09-Limited Dependent Variable Models

8/3/2019 09-Limited Dependent Variable Models

http://slidepdf.com/reader/full/09-limited-dependent-variable-models 1/71

1

Limited Dependent VariableModels

EMET 8002

Lecture 9

August 27, 2009



2

Limited Dependent Variables

A limited dependent variable is a dependentvariable whose range is restricted

For example: Any indicator variable such as whether or not a

household is poor (i.e., 0 or 1)

Test scores (generally bound by 0 and 100)

The number of children born to a woman is a non-negative integer



3

Outline

Logit and probit models for binary dependentvariables

Tobit model for corner solutions



4

Why do we care?

Let’s start with a review of the linear probabilitymodel to examine some of its shortcomings

The model is given by:

where

0 1 1 ...k k

y x x u β β β = + + + +

( ) ( ) 0 1 11| | ...k k

P y E y x x β β β = = = + + +x x



5

Linear Probability Model

There will be three undesirable features of this model:

1.

The error term will not be homoskedastic. This violatesassumption LMR.4. Our OLS estimates will still be unbiased,but the standard errors are incorrect. Nonetheless, it is

easy to adjust for heteroskedasticity of unknown form.

2.

We can get predictions that are either greater than 1 orless than 0!

3.

The independent variables cannot be linearly related to thedependent variable for all possible values.



6


Example

Let’s look at how being in the labour force isinfluenced by various determinants:

Husband’s earnings

Years of education

Previous labour market experience

Age

Number of children less than 6 years old

Number of children between 6 and 18 years of age



7


ExampleCoefficientestimate

Usual standarderrors

Robust standarderrors

Husband’s income -0.0034 0.0014 0.0015

Years of education

0.038 0.007 0.007

Experience 0.039 0.006 0.006

Experience2 -0.00060 0.00018 0.00019

Age -0.016 0.002 0.002

# kids <= 6 yearsold

-0.262 0.034 0.032

# kids > 6 yearsold 0.013 0.013 0.014



8


Example

Using standard errors that are robust to unknownheteroskedasticity is simple and does notsubstantially change the reported standard errors

Interpreting the coefficients:

All else equal, an extra year of education increases theprobability of participating in the labour force by 0.038(3.8%)

All else equal, an additional child 6 years of age or lessdecreases the probability of working by 0.262



9


Example Predicted probabilities:

Sometimes we obtain predicted probabilities that are outsideof the range [0,1]. In this sample, 33 of the 753observations produce predicted probabilities outside of [0,1].

For example, consider the following observation: Husband’s earnings = 17.8

Years of education = 17

Previous labour market experience = 15

Age = 32

Number of children less than 6 years old = 0

Number of children between 6 and 18 years of age = 1

The predicted probability is 1.13!!



10


Example

An additional problem is that probabilities cannot belinearly related to the independent variables for allpossible values

For example, consider the estimate of the marginaleffect of increasing the number of children 6 years of age or younger. It is estimated to be -0.262. Thismeans that if this independent variable increased from

0 to 4, the probability of being in the labour marketwould fall by 1.048, which is impossible!



11


It is still a useful model to estimate, especially sincethe estimate coefficients are much easier to interpretthan the nonlinear models that we are going tointroduce shortly

Plus, it usually works well for values of theindependent variables that are close to the respectivemeans (i.e., outlying values of x cause problems)



12

Limited Dependent Variables

Models

In this lecture we’re going to cover estimationtechniques that will better address the nature of thedependent variable

Logit & Probit

Tobit



13

Logit and Probit Models for

Binary Response

We’re going to prevent predicted values from everfalling outside the range [0,1] by estimating anonlinear regression:

where 0<G(z)<1 for all real numbers z

The two most commonly used functions for G(.) arethe logit model and the probit model:

( ) ( )01|P y G β = = +x xβ

( )( )

( )

( )

( ) ( )

exp

1 exp

zG z z

zG z z

= = Λ

+= Φ



14


Binary Response

Logit and probit models can be derived from anunderlying latent variable model

i.e., an unobserved variable

We assume that e is independent of x and that eeither has the standard logistic distribution or thestandard normal distribution

Under either assumption e is symmetricallydistributed about 0, which implies that 1-G(-z)=G(z)for all real numbers z

* *

0 , 1 0 y e y y β ⎡ ⎤= + + = >⎣ ⎦xβ



15


Binary Response

We can now derive the response probability for y:( ) ( )( )

( )( )( )

( )

*

0

0

0

0

1| 0 |

0 |

|

1

P y P y

P e

P e

G

G

β

β

β

β

= = >

= + + >

= > − +

⎡ ⎤= − − +⎣ ⎦

= +

x x

xβ x

xβ x

xβ

xβ



16


Binary Response In most applications of binary response models our main

interest is to explain the effects of the x’s on the responseprobability P(y=1|x)

The latent variable interpretation tends to give the impression

that we are interested in the effects of the x’s on y* For probit and logit models, the direction of the effect of the x’s

on E(y*|x) and E(y|x)=P(y=1|x) are the same

In most applications however, the latent variable does not have

a well-defined unit of measurement which limits itsinterpretation. Nonetheless, in some examples this is a veryuseful tool for thinking about the problem.



17


Binary Response

The sign of the coefficients will tell us the direction of the partial effect of x j on P(y=1|x)

However, unlike the linear probability model, themagnitudes of the coefficients are not especiallyuseful

If x j is a roughly continuous variable, its partial effectis given by: ( ) ( )

j

j

p dG z

x dz

β ∂

=

∂

x



18


Binary Response In the linear probability model the derivative of G was simply 1,

since G(z)=z in the linear probability model.

In other words, we can move from this nonlinear functionback to the linear model by simply assuming G(z)=z.

For both the logit and the probit models g(z)=dG(z)/dz isalways positive (since G is the cumulative distribution function,g is the probability density function). Thus, the sign of β j is the

same as the sign of the partial effect.

The magnitude of the partial effect is influenced by the entirevector of x’s



19


Binary Response

Nonetheless, the relative effect of any twocontinuous explanatory variables do not depend on x

The ratio of the partial effects for x j and xh is β j /βh,which does not depend on x



20


Binary Response Suppose x1 is a discrete variable, its partial effect of going from

c to c+1 is given by:

Again, this effect depends on x

Note, however, that the sign of β1 is enough to know whetherthe discrete variable has a positive or negative effect

This is because G() is strictly increasing

( )( )

( )

0 1 2 2

0 1 2 2

1 ...

...

k k

k k

G c x x

G c x x

β β β β

β β β β

+ + + + + −

+ + + +



21


Binary Response

We use Maximum Likelihood Estimation, whichalready takes into consideration theheteroskedasticity inherent in the model

Assume that we have a random sample of size n

To obtain the maximum likelihood estimator,conditional on the explanatory variables, we need thedensity of yi given xi

( ) ( ) ( )

1

| ; 1 , 0,1

y y

i i i f y G G y

−

⎡ ⎤ ⎡ ⎤= − =⎣ ⎦ ⎣ ⎦x

βx

βx

β



22


Binary Response

When y=1: f(y| xi:β)=G( x

iβ)

When y=0: f(y| xi:β)=1-G( xiβ)

The log-likelihood function for observation i isgiven by:

The log-likelihood for a sample of size n is obtainedby summing this expression over all observations

( ) ( ) ( ) ( )log 1 log 1i i i i i

l y G y G⎡ ⎤ ⎡ ⎤= + − −⎣ ⎦ ⎣ ⎦β x β x β

( ) ( )1

n

i

i

L l

=

= ∑β β



23


Binary Response The MLE of β maximizes this log-likelihood

If G is the standard logit cdf, then we get the logitestimator

If G is the standard normal cdf, then we get the

probit estimator

Under general conditions, the MLE is:

Consistent Asymptotically normal

Asymptotically efficient



24

Inference in Probit and Logit Models

Standard regression software, such as Stata, willautomatically report asymptotic standard errors forthe coefficients

This means we can construct (asymptotic) t-tests forstatistical significance in the usual way:

( )ˆ ˆ j j jt se β β =



25

Logit

and Probit

Models for Binary

Response: Testing Multiple Hypotheses

We can also test for multiple exclusion restrictions(i.e., two or more regression parameters are equal to0)

There are two options commonly used:

A Wald test

A likelihood ratio test



26

Logit

and Probit

Models for Binary


Wald test:

In the linear model, the Wald statistic, can betransformed to be essentially the same as the Fstatistic

The formula can be found in Wooldridge (2002,Chapter 15)

It has an asymptotic chi-squared distribution, with

degrees of freedom equal to the number of restrictionsbeing tested

In Stata we can use the “test” command followingprobit or logit estimation



27

Logit

and Probit

Models for Binary


Likelihood ratio (LR) test If both the restricted and unrestricted models are easy to

compute (as is the case when testing exclusion restrictions),then the LR test is very attractive

It is based on the difference in the log-likelihood functions

for the restricted and unrestricted models Because the MLE maximizes the log-likelihood function,

dropping variables generally leads to a smaller log-likelihood(much in the same way are dropping variables in a liner modelleads to a smaller R 2)

The likelihood ratio statistic is given by:

It is asymptotically chi-squared with degrees of freedomequal to the number of restrictions

can use lrtest in Stata

( )2 ur r LR L L= −



28

Logit and Probit Models for BinaryResponse: Interpreting Probit

and Logit

Estimates

Recall that unlike the linear probability model, theestimated coefficients from Probit or Logit estimationdo not tell us the magnitude of the partial effect of achange in an independent variable on the predicted

probability

This depends not just on the coefficient estimates,

but also on the values of all the independentvariables and the coefficients



29


and Logit

Estimates

For roughly continuous variables the marginal effectis approximately by:

For discrete variables the estimated change in thepredicted probability is given by:

( ) ( )0ˆ ˆ ˆˆ 1|

j jP y g x β β ⎡ ⎤Δ = ≈ + Δ

⎣ ⎦x xβ

( )( )( )

0 1 2 2

0 1 2 2

ˆ ̂ ˆ ˆ1 ...

ˆ ˆ ˆ ˆ...

k k

k k

G c x x

G c x x

β β β β

β β β β

+ + + + + −+ + + +



30


and Logit

Estimates

Thus, we need to pick “interesting” value of x atwhich to evaluate the partial effects Often the sample averages are used. Thus, we obtain

the partial effect at the average (PEA)

We could also use lower or upper quartiles, forexample, to see how the partial effects change assome elements of x get large or small

If xk is a binary variable, then it often makes sense touse a value of 0 or 1 in the partial effect equation,

rather than the average value of xk



31


and Logit

Estimates

An alternative approach is to calculate the averagepartial effect (APE)

For a continuous explanatory variable, x j, the APE is:

The two scale factors (at the mean for PEA andaveraged over the sample for the APE) differ sincethe first uses a nonlinear function of the average and

the second uses the average of a nonlinear function

( ) ( )1 1

0 0

1 1

ˆ ˆ ̂ ˆ ˆ ̂n n

i j i j

i i

n g n g β β β β − −

= =

⎡ ⎤ ⎡ ⎤+ = +⎣ ⎦ ⎣ ⎦∑ ∑x β x β



32

Example 17.1: Married Women’s

Labour Force Participation

We are going to use the data in MROZ.RAW toestimate a labour force participation for women usinglogit and probit estimation.

The explanatory variables are nwifeinc, educ, exper,

exper2, age, kidslt6, kidsge6

probit inlf nwifeinc educ exper expersq age kidslt6kidsge6



33

Example 17.1Independentvariable

Coefficient Estimates

OLS

(robust stderr)

Probit Logit

Husband’s income -0.0034

(0.0015)

-0.012

(0.005)

-0.021

(0.008)

Years of education

0.038

(0.007)

0.131

(0.025)

0.221

(0.043)

Age -0.016

(0.002)

-0.053

(0.008)

-0.088

(0.014)# kids <= 6 yearsold

-0.262

(0.032)

-0.868

(0.119)

-1.44

(0.20)

# kids > 6 years

old

0.013

(0.014)

0.036

(0.043)

0.060

(0.075)



34

Example 17.1

True or false:

The Probit and Logit model estimates suggest that thelinear probability model was underestimating thenegative impact of having young children on the

probability of women participating in the labour force.



35

Example 17.1

How does the predicted probability change as thenumber of young children increases from 0 to 1?What about from 1 to 2?

We’ll evaluate the effects at:

Husband’s income=20.13

Education=12.3

Experience=10.6

Age=42.5 # older children=1

These are all close to the sample averages



36

Example 17.1 From the probit estimates:

Going from 0 to 1 small child decreases the probability of labour force participation by 0.334

Going from 1 to 2 small child decreases the probability of labour force participation by 0.256

Notice that the impact of one extra child is now nonlinear (thereis a diminishing impact). This differs from the linear probabilitymodel which says any increase of one young child has the same

impact.



37

Logit

and Probit

Models for Binary

Response

Similar to linear models, we have to be concerned with

endogenous explanatory variables. We don’t have time to coverthis so see Wooldridge (2002, Chapter 15) for a discussion

We need to be concerned with heteroskedasticity in probit andlogit models. If var(e| x) depends on x then the responseprobability no longer has the form G(β0+β x) implying that moregeneral estimation techniques are required

The linear probability can be applied to panel data, typically

estimated using fixed effects Logit and probit models with unobserved effects are difficult

to estimate and interpret (see Wooldridge (2002, Chapter15))



38

The Tobit Model for Corner

Solution Responses

Often in economics we observes variables for which 0(or some other fixed number) is in an optimaloutcome for some units of observations, but a rangeof positive outcomes prevail for other observations

For example:

Number of hours worked annually

Trade flows

Hours spent on the internet Grade on a test (may be grouped at both 0 and 100)



39


Solution Responses

Let y be a variable that is roughly continuous overstrictly positive values but that takes on zero with apositive probability

Similar to the binary dependent variable context wecan use a linear model and this might not be so badfor observations that are close to the mean, but we

may obtain negative fitted values and thereforenegative predictions for y



40


Solution Responses We often express the observed outcome, y, in terms

of an unobserved latent variable, say y*

We now need to think about how to estimate this

model. There are two cases to consider: When y=0

When y>0

( )

( )

2* , | ~ 0,

max 0, *

y u u N

y y

σ = +

=

xβ x





42


Solution Responses What is the probability that y>0 conditional on the

explanatory variables?

Since y is continuous for values greater than 0, theprobability is simply the density of the normalvariable u

We can now put together these two pieces to formthe log-likelihood function for the Tobit model (seeequation 17.22 in Wooldridge)





44

Interpreting Tobit estimates

Take home message: Conditional expectations in theTobit are much more complicated than in the linearmodel

E(y|x) is a nonlinear of function of both x and β.Moreover, this conditional expectation can be shownto be positive for any values of x and β.

( ) ( )

( ) ( ) ( )

| 0, /

| / /

E y y

E y

σλ σ

σ σφ σ

> = +

= Φ +

x xβ xβ

x xβ xβ xβ



45

Interpreting Tobit estimates To examine partial effects, we should consider two cases:

When x j is continuous When x j is discrete

When x j is continuous we can use calculus to solve for the

partial effects:

Like in probit or logit models, the partial effect will depend onall explanatory variables and parameters

( )( ) ( ){ }

( )( )

| 0,1

|

j

j

j

j

E y y

x

E y

x

β λ σ σ λ σ

β σ

∂ >⎡ ⎤= − +⎣ ⎦∂

∂= Φ

∂

x

xβ xβ xβ

x

xβ



46

Interpreting Tobit estimates When x

j

is discrete we estimate the partial effect asthe difference:

( ) ( )

( ) ( )

| 0, , 1 | 0, ,

| , 1 | ,

j j j j

j j j j

E y y x c E y y x c

E y x c E y x c

− −

− −

> = + − > =

= + − =

x x

x x



47

Interpreting Tobit estimates Just like the probit and logit models, there are two

common approaches for evaluating the partialeffects:

Partial Effect at the Average (PEA)

Evaluate the expressions at the same average

Average Partial Effect (APE)

Calculate the mean over the values for the entire sample



48

Example 17.2: Women’s

annual labour supply We can use the same dataset, MROZ.RAW, that we

used to estimate the probability of womenparticipating in the labour force to estimate theimpact of various explanatory variables on the total

number of hours worked

Of the 753 women in the sample:

428 worked for a wage during the year

325 worked zero hours in the labour market



49

Tobit example: Women’s

annual labour supply reg hours nwifeinc educ exper expersq age kidslt6

kidsge6

tobit hours nwifeinc educ exper expersq age kidslt6kidsge6, ll(0)



50


annual labour supplyCoefficient Estimates

OLS Tobit

Husband’s income -3.45

(2.54)

-8.81

(4.46)

Years of education 28.76(12.95)

80.65(21.58)

Age -30.51

(4.36)

-54.41

(7.42)

# kids <= 6 years old -442.09

(58.85)

-894.02

(111.88)

# kids > 6 years old -32.78

(23.18)

-16.22

(38.64)Sigma 1122.022

(41.58)



51


annual labour supply The Tobit coefficient estimates all have the same sign

as the OLS coefficients

The pattern of statistical significance is also verysimilar

Remember though, we cannot directly compare theOLS and Tobit coefficients in terms of their effect onhours worked

b l ’



52


annual labour supply Let’s construct some marginal effects for some of the

discrete variables

First, the means of the explanatory variables:

Husband’s income: 20.12896

Education: 12.28685

Experience: 10.63081

Age: 42.53785

# young children: 0.2377158

# older children: 1.353254

b l ’



53


annual labour supply Recall the formula:

We can use this to answer the following question: What is theimpact of moving from 0 to 1 young children on the total

number of hours worked? We’ll evaluate for a hypothetical person close to the mean

values: Husband’s income: 20.12896

Education: 12 Experience: 11

Age: 43

# older children: 1

( ) ( ) ( )| / / E y σ σφ σ = Φ +x xβ xβ xβ

T bi l W ’



54


annual labour supply xβ(#young=0,means)=624.64

xβ(#young=1,means)=-269.38

xβ(#young=0,means) / σ=0.5567

xβ(#young=1,means) / σ=-0.2401

φ(#young=0,means)=0.3417

φ(#young=1,means)=0.3876

Φ(#young=0,means)=0.7111

Φ(#young=1,means)=0.4051

T bi l W ’



55


annual labour supply E(y|#young=0,means)=827.6

E(y|#young=1,means)=325.8

E(y|#young=0,means)-E(y|#young=1,means)=502

Thus, for a hypothetical “average” woman, going from 0 youngchildren to 1 young child would decrease hours worked by 502hours. This is larger than the OLS estimate of a 442 hourdecrease.

We could do the same thing to look at the impact of adding asecond young child.



56

Specification Issues The Tobit model relies on the assumptions of normality and

homoskedasticity in the latent variable model

Recall, using OLS we did not need to assume a distributionalform for the error term in order to have unbiased (or consistent)estimates of the parameters.

Thus, although using Tobit may provide us with a more realisticdescription of the data (for example, no negative predictedvalues) we have to make stronger assumptions than when using

OLS.

In a Tobit model, if any of the assumptions fail, it is hard toknow what the estimated coefficients mean.



57

Specification Issues One important limitation of Tobit models is that the expectation of y,

conditional on a positive value, is closely linked to the probability thaty>0

The effect of x j on P(y>0| x) is proportional to β j, as is the effect onE(y|y>0, x). Moreover, for both expressions the factor multiplying β j is

positive.

Thus, if you want a model where an explanatory variable has oppositeeffects on P(y>0| x) and E(y|y>0, x), then Tobit is inappropriate.

One way to informally evaluate a Tobit model is to estimate a probitmodel where: w=1 if y>0 w=0 if y=0



58

Specification Issues The coefficient on x j in the above probit model, say

γ j, is directly related to the coefficient on x j in theTobit model, β j:

Thus, we can look to see if the estimated valuesdiffer.

For example, if the estimates differ in sign, this maysuggest that the Tobit model is in appropriate

j jγ β σ =

S ifi ti I A l



59

Specification Issues: Annual

hours worked example From our previous examples, we estimated the probit coefficient on the

variable # of young children to be -0.868

In the Tobit model, we estimated β j /σ=-0.797 for the variable # of young children

This is not a very large difference, but it suggests that having a youngchild impacts the initial labour force participation decision more thanhow many hours a woman works, once she is in the labour force

The Tobit model effectively averages this two effects:

The impact on the probability of working

The impact on the number of hours worked, conditional on working



60

Specification Issues If we find evidence that the Tobit model is

inappropriate, we can use hurdle or two-part models

These models have the feature that P(y>0|x) andE(y|y>0,x) depend on different parameters and thusx j can have dissimilar effects on the two functions(see Wooldridge (2002, Chapter 16))



61

Practice questions 17.2, 17.3

C17.1, C17.2, C17.3



62

Computer Exercise C17.2 Use the data in LOANAPP.RAW for this exercise.

Estimate a probit model of approve on white. Findthe estimated probability of loan approval for bothwhites and nonwhites. How do these compare to thelinear probability model estimates?

probit approve white

regress approve white



63

Computer Exercise C17.2Probit LPM

White 0.784

(0.087)

0.201

(0.020)

Constant 0.547

(0.075)

0.708

(0.018)

• As there is only one explanatory variable and it takes only twovalues, there are only two different predicted probabilities: theestimated loan approval probabilities for white and nonwhite

applicants

•Hence, the predicted probabilities, whether we use a probit, logit, orLPM model are simply the cell frequencies:

•0.708 for nonwhite applicants•0.908 for white applicants



64

Computer Exercise C17.2 We can do this in Stata using the following

commands following the probit estimation:

predict phat

summarize phat if white==1

summarize phat if white==0



65

Computer Exercise C17.2 Now add the variables hrat, obrat, loanprc, unem,

male, married, dep, sch, cosign, chist, pubrec,mortlat1, mortlat2, and vr to the probit model. Isthere statistically significant evidence of

discrimination against nonwhites?



66

Computer Exercise C17.2

approve

Coef.

Std. Err.

z

P>z

[95% Conf.Interval]

white

.5202525

.0969588

5.37

0.000

.3302168

.7102883

hrat

.0078763

.0069616

1.13

0.258

-.0057682

.0215209

obrat

-.0276924

.0060493

-4.58

0.000

-.0395488

-.015836

loanprc

-1.011969

.2372396

-4.27

0.000

-1.47695

-.5469881

unem -.0366849 .0174807 -2.10 0.036 -.0709464 -.0024234male

-.0370014

.1099273

-0.34

0.736

-.2524549

.1784521

married

.2657469

.0942523

2.82

0.005

.0810159

.4504779

dep

-.0495756

.0390573

-1.27

0.204

-.1261266

.0269753

sch

.0146496

.0958421

0.15

0.879

-.1731974

.2024967

cosign

.0860713

.2457509

0.35

0.726

-.3955917

.5677343

chist

.5852812

.0959715

6.10

0.000

.3971805

.7733818

pubrec

-.7787405

.12632

-6.16

0.000

-1.026323

-.5311578

mortlat1

-.1876237

.2531127

-0.74

0.459

-.6837153

.308468

mortlat2

-.4943562

.3265563

-1.51

0.130

-1.134395

.1456823

vr

-.2010621

.0814934

-2.47

0.014

-.3607862

-.041338

_cons

2.062327

.3131763

6.59

0.000

1.448512

2.676141



67

Computer Exercise C17.2 Estimate the previous model by logit. Compare the

coefficient on white to the probit estimate.



68

Computer Exercise C17.2approve

Coef.

Std. Err.

z

P>z

[95% Conf.Interval]

white

.9377643

.1729041

5.42

0.000

.5988784

1.27665

hrat

.0132631

.0128802

1.03

0.303

-.0119816

.0385078

obrat

-.0530338

.0112803

-4.70

0.000

-.0751427

-.0309249

loanprc

-1.904951

.4604412

-4.14

0.000

-2.807399

-1.002503

unem

-.0665789

.0328086

-2.03

0.042

-.1308825

-.0022753

male

-.0663852

.2064288

-0.32

0.748

-.4709781

.3382078

married

.5032817

.177998

2.83

0.005

.1544121

.8521513

dep

-.0907336

.0733341

-1.24

0.216

-.2344657

.0529986

sch

.0412287

.1784035

0.23

0.817

-.3084356

.3908931

cosign

.132059

.4460933

0.30

0.767

-.7422677

1.006386

chist

1.066577

.1712117

6.23

0.000

.731008

1.402146

pubrec

-1.340665

.2173657

-6.17

0.000

-1.766694

-.9146363

mortlat1

-.3098821

.4635193

-0.67

0.504

-1.218363

.598599

mortlat2

-.8946755

.5685807

-1.57

0.116

-2.009073

.2197222

vr

-.3498279

.1537248

-2.28

0.023

-.6511231

-.0485328

_cons

3.80171

.5947054

6.39

0.000

2.636109

4.967311



69

Computer Exercise C17.2 Use the average partial effect (APE) to calculate the

size of discrimination for the probit and logitestimates.



70

Computer Exercise C17.2 This can be done in Stata using the user-written

command margeff For dummy variables the APE is calculated as a

discrete change in the dependent variable as thedummy variable changes from 0 to 1 (see Cameronand Trivedi, 2009, Chapter 14)

probit

...

margeff

logit ...

margeff



Computer Exercise C17.2 Average Partial Effect of being White on Loan

Approval

Probit Logit OLS

White 0.104(0.023)

0.101(0.022)

0.129(0.020)

Partial Effect at the Average

White 0.106

(0.024)

0.097

(0.022)

0.129

(0.020)

Documents

09-Limited Dependent Variable Models