In previous lecture, we dealt with the unboundedness
problem of LPM using the logit model.
In this lecture, we will consider another alternative, i.e.
the probit model.
Adapted
from
“Introduction
to
Econometrics” by Christopher Dougherty
1
BINARY CHOICE MODELS: PROBIT ANALYSIS
1.00
1
1 2Z 2
f (Z ) 
e
2
0.3
0.50
0.2
0.25
Z   1   2 X 2  ...   k X k
0.00
Marginal effect
Cumulative effect
0.75
0.4
0.1
0
-3
-2
-1
0
1
2
Z
In the case of probit analysis, the sigmoid function is the cumulative standardized normal
distribution. The maximum likelihood principle is again used to obtain estimates of the
parameters.
2
Estimating the probability of success
Z   1   2 X 2  ...   k X k
Suppose that the probit equation yields a Z = +0.2171. Since Z is positive, the
area in the larger portion of the curve is 0.5859, or a prediction of a 58.59%
success rate [You can use a standard normal table or Excel function
NORMSDIST].
Area = 0.5859
3
Z= + 0.2171
Quantifying the Marginal Effect
We will do this theoretically for the general case where Z is a
function of several explanatory variables.
p  F (Z )
Z   1   2 X 2  ...  k X k
Since p is a function of Z, and Z is a function of the X variables, the
marginal effect of Xi on p can be written as:
p
dp Z


X i dZ X i
4
(1)
dp
dZ
1
dp
1 2Z2
 f (Z ) 
e
dZ
2
(2)
The marginal effect of Z on p is given by the
standardized normal distribution.
Z
X i
Z   1   2 X 2  ...  k X k
Z
 i
X i
The marginal effect of Xi on Z is given by i.
5
(3)
p
dp Z


X i
dZ X i
 1  12 Z 2 
i
 
e

2



Hence we obtain an expression for the
marginal effect of Xi on p.
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.
6
ILLUSTRATION
•
Why do some people graduate from high school while others drop
out?
Here we use the same multivariate example as in the case of logit
model (see Illustration 2 in logit lecture slides), so as to facilitate
comparison.
7
. 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
------------------------------------------------------------------------------
8
. 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 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.
We will estimate the marginal effects, putting all the explanatory variables
equal to their sample means.
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Step 1: Calculate Z, when the X variables are equal to their sample
means.
Xi
i
i X i
ASVABC
51.36
0.065
3.328
SM
11.58
–0.008
–0.094
SF
11.84
0.006
0.066
MALE
0.50
0.063
0.032
Constant
1.00
–1.451
–1.451
Total
1.881
Z  1   2 X 2  ... k X k
 1.881
10
Step 2: Calculate
dp
dZ
1
dp
1 2Z2
 f (Z ) 
e
 0.068
dZ
2
p
Step 3: Calculate
X i
Note that:
Z
 i
X i
p dp Z


X i dZ X i
11
dp
dZ
p dp

 i
X i dZ
0.065
0.068
0.004
SM
–0.008
0.068
-0.001
SF
0.006
0.068
0.000
MALE
0.063
0.068
0.004
i
ASVABC
We see that a one-point increase in ASVABC increases the
probability of graduating from high school by about 0.004, i.e. 0.4%.
Mother's schooling (SM) has negligible effect and father's schooling
(SF) has no discernible effect at all.
Males have 0.4 percent higher probability of graduating than
females.
12
What is the probability of graduating when ASVABC equal to (a) 30
(b) 50 ? Set the values of other X variables equal to their sample
means.
Xi
ASVABC
30
SM
SF
i
i X i
0.065
1.95
11.58
–0.008
–0.094
11.84
0.006
0.066
MALE
0.50
0.063
0.032
Constant
1.00
–1.451
–1.451
Total
Z  1   2 X 2  ... k X k
 0.503
NORMSDIST (0.503)
=0.6925
0.503
When ASVABC = 30, the probability of graduating is 69.25%.
13
When ASVABC = 50, the probability of graduating is 96.43%.
Xi
ASVABC
50
SM
SF
i
i X i
0.065
3.25
11.58
–0.008
–0.094
11.84
0.006
0.066
MALE
0.50
0.063
0.032
Constant
1.00
–1.451
–1.451
Total
Z  1   2 X 2  ... k X k
 1.803
NORMSDIST (0.503)
=0.9643
1.803
14
Logit versus Probit
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.
Nevertheless the estimates of the marginal effects are usually similar.
15
However, if the outcomes in the sample are divided between a large
majority and a small minority, they can differ.
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.
This is the case here, but even so the estimates are identical to three
decimal places.
16
So, logit or probit?
The logit model is easier to compute, and used to be more popular
than the probit model.
Probit model is theoretically more appealing as it is based on normal
distribution. However, it uses more computer time.
Given computer technology advanced nowadays, the choice
between the logit model and probit model is a matter of taste.
17