BINARY CHOICE MODELS: PROBIT ANALYSIS
1.00
F (Z )
0.4
p  F (Z )
0.3
0.50
0.2
0.25
Marginal effect
Cumulative effect
0.75
0.1
0.00
0
-3
-2
-1
0
1
2
Z
Z   1   2 X 2  ...   k X k
In the case of probit analysis, the sigmoid function is the cumulative standardized normal
distribution.
1
BINARY CHOICE MODELS: PROBIT ANALYSIS
1.00
1
1 2Z 2
f (Z ) 
e
2
0.3
0.50
0.2
0.25
Marginal effect
Cumulative effect
0.75
0.4
0.1
0.00
0
-3
-2
-1
0
1
2
Z
Z   1   2 X 2  ...   k X k
The maximum likelihood principle is again used to obtain estimates of the parameters.
2
BINARY CHOICE MODELS: PROBIT ANALYSIS
. probit GRAD ASVABC SM SF MALE
Iteration
Iteration
Iteration
Iteration
Iteration
0:
1:
2:
3:
4:
log
log
log
log
log
likelihood
likelihood
likelihood
likelihood
likelihood
Probit estimates
Log likelihood = -96.624926
=
=
=
=
=
-118.67769
-98.195303
-96.666096
-96.624979
-96.624926
Number of obs
LR chi2(4)
Prob > chi2
Pseudo R2
=
=
=
=
540
44.11
0.0000
0.1858
-----------------------------------------------------------------------------GRAD |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------ASVABC |
.0648442
.0120378
5.39
0.000
.0412505
.0884379
SM | -.0081163
.0440399
-0.18
0.854
-.094433
.0782004
SF |
.0056041
.0359557
0.16
0.876
-.0648677
.0760759
MALE |
.0630588
.1988279
0.32
0.751
-.3266368
.4527544
_cons | -1.450787
.5470608
-2.65
0.008
-2.523006
-.3785673
------------------------------------------------------------------------------
Here is the result of the probit regression using the example of graduating from high
school.
3
BINARY CHOICE MODELS: PROBIT ANALYSIS
. probit GRAD ASVABC SM SF MALE
Iteration
Iteration
Iteration
Iteration
Iteration
0:
1:
2:
3:
4:
log
log
log
log
log
likelihood
likelihood
likelihood
likelihood
likelihood
Probit estimates
Log likelihood = -96.624926
=
=
=
=
=
-118.67769
-98.195303
-96.666096
-96.624979
-96.624926
Number of obs
LR chi2(4)
Prob > chi2
Pseudo R2
=
=
=
=
540
44.11
0.0000
0.1858
-----------------------------------------------------------------------------GRAD |
Coef.
Std. Err.
z
P>|z|
[95% Conf. Interval]
-------------+---------------------------------------------------------------ASVABC |
.0648442
.0120378
5.39
0.000
.0412505
.0884379
SM | -.0081163
.0440399
-0.18
0.854
-.094433
.0782004
SF |
.0056041
.0359557
0.16
0.876
-.0648677
.0760759
MALE |
.0630588
.1988279
0.32
0.751
-.3266368
.4527544
_cons | -1.450787
.5470608
-2.65
0.008
-2.523006
-.3785673
------------------------------------------------------------------------------
As with logit analysis, the coefficients have no direct interpretation. However, we can use
them to quantify the marginal effects of the explanatory variables on the probability of
graduating from high school.
4
BINARY CHOICE MODELS: PROBIT ANALYSIS
p  F (Z )
Z   1   2 X 2  ... k X k
 1  12 Z 2 
p dp Z
  i

 f ( Z )  i  
e
X i dZ X i
 2

As with logit analysis, the marginal effect of Xi on p can be written as the product of the
marginal effect of Z on p and the marginal effect of Xi on Z.
5
BINARY CHOICE MODELS: PROBIT ANALYSIS
p  F (Z )
Z   1   2 X 2  ... k X k
1
dp
1 2Z 2
f (Z ) 

e
dZ
2
 1  12 Z 2 
p dp Z
  i

 f ( Z )  i  
e
X i dZ X i
 2

The marginal effect of Z on p is given by the standardized normal distribution. The marginal
effect of Xi on Z is given by i.
6
BINARY CHOICE MODELS: PROBIT ANALYSIS
p  F (Z )
Z   1   2 X 2  ... k X k
1
dp
1 2Z 2
f (Z ) 

e
dZ
2
 1  12 Z 2 
p dp Z
  i

 f ( Z )  i  
e
X i dZ X i
 2

As with logit analysis, the marginal effects vary with Z. A common procedure is to evaluate
them for the value of Z given by the sample means of the explanatory variables.
7
BINARY CHOICE MODELS: PROBIT ANALYSIS
. sum GRAD ASVABC SM SF MALE
Variable |
Obs
Mean
Std. Dev.
Min
Max
-------------+-------------------------------------------------------GRAD |
540
.9425926
.2328351
0
1
ASVABC |
540
51.36271
9.567646
25.45931
66.07963
SM |
540
11.57963
2.816456
0
20
SF |
540
11.83704
3.53715
0
20
MALE |
540
.5
.5004636
0
1
As with logit analysis, the marginal effects vary with Z. A common procedure is to evaluate
them for the value of Z given by the sample means of the explanatory variables.
8
BINARY CHOICE MODELS: PROBIT ANALYSIS
Probit: Marginal Effects
mean
b
product
f(Z)
f(Z)b
ASVABC
51.36
0.065
3.328
0.068
0.004
SM
11.58
–0.008
–0.094
0.068
–0.001
SF
11.84
0.006
0.066
0.068
0.000
MALE
0.50
0.063
0.032
0.068
0.004
constant
1.00
–1.451
–1.451
Total
Z   1   2 X 2  ... k X k
 1.881
1.881
In this case Z is equal to 1.881 when the X variables are equal to their sample means.
9
BINARY CHOICE MODELS: PROBIT ANALYSIS
Probit: Marginal Effects
mean
b
product
f(Z)
f(Z)b
ASVABC
51.36
0.065
3.328
0.068
0.004
SM
11.58
–0.008
–0.094
0.068
–0.001
SF
11.84
0.006
0.066
MALE
0.50
0.063
0.032
constant
1.00
–1.451
–1.451
Total
1  12 Z 2
0.000
f ( Z0.068
)
e
 0.068
2
0.068
0.004
1.881
We then calculate f(Z).
10
BINARY CHOICE MODELS: PROBIT ANALYSIS
Probit: Marginal Effects
mean
b
product
f(Z)
f(Z)b
ASVABC
51.36
0.065
3.328
0.068
0.004
SM
11.58
–0.008
–0.094
0.068
–0.001
SF
11.84
0.006
0.066
0.068
0.000
MALE
0.50
0.063
0.032
0.068
0.004
constant
1.00
–1.451
–1.451
Total
1.881
p dp Z

 f ( Z ) i
X i dZ X i
The estimated marginal effects are f(Z) multiplied by the respective coefficients. We see
that a one-point increase in ASVABC increases the probability of graduating from high
school by 0.4 percent.
11
BINARY CHOICE MODELS: PROBIT ANALYSIS
Probit: Marginal Effects
mean
b
product
f(Z)
f(Z)b
ASVABC
51.36
0.065
3.328
0.068
0.004
SM
11.58
–0.008
–0.094
0.068
–0.001
SF
11.84
0.006
0.066
0.068
0.000
MALE
0.50
0.063
0.032
0.068
0.004
constant
1.00
–1.451
–1.451
Total
1.881
p dp Z

 f ( Z ) i
X i dZ X i
Every extra year of schooling of the mother decreases the probability of graduating by 0.1
percent. Father's schooling has no discernible effect. Males have 0.4 percent higher
probability than females.
12
BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit
Probit
Linear
f(Z)b
f(Z)b
b
0.004
0.004
0.007
SM
–0.001
–0.001
–0.002
SF
0.000
0.000
0.001
MALE
0.004
0.004
–0.007
ASVABC
The logit and probit results are displayed for comparison. The coefficients in the
regressions are very different because different mathematical functions are being fitted.
13
BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit
Probit
Linear
f(Z)b
f(Z)b
b
0.004
0.004
0.007
SM
–0.001
–0.001
–0.002
SF
0.000
0.000
0.001
MALE
0.004
0.004
–0.007
ASVABC
Nevertheless the estimates of the marginal effects are usually similar.
14
BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit
Probit
Linear
f(Z)b
f(Z)b
b
0.004
0.004
0.007
SM
–0.001
–0.001
–0.002
SF
0.000
0.000
0.001
MALE
0.004
0.004
–0.007
ASVABC
However, if the outcomes in the sample are divided between a large majority and a small
minority, they can differ.
15
BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit
Probit
Linear
f(Z)b
f(Z)b
b
0.004
0.004
0.007
SM
–0.001
–0.001
–0.002
SF
0.000
0.000
0.001
MALE
0.004
0.004
–0.007
ASVABC
This is because the observations are then concentrated in a tail of the distribution.
Although the logit and probit functions share the same sigmoid outline, their tails are
somewhat different.
16
BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit
Probit
Linear
f(Z)b
f(Z)b
b
0.004
0.004
0.007
SM
–0.001
–0.001
–0.002
SF
0.000
0.000
0.001
MALE
0.004
0.004
–0.007
ASVABC
This is the case here, but even so the estimates are identical to three decimal places.
According to a leading authority, Amemiya, there are no compelling grounds for preferring
logit to probit or vice versa.
17
BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit
Probit
Linear
f(Z)b
f(Z)b
b
0.004
0.004
0.007
SM
–0.001
–0.001
–0.002
SF
0.000
0.000
0.001
MALE
0.004
0.004
–0.007
ASVABC
Finally, for comparison, the estimates for the corresponding regression using the linear
probability model are displayed.
18
BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit
Probit
Linear
f(Z)b
f(Z)b
b
0.004
0.004
0.007
SM
–0.001
–0.001
–0.002
SF
0.000
0.000
0.001
MALE
0.004
0.004
–0.007
ASVABC
If the outcomes are evenly divided, the LPM coefficients are usually similar to those for logit
and probit. However, when one outcome dominates, as in this case, they are not very good
approximations.
19
Binary Response Models:
Interpretation II
• Probit: g(0)=.4
• Logit: g(0)=.25
• Linear probability model: g(0)=1
– To make the logit and probit slope estimates
comparable, we can multiply the probit
estimates by .4/.25=1.6.
– The logit slope estimates should be divided by
4 to make them roughly comparable to the
LPM (Linear Probability Model) estimates.