1. EC220 - Introduction to econometrics (chapter 10)
Christopher Dougherty
EC220 - Introduction to econometrics (chapter 10)
Slideshow: binary choice probit models
Original citation:
Dougherty, C. (2012) EC220 - Introduction to econometrics (chapter 10). [Teaching Resource]
© 2012 The Author
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Available in LSE Learning Resources Online: May 2012
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2. BINARY CHOICE MODELS: PROBIT ANALYSIS
In the case of probit analysis, the sigmoid function is the cumulative standardized normal distribution. 1

3. BINARY CHOICE MODELS: PROBIT ANALYSIS
The maximum likelihood principle is again used to obtain estimates of the parameters. 2

4. BINARY CHOICE MODELS: PROBIT ANALYSIS
. probit GRAD ASVABC SM SF MALE
Iteration 0: log likelihood =
Iteration 1: log likelihood =
Iteration 2: log likelihood =
Iteration 3: log likelihood =
Iteration 4: log likelihood =
Probit estimates Number of obs =
LR chi2(4) =
Prob > chi2 =
Log likelihood = Pseudo R =
GRAD | Coef. Std. Err z P>|z| [95% Conf. Interval]
ASVABC |
SM |
SF |
MALE |
_cons |
Here is the result of the probit regression using the example of graduating from high school. 3

5. BINARY CHOICE MODELS: PROBIT ANALYSIS
. probit GRAD ASVABC SM SF MALE
Iteration 0: log likelihood =
Iteration 1: log likelihood =
Iteration 2: log likelihood =
Iteration 3: log likelihood =
Iteration 4: log likelihood =
Probit estimates Number of obs =
LR chi2(4) =
Prob > chi2 =
Log likelihood = Pseudo R =
GRAD | Coef. Std. Err z P>|z| [95% Conf. Interval]
ASVABC |
SM |
SF |
MALE |
_cons |
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

6. BINARY CHOICE MODELS: PROBIT ANALYSIS
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

7. BINARY CHOICE MODELS: PROBIT ANALYSIS
The marginal effect of Z on p is given by the standardized normal distribution. The marginal effect of Xi on Z is given by bi. 6

8. BINARY CHOICE MODELS: PROBIT ANALYSIS
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

9. BINARY CHOICE MODELS: PROBIT ANALYSIS
. sum GRAD ASVABC SM SF MALE
Variable | Obs Mean Std. Dev Min Max
GRAD |
ASVABC |
SM |
SF |
MALE |
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

10. BINARY CHOICE MODELS: PROBIT ANALYSIS
Probit: Marginal Effects
mean b product f(Z) f(Z)b
ASVABC
SM –0.008 – –0.001 SF
MALE
constant 1.00 –1.451 –1.451
Total
In this case Z is equal to when the X variables are equal to their sample means. 9

11. BINARY CHOICE MODELS: PROBIT ANALYSIS
Probit: Marginal Effects
mean b product f(Z) f(Z)b
ASVABC
SM –0.008 – –0.001 SF
MALE
constant 1.00 –1.451 –1.451
Total
We then calculate f(Z). 10

12. BINARY CHOICE MODELS: PROBIT ANALYSIS
Probit: Marginal Effects
mean b product f(Z) f(Z)b
ASVABC
SM –0.008 – –0.001 SF
MALE
constant 1.00 –1.451 –1.451
Total
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

13. BINARY CHOICE MODELS: PROBIT ANALYSIS
Probit: Marginal Effects
mean b product f(Z) f(Z)b
ASVABC
SM –0.008 – –0.001 SF
MALE
constant 1.00 –1.451 –1.451
Total
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

14. BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit Probit Linear
f(Z)b f(Z)b b
ASVABC
SM –0.001 –0.001 –0.002 SF
MALE –0.007
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

15. BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit Probit Linear
f(Z)b f(Z)b b
ASVABC
SM –0.001 –0.001 –0.002 SF
MALE –0.007
Nevertheless the estimates of the marginal effects are usually similar. 14

16. BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit Probit Linear
f(Z)b f(Z)b b
ASVABC
SM –0.001 –0.001 –0.002 SF
MALE –0.007
However, if the outcomes in the sample are divided between a large majority and a small minority, they can differ. 15

17. BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit Probit Linear
f(Z)b f(Z)b b
ASVABC
SM –0.001 –0.001 –0.002 SF
MALE –0.007
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

18. BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit Probit Linear
f(Z)b f(Z)b b
ASVABC
SM –0.001 –0.001 –0.002 SF
MALE –0.007
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

19. BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit Probit Linear
f(Z)b f(Z)b b
ASVABC
SM –0.001 –0.001 –0.002 SF
MALE –0.007
Finally, for comparison, the estimates for the corresponding regression using the linear probability model are displayed. 18

20. BINARY CHOICE MODELS: PROBIT ANALYSIS
Logit Probit Linear
f(Z)b f(Z)b b
ASVABC
SM –0.001 –0.001 –0.002 SF
MALE –0.007
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

21. Copyright Christopher Dougherty 2011.
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Subject to respect for copyright and, where appropriate, attribution, they may be used as a resource for teaching an econometrics course. There is no need to refer to the author.
The content of this slideshow comes from Section 10.3 of C. Dougherty, Introduction to Econometrics, fourth edition 2011, Oxford University Press.
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Individuals studying econometrics on their own and who feel that they might benefit from participation in a formal course should consider the London School of Economics summer school course
EC212 Introduction to Econometrics
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20 Elements of Econometrics