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```					Limited Dependent Variable
Models

EMET 8002
Lecture 9
August 27, 2009

1
Limited Dependent Variables
A limited dependent variable is a dependent
variable 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

2
Outline
Logit and probit models for binary dependent
variables

Tobit model for corner solutions

3
Why do we care?
model to examine some of its shortcomings

The model is given by:
y = β 0 + β1 x1 + ... + β k xk + u

where
P ( y = 1| x ) = E ( y | x ) = β 0 + β1 x1 + ... + β k xk

4
Linear Probability Model
There will be three undesirable features of this model:
1. The error term will not be homoskedastic. This violates
assumption 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 or
less than 0!

3.   The independent variables cannot be linearly related to the
dependent variable for all possible values.

5
Linear Probability Model
Example
Let’s look at how being in the labour force is
influenced 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

6
Linear Probability Model
Example
Coefficient   Usual standard   Robust standard
estimate         errors           errors
Husband’s income     -0.0034         0.0014            0.0015

Years of              0.038           0.007            0.007
education
Experience            0.039           0.006            0.006

Experience2         -0.00060         0.00018          0.00019

Age                   -0.016          0.002            0.002

# kids <= 6 years     -0.262          0.034            0.032
old
# kids > 6 years      0.013           0.013            0.014
old
7
Linear Probability Model
Example
Using standard errors that are robust to unknown
heteroskedasticity is simple and does not
substantially change the reported standard errors

Interpreting the coefficients:
All else equal, an extra year of education increases the
probability of participating in the labour force by 0.038
(3.8%)
All else equal, an additional child 6 years of age or less
decreases the probability of working by 0.262

8
Linear Probability Model
Example
Predicted probabilities:
Sometimes we obtain predicted probabilities that are outside
of the range [0,1]. In this sample, 33 of the 753
observations 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!!

9
Linear Probability Model
Example
An additional problem is that probabilities cannot be
linearly related to the independent variables for all
possible values
For example, consider the estimate of the marginal
effect of increasing the number of children 6 years of
age or younger. It is estimated to be -0.262. This
means that if this independent variable increased from
0 to 4, the probability of being in the labour market
would fall by 1.048, which is impossible!

10
Linear Probability Model
It is still a useful model to estimate, especially since
the estimate coefficients are much easier to interpret
than the nonlinear models that we are going to
introduce shortly
Plus, it usually works well for values of the
independent variables that are close to the respective
means (i.e., outlying values of x cause problems)

11
Limited Dependent Variables
Models
In this lecture we’re going to cover estimation
techniques that will better address the nature of the
dependent variable
Logit & Probit
Tobit

12
Logit and Probit Models for
Binary Response
We’re going to prevent predicted values from ever
falling outside the range [0,1] by estimating a
nonlinear regression:
P ( y = 1| x ) = G ( β 0 + xβ )
where 0<G(z)<1 for all real numbers z

The two most commonly used functions for G(.) are
the logit model and the probit model:
exp ( z )
G(z) =               = Λ(z)
1 + exp ( z )
G(z) = Φ(z)                                13
Logit and Probit Models for
Binary Response
Logit and probit models can be derived from an
underlying latent variable model
i.e., an unobserved variable
y* = β 0 + xβ + e,
y = 1 ⎡ y* > 0 ⎤
⎣        ⎦
We assume that e is independent of x and that e
either has the standard logistic distribution or the
standard normal distribution
Under either assumption e is symmetrically
distributed about 0, which implies that 1-G(-z)=G(z)
for all real numbers z

14
Logit and Probit Models for
Binary Response
We can now derive the response probability for y:
P ( y = 1| x ) = P ( y* > 0 | x )
= P ( β 0 + xβ + e > 0 | x )
= P ( e > − ( β 0 + xβ ) | x )
= 1 − G ⎡ − ( β 0 + xβ ) ⎤
⎣                ⎦
= G ( β 0 + xβ )

15
Logit and Probit Models for
Binary Response
In most applications of binary response models our main
interest is to explain the effects of the x’s on the response
probability 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 its
interpretation. Nonetheless, in some examples this is a very
useful tool for thinking about the problem.

16
Logit and Probit Models for
Binary Response
The sign of the coefficients will tell us the direction of
the partial effect of xj on P(y=1|x)

However, unlike the linear probability model, the
magnitudes of the coefficients are not especially
useful

If xj is a roughly continuous variable, its partial effect
is given by:      ∂p ( x ) dG ( z )
=         βj
∂x j    dz
17
Logit and Probit Models for
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 function
back to the linear model by simply assuming G(z)=z.

For both the logit and the probit models g(z)=dG(z)/dz is
always positive (since G is the cumulative distribution function,
g is the probability density function). Thus, the sign of βj is the
same as the sign of the partial effect.

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

18
Logit and Probit Models for
Binary Response
Nonetheless, the relative effect of any two
continuous explanatory variables do not depend on x

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

19
Logit and Probit Models for
Binary Response
Suppose x1 is a discrete variable, its partial effect of going from
c to c+1 is given by:
G ( β 0 + β1 ( c + 1) + β 2 x2 + ... + β k xk ) −
G ( β 0 + β1c + β 2 x2 + ... + β k xk )
Again, this effect depends on x

Note, however, that the sign of β1 is enough to know whether
the discrete variable has a positive or negative effect
This is because G() is strictly increasing

20
Logit and Probit Models for
Binary Response
We use Maximum Likelihood Estimation, which
heteroskedasticity 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 the
density of yi given xi
1− y
f ( y | xi ; β ) = ⎡G ( xi β ) ⎦ ⎣1 − G ( xi β ) ⎦
y
⎣           ⎤ ⎡               ⎤      , y = 0,1
21
Logit and Probit Models for
Binary Response
When y=1: f(y|xi:β)=G(xiβ)
When y=0: f(y|xi:β)=1-G(xiβ)

The log-likelihood function for observation i is
given by:
li ( β ) = yi log ⎡G ( xi β ) ⎤ + (1 − yi ) log ⎡1 − G ( xi β ) ⎤
⎣           ⎦                 ⎣               ⎦
The log-likelihood for a sample of size n is obtained
by summing this expression over all observations
n
L ( β ) = ∑ li ( β )
i =1

22
Logit and Probit Models for
Binary Response
The MLE of β maximizes this log-likelihood
If G is the standard logit cdf, then we get the logit
estimator
If G is the standard normal cdf, then we get the
probit estimator

Under general conditions, the MLE is:
Consistent
Asymptotically normal
Asymptotically efficient

23
Inference in Probit and Logit
Models
Standard regression software, such as Stata, will
automatically report asymptotic standard errors for
the coefficients

This means we can construct (asymptotic) t-tests for
statistical significance in the usual way:
ˆ     ( )
ˆ
t j = β j se β j

24
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 to
0)

There are two options commonly used:
A Wald test
A likelihood ratio test

25
Logit and Probit Models for Binary
Response: Testing Multiple Hypotheses

Wald test:
In the linear model, the Wald statistic, can be
transformed to be essentially the same as the F
statistic
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 restrictions
being tested
In Stata we can use the “test” command following
probit or logit estimation

26
Logit and Probit Models for Binary
Response: Testing Multiple Hypotheses
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 model
The likelihood ratio statistic is given by:
LR = 2 ( Lur − Lr )
It is asymptotically chi-squared with degrees of freedom
equal to the number of restrictions
can use lrtest in Stata

27
Logit and Probit Models for Binary
Response: Interpreting Probit and Logit
Estimates

Recall that unlike the linear probability model, the
estimated coefficients from Probit or Logit estimation
do not tell us the magnitude of the partial effect of a
change in an independent variable on the predicted
probability

This depends not just on the coefficient estimates,
but also on the values of all the independent
variables and the coefficients

28
Logit and Probit Models for Binary
Response: Interpreting Probit and Logit
Estimates

For roughly continuous variables the marginal effect
is approximately by:
(          )
ΔP ( y = 1| x ) ≈ ⎡ g β 0 + xβ β j ⎤ Δx j
ˆ
⎣
ˆ      ˆ ˆ
⎦

For discrete variables the estimated change in the
predicted probability is given by:
(                                         )
G β 0 + β1 ( c + 1) + β 2 x2 + ... + β k xk −
ˆ     ˆ            ˆ              ˆ

G(βˆ
0
ˆ    ˆ              ˆ
+ β1c + β 2 x2 + ... + β k xk   )
29
Logit and Probit Models for Binary
Response: Interpreting Probit and Logit
Estimates
Thus, we need to pick “interesting” value of x at
which 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, for
example, to see how the partial effects change as
some elements of x get large or small

If xk is a binary variable, then it often makes sense to
use a value of 0 or 1 in the partial effect equation,
rather than the average value of xk

30
Logit and Probit Models for Binary
Response: Interpreting Probit and Logit
Estimates

An alternative approach is to calculate the average
partial effect (APE)

For a continuous explanatory variable, xj, the APE is:
(         )               (        )
n                             n
n −1 ∑ ⎡ g β 0 + xi β β j ⎤ = n −1 ∑ ⎡ g β 0 + xi β ⎤β j
ˆ       ˆ ˆ                   ˆ       ˆ ˆ
i =1
⎣               ⎦        i =1
⎣           ⎦

The two scale factors (at the mean for PEA and
averaged over the sample for the APE) differ since
the first uses a nonlinear function of the average and
the second uses the average of a nonlinear function
31
Example 17.1: Married Women’s
Labour Force Participation
We are going to use the data in MROZ.RAW to
estimate a labour force participation for women using
logit and probit estimation.
The explanatory variables are nwifeinc, educ, exper,
exper2, age, kidslt6, kidsge6
probit inlf nwifeinc educ exper expersq age kidslt6
kidsge6

32
Example 17.1
Independent                           Coefficient Estimates
variable
OLS                 Probit            Logit
(robust stderr)
Husband’s income        -0.0034              -0.012            -0.021
(0.0015)             (0.005)           (0.008)
Years of                0.038                0.131             0.221
education              (0.007)              (0.025)           (0.043)
Age                     -0.016               -0.053            -0.088
(0.002)              (0.008)           (0.014)
# kids <= 6 years       -0.262               -0.868            -1.44
old                    (0.032)              (0.119)           (0.20)
# kids > 6 years        0.013                0.036             0.060
old                    (0.014)              (0.043)           (0.075)
33
Example 17.1
True or false:
The Probit and Logit model estimates suggest that the
linear probability model was underestimating the
negative impact of having young children on the
probability of women participating in the labour force.

34
Example 17.1
How does the predicted probability change as the
number 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

35
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 (there
is a diminishing impact). This differs from the linear probability
model which says any increase of one young child has the same
impact.

36
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 cover
this so see Wooldridge (2002, Chapter 15) for a discussion

We need to be concerned with heteroskedasticity in probit and
logit models. If var(e|x) depends on x then the response
probability no longer has the form G(β0+βx) implying that more
general 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, Chapter
15))

37
The Tobit Model for Corner
Solution Responses
Often in economics we observes variables for which 0
(or some other fixed number) is in an optimal
outcome for some units of observations, but a range
of positive outcomes prevail for other observations
For example:
Number of hours worked annually
Hours spent on the internet
Grade on a test (may be grouped at both 0 and 100)

38
The Tobit Model for Corner
Solution Responses
Let y be a variable that is roughly continuous over
strictly positive values but that takes on zero with a
positive probability

Similar to the binary dependent variable context we
can use a linear model and this might not be so bad
for observations that are close to the mean, but we
may obtain negative fitted values and therefore
negative predictions for y

39
The Tobit Model for Corner
Solution Responses
We often express the observed outcome, y, in terms
of an unobserved latent variable, say y*
y* = xβ + u , u | x ~ N ( 0, σ 2 )
y = max ( 0, y *)

We now need to think about how to estimate this
model. There are two cases to consider:
When y=0
When y>0

40
The Tobit Model for Corner
Solution Responses
the probability that y=0 conditional on the
explanatory variables?
P ( y = 0 | x ) = P ( y* < 0 | x )   Definition of y

= P ( xβ + u < 0 | x )               Definition of y*

= P ( u < − xβ | x )
= P ( u σ < − xβ σ | x )             Creating a standard normal variable

= Φ ( − xβ σ )                       The normal CDF

= 1 − Φ ( xβ σ )
41
The Tobit Model for Corner
Solution Responses
What is the probability that y>0 conditional on the
explanatory variables?

Since y is continuous for values greater than 0, the
probability is simply the density of the normal
variable u

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

42
Interpreting Tobit estimates
Given standard regression packages, it is straight forward to
estimate a Tobit model using maximum likelihood (the details of
the formulation are available in Wooldridge (2002, Chapter 16))

The underlying model tells us that βj measures the partial effect
of xj on y*, the latent variable. However, we’re usually
interested in the observed outcome y, not y*

In the Tobit model two conditional expectations are generally of
interest:
E(y|y>0,x)
E(y|x)

43
Interpreting Tobit estimates
E ( y | y > 0, x ) = xβ + σλ ( xβ / σ )
E ( y | x ) = Φ ( xβ / σ ) xβ + σφ ( xβ / σ )

Take home message: Conditional expectations in the
Tobit are much more complicated than in the linear
model

E(y|x) is a nonlinear of function of both x and β.
Moreover, this conditional expectation can be shown
to be positive for any values of x and β.
44
Interpreting Tobit estimates
To examine partial effects, we should consider two cases:
When xj is continuous
When xj is discrete

When xj is continuous we can use calculus to solve for the
partial effects:

∂E ( y | y > 0, x )
∂x j
{                                  }
= β j 1 − λ ( xβ σ ) ⎡ xβ σ + λ ( xβ σ ) ⎤
⎣                   ⎦

∂E ( y | x )
= β j Φ ( xβ σ )
∂x j
Like in probit or logit models, the partial effect will depend on
all explanatory variables and parameters
45
Interpreting Tobit estimates
When xj is discrete we estimate the partial effect as
the difference:
E ( y | y > 0, x − j , x j = c + 1) − E ( y | y > 0, x − j , x j = c )
E ( y | x − j , x j = c + 1) − E ( y | x − j , x j = c )

46
Interpreting Tobit estimates
Just like the probit and logit models, there are two
common approaches for evaluating the partial
effects:
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

47
Example 17.2: Women’s
annual labour supply
We can use the same dataset, MROZ.RAW, that we
used to estimate the probability of women
participating in the labour force to estimate the
impact 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

48
Tobit example: Women’s
annual labour supply
reg hours nwifeinc educ exper expersq age kidslt6
kidsge6

tobit hours nwifeinc educ exper expersq age kidslt6
kidsge6, ll(0)

49
Tobit example: Women’s
annual labour supply
Coefficient Estimates

OLS               Tobit

Husband’s income         -3.45             -8.81
(2.54)            (4.46)
Years of education       28.76            80.65
(12.95)          (21.58)
Age                     -30.51            -54.41
(4.36)            (7.42)
# kids <= 6 years old   -442.09           -894.02
(58.85)          (111.88)
# kids > 6 years old     -32.78           -16.22
(23.18)          (38.64)
Sigma                                   1122.022
50
(41.58)
Tobit example: Women’s
annual labour supply
The Tobit coefficient estimates all have the same sign
as the OLS coefficients

The pattern of statistical significance is also very
similar

Remember though, we cannot directly compare the
OLS and Tobit coefficients in terms of their effect on
hours worked

51
Tobit example: Women’s
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

52
Tobit example: Women’s
annual labour supply
Recall the formula:
E ( y | x ) = Φ ( xβ / σ ) xβ + σφ ( xβ / σ )

We can use this to answer the following question: What is the
impact 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

53
Tobit example: Women’s
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

54
Tobit example: Women’s
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 young
children to 1 young child would decrease hours worked by 502
hours. This is larger than the OLS estimate of a 442 hour
decrease.

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

55
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 distributional
form 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 realistic
description of the data (for example, no negative predicted
values) we have to make stronger assumptions than when using
OLS.

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

56
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 that
y>0

The effect of xj on P(y>0|x) is proportional to βj, as is the effect on
E(y|y>0,x). Moreover, for both expressions the factor multiplying βj is
positive.

Thus, if you want a model where an explanatory variable has opposite
effects 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 probit
model where:
w=1 if y>0
w=0 if y=0

57
Specification Issues
The coefficient on xj in the above probit model, say
γj, is directly related to the coefficient on xj in the
Tobit model, βj:
γ j = βj σ

Thus, we can look to see if the estimated values
differ.
For example, if the estimates differ in sign, this may
suggest that the Tobit model is in appropriate

58
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 βj/σ=-0.797 for the variable # of
young children

This is not a very large difference, but it suggests that having a young
child impacts the initial labour force participation decision more than
how 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

59
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) and
E(y|y>0,x) depend on different parameters and thus
xj can have dissimilar effects on the two functions
(see Wooldridge (2002, Chapter 16))

60
Practice questions
17.2, 17.3
C17.1, C17.2, C17.3

61
Computer Exercise C17.2
Use the data in LOANAPP.RAW for this exercise.

Estimate a probit model of approve on white. Find
the estimated probability of loan approval for both
whites and nonwhites. How do these compare to the
linear probability model estimates?

probit approve white
regress approve white

62
Computer Exercise C17.2
Probit             LPM
White                  0.784             0.201
(0.087)           (0.020)
Constant               0.547             0.708
(0.075)           (0.018)
•As there is only one explanatory variable and it takes only two
values, there are only two different predicted probabilities: the
estimated loan approval probabilities for white and nonwhite
applicants
•Hence, the predicted probabilities, whether we use a probit, logit, or
LPM model are simply the cell frequencies:
•0.708 for nonwhite applicants
•0.908 for white applicants                                           63
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

64
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. Is
there statistically significant evidence of
discrimination against nonwhites?

65
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   -.0024234
male       -.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

66
Computer Exercise C17.2
Estimate the previous model by logit. Compare the
coefficient on white to the probit estimate.

67
Computer Exercise C17.2
approve    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

68
Computer Exercise C17.2
Use the average partial effect (APE) to calculate the
size of discrimination for the probit and logit
estimates.

69
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 the
dummy variable changes from 0 to 1 (see Cameron
and Trivedi, 2009, Chapter 14)

probit ...
margeff
logit ...
margeff

70
Computer Exercise C17.2
Average Partial Effect of being White on Loan
Approval
Probit       Logit      OLS

White          0.104       0.101      0.129
(0.023)     (0.022)    (0.020)
Partial Effect at the Average

White          0.106       0.097      0.129
(0.024)     (0.022)    (0.020)

71

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