~D| +) 10.00%False - rate for true D Pr( -| D) 10.00%False + rate for true ~D Pr( +|~D) 9.09% Negative predictive value Pr(~D| -) 90.91%Positive predictive value Pr( D| +) 90.00%Specificity Pr( -|~D) 90.91%Sensitivity Pr( +| D) 90.00% True D defined as auto != 0Classified + if predicted Pr(D) >= .5

Total 10 11 21 - 1 10 11 + 9 1 10 Classified D ~D Total True

Probit model for auto

. lstat

So in STATAThe “ones” do not Really have to be Actual ones, justNon-zerosIN GRETL if you do not have a binary Dependent variableIt is assumed Ordered unless specified multinomial. If not discrete: error!

But be very careful with these measures!

More diagnostics (STATA only) To compute the deviance of the

residuals: predict “newname”, deviance The deviance for a logit model is like

the RSS in OLS. The smaller the deviance the better the fit.

And (Logit only) to combine with information about leverage:predict “newnamedelta”, ddeviance

(A recommended cut-off value for the ddeviance is 4)

More diagnostics

13. .0708038 pred

. list pred if delta>4

. predict delta, ddeviance

. predict dev, deviance

. predict pred, p

_cons -.2375754 .7504766 -0.32 0.752 -1.708483 1.233332 dtime .0531098 .0206423 2.57 0.010 .0126517 .093568 auto Coef. Std. Err. z P>|z| [95% Conf. Interval]

Log likelihood = -6.1660422 Pseudo R2 = 0.5757 Prob > chi2 = 0.0000 LR chi2(1) = 16.73Logistic regression Number of obs = 21

. logit auto dtime, nolog

Probit versus Logit

bic 13.5 13.7 aic 10.3 10.5 N 21 21 df chi2 24.7 24.5 _cons -4.73 -8.15 bustime .103 .184 dtime -.0052 -.0044 Variable probit logit

Why does rule of thumb not work for dtime

Probit versus Logit

A matter of taste nowadays, since we all have good computers

The underlying distributions share the mean of zero but have different variances: Logit And normal 1

So estimated slope coefficients differ by a factor of about 1.8 ( ) . Logit ones are bigger

2

3

3

More on Probit versus Logit Watch out for “perfect predictions” Luckily STATA will flag them for you and drop the

culprit observations

Gretl has a mechanism for preventing the algorithm from iterating endlessly in search of a nonexistent maximum. One sub-case of interest is when the perfect prediction problem arises because of a single binary explanatory variable. In this case, the offending variable is dropped from the model and estimation proceeds with the reduced specification.

More on Probit versus Logit However, it may happen that no single

“perfect classifier” exists among the regressors, in which case estimation is simply impossible and the algorithm stops with an error.

If this happens, unless your model is trivially mis-specified (like predicting if a country is an oil exporter on the basis of oil revenues), it is normally a small-sample problem: you probably just don’t have enough data to estimate your model. You may want to drop some of your explanatory variables.

More on Probit versus Logit Learn about the test (Wald tests based on chi-

2) and lrtest commands (LR tests), so you can test hypotheses as we did with t-tests and F tests in OLS

They are asymptotically equivalent but can differ in small samples

More on Probit versus Logit Learn about the many extra STATA

capabilities, if you use it, that will make your postestimation life much easier

Long and Freese’s book is a great resource GRETL is more limited but doing things by

hand for now will actually be a good thing!

47Principles of Econometrics, 3rd Edition

SDofX = standard deviation of X e^bStdX = exp(b*SD of X) = change in odds for SD increase in X e^b = exp(b) = factor change in odds for unit increase in X P>|z| = p-value for z-test z = z-score for test of b=0 b = raw coefficient dtime 0.05311 2.573 0.010 1.0545 20.5426 56.9104 auto b z P>|z| e^b e^bStdX SDofX

Odds of: 1 vs 0

logit (N=21): Factor Change in Odds

. listcoef, help

_cons -.2375754 .7504766 -0.32 0.752 -1.708483 1.233332 dtime .0531098 .0206423 2.57 0.010 .0126517 .093568 auto Coef. Std. Err. z P>|z| [95% Conf. Interval]


. logit auto dtime, nolog

For example

48Principles of Econometrics, 3rd Edition

For example

female 1.918168 .6400451 1.95 0.051 .9973827 3.689024 honcomp Odds Ratio Std. Err. z P>|z| [95% Conf. Interval]


Iteration 3: log likelihood = -113.6769 Iteration 2: log likelihood = -113.67691 Iteration 1: log likelihood = -113.68907 Iteration 0: log likelihood = -115.64441

. logit honcomp female, or

_cons -1.400088 .2631619 -5.32 0.000 -1.915875 -.8842998 female .6513706 .3336752 1.95 0.051 -.0026207 1.305362 honcomp Coef. Std. Err. z P>|z| [95% Conf. Interval]


More on Probit versus Logit Stata users? Go through a couple of examples

available online with your own STATA session connected to the internet. Examples:

http://www.ats.ucla.edu/stat/stata/dae/probit.htm http://www.ats.ucla.edu/stat/stata/dae/logit.htm

http://www.ats.ucla.edu/stat/stata/output/old/lognoframe.htm

http://www.ats.ucla.edu/stat/stata/output/stata_logistic.htm

http://www.ats.ucla.edu/stat/stata/dae/logit.htm



Keywords

Slide 16-50Principles 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

References

Long, S. and J. Freese for all topics (available on Google!)

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Multinomial Logit Conditional Logit

Chapter 16

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