1. BINARY CHOICE MODELS: LOGIT ANALYSIS
Y, p A 1
1 – b1 – b2Xi
b1 +b2Xi b1
b1 + b2Xi B Xi X
The linear probability model may make the nonsense predictions that an event will occur with probability greater than 1 or less than 0. 1

2. BINARY CHOICE MODELS: LOGIT ANALYSIS
The usual way of avoiding this problem is to hypothesize that the probability is a sigmoid (S-shaped) function of Z, F(Z), where Z is a function of the explanatory variables. 2

3. BINARY CHOICE MODELS: LOGIT ANALYSIS
Several mathematical functions are sigmoid in character. One is the logistic function shown here. As Z goes to infinity, e–Z goes to 0 and p goes to 1 (but cannot exceed 1). As Z goes to minus infinity, e–Z goes to infinity and p goes to 0 (but cannot be below 0). 3

4. BINARY CHOICE MODELS: LOGIT ANALYSIS
The model implies that, for values of Z less than –2, the probability of the event occurring is low and insensitive to variations in Z. Likewise, for values greater than 2, the probability is high and insensitive to variations in Z. 4

5. BINARY CHOICE MODELS: LOGIT ANALYSIS
To obtain an expression for the sensitivity, we differentiate F(Z) with respect to Z. The box gives the general rule for differentiating a quotient. 5

6. BINARY CHOICE MODELS: LOGIT ANALYSIS
We apply the rule to the expression for F(Z). 6

7. BINARY CHOICE MODELS: LOGIT ANALYSIS
The sensitivity, as measured by the slope, is greatest when Z is 0. The marginal function, f(Z), reaches a maximum at this point. 7

8. BINARY CHOICE MODELS: LOGIT ANALYSIS
For a nonlinear model of this kind, maximum likelihood estimation is much superior to the use of the least squares principle for estimating the parameters. More details concerning its application are given at the end of this sequence. 8

9. BINARY CHOICE MODELS: LOGIT ANALYSIS
We will apply this model to the graduating from high school example described in the linear probability model sequence. We will begin by assuming that ASVABC is the only relevant explanatory variable, so Z is a simple function of it. 9

10. BINARY CHOICE MODELS: LOGIT ANALYSIS
. logit GRAD ASVABC
Iteration 0: log likelihood =
Iteration 1: log likelihood =
Iteration 2: log likelihood =
Iteration 3: log likelihood =
Iteration 4: log likelihood =
The Stata command is logit, followed by the outcome variable and the explanatory variable(s). Maximum likelihood estimation is an iterative process, so the first part of the output will be like that shown. 10

11. BINARY CHOICE MODELS: LOGIT ANALYSIS
. logit GRAD ASVABC
Iteration 0: log likelihood =
Iteration 1: log likelihood =
Iteration 2: log likelihood =
Iteration 3: log likelihood =
Iteration 4: log likelihood =
Logit estimates Number of obs =
LR chi2(1) =
Prob > chi2 =
Log likelihood = Pseudo R =
GRAD | Coef. Std. Err z P>|z| [95% Conf. Interval]
ASVABC |
_cons |
In this case the coefficients of the Z function are as shown. 11

12. BINARY CHOICE MODELS: LOGIT ANALYSIS
Since there is only one explanatory variable, we can draw the probability function and marginal effect function as functions of ASVABC. 12

13. BINARY CHOICE MODELS: LOGIT ANALYSIS
We see that ASVABC has its greatest effect on graduating when it is below 40, that is, in the lower ability range. Any individual with a score above the average (50) is almost certain to graduate. 13

14. BINARY CHOICE MODELS: LOGIT ANALYSIS
. logit GRAD ASVABC
Iteration 0: log likelihood =
Iteration 1: log likelihood =
Iteration 2: log likelihood =
Iteration 3: log likelihood =
Iteration 4: log likelihood =
Logit estimates Number of obs =
LR chi2(1) =
Prob > chi2 =
Log likelihood = Pseudo R =
GRAD | Coef. Std. Err z P>|z| [95% Conf. Interval]
ASVABC |
_cons |
The t statistic indicates that the effect of variations in ASVABC on the probability of graduating from high school is highly significant. 14

15. BINARY CHOICE MODELS: LOGIT ANALYSIS
. logit GRAD ASVABC
Iteration 0: log likelihood =
Iteration 1: log likelihood =
Iteration 2: log likelihood =
Iteration 3: log likelihood =
Iteration 4: log likelihood =
Logit estimates Number of obs =
LR chi2(1) =
Prob > chi2 =
Log likelihood = Pseudo R =
GRAD | Coef. Std. Err z P>|z| [95% Conf. Interval]
ASVABC |
_cons |
Strictly speaking, the t statistic is valid only for large samples, so the normal distribution is the reference distribution. For this reason the statistic is denoted z in the Stata output. This z has nothing to do with our Z function. 15

16. BINARY CHOICE MODELS: LOGIT ANALYSIS
The coefficients of the Z function do not have any direct intuitive interpretation. 16

17. BINARY CHOICE MODELS: LOGIT ANALYSIS
However, we can use them to quantify the marginal effect of a change in ASVABC on the probability of graduating. We will do this theoretically for the general case where Z is a function of several explanatory variables. 17

18. BINARY CHOICE MODELS: LOGIT ANALYSIS
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 the product of the marginal effect of Z on p and the marginal effect of Xi on Z. 18

19. BINARY CHOICE MODELS: LOGIT ANALYSIS
We have already derived an expression for dp/dZ. The marginal effect of Xi on Z is given by its b coefficient. 19

20. BINARY CHOICE MODELS: LOGIT ANALYSIS
Hence we obtain an expression for the marginal effect of Xi on p. 20

21. BINARY CHOICE MODELS: LOGIT ANALYSIS
The marginal effect is not constant because it depends on the value of Z, which in turn depends on the values of the explanatory variables. A common procedure is to evaluate it for the sample means of the explanatory variables. 21

22. BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev Min Max
GRAD |
ASVABC |
The sample mean of ASVABC in this sample is 22

23. BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev Min Max
GRAD |
ASVABC |
Logit estimates Number of obs =
LR chi2(1) =
Prob > chi2 =
Log likelihood = Pseudo R =
GRAD | Coef. Std. Err z P>|z| [95% Conf. Interval]
ASVABC |
_cons |
When evaluated at the mean, Z is equal to 23

24. BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev Min Max
GRAD |
ASVABC |
e–Z is Hence F(Z) is There is 97.1 percent probability that an individual with average ASVABC will graduate from high school. 24

25. BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev Min Max
GRAD |
ASVABC |
f(Z) is 25

26. BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev Min Max
GRAD |
ASVABC |
The marginal effect, evaluated at the mean, is therefore This implies that a one point increase in ASVABC would increase the probability of graduating from high school by 0.4 percent. 26

27. BINARY CHOICE MODELS: LOGIT ANALYSIS
0.971
0.004
51.36
In this example, the marginal effect at the mean of ASVABC is very low. The reason is that anyone with an average score is almost certain to graduate anyway. So an increase in the score has little effect. 27

28. BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev Min Max
GRAD |
ASVABC |
To show that the marginal effect varies, we will also calculate it for ASVABC equal to 30. For this value of ASVABC, the probability of graduating is only 66.9 percent. 28

29. BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC
Variable | Obs Mean Std. Dev Min Max
GRAD |
ASVABC |
For ASVABC equal to 30, a one point increase in ASVABC increases the probability of graduating by 2.9 percent. 29

30. BINARY CHOICE MODELS: LOGIT ANALYSIS
0.029
0.669
For an individual with a score of 30, with only a 67 percent probability of graduating, an increase in the score has a relatively large impact. 30

31. BINARY CHOICE MODELS: LOGIT ANALYSIS
. logit 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 =
Iteration 5: log likelihood =
Logit 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 output for a model with a somewhat better specification. 31

32. BINARY CHOICE MODELS: LOGIT ANALYSIS
. sum GRAD ASVABC SM SF MALE
Variable | Obs Mean Std. Dev Min Max
GRAD |
ASVABC |
SM |
SF |
MALE |
We will estimate the marginal effects, putting all the explanatory variables equal to their sample means. 32

33. BINARY CHOICE MODELS: LOGIT ANALYSIS
Logit: Marginal Effects
mean b product
ASVABC
SM –0.023 –0.269 SF
MALE
constant 1.00 –3.252 –3.252
Total
The first step is to calculate Z, when the X variables are equal to their sample means. 33

34. BINARY CHOICE MODELS: LOGIT ANALYSIS
Logit: Marginal Effects
mean b product
ASVABC
SM –0.023 –0.269 SF
MALE
constant 1.00 –3.252 –3.252
Total
We then calculate f(Z). 34

35. BINARY CHOICE MODELS: LOGIT ANALYSIS
Logit: Marginal Effects
mean b product f(Z) f(Z)b
ASVABC
SM –0.023 – –0.001 SF
MALE
constant 1.00 –3.252 –3.252
Total
The estimated marginal effects are f(Z) multiplied by the respective coefficients. We see that the effect of ASVABC is about the same as before. Mother's schooling has negligible effect and father's schooling has no discernible effect at all. 35

36. BINARY CHOICE MODELS: LOGIT ANALYSIS
Logit: Marginal Effects
mean b product f(Z) f(Z)b
ASVABC
SM –0.023 – –0.001 SF
MALE
constant 1.00 –3.252 –3.252
Total
Males have 0.4 percent higher probability of graduating than females. These effects would all have been larger if they had been evaluated at a lower ASVABC score. 36

37. BINARY CHOICE MODELS: LOGIT ANALYSIS
Individuals who graduated: outcome probability
This sequence will conclude with an outline explanation of how the model is fitted using maximum likelihood estimation. 37

38. BINARY CHOICE MODELS: LOGIT ANALYSIS
Individuals who graduated: outcome probability
In the case of an individual who graduated, the probability of that outcome is F(Z). We will give subscripts 1, ..., s to the individuals who graduated. 38

39. BINARY CHOICE MODELS: LOGIT ANALYSIS
Individuals who graduated: outcome probability
Individuals who did not graduate: outcome probability
In the case of an individual who did not graduate, the probability of that outcome is 1 – F(Z). We will give subscripts s+1, ..., n to these individuals. 39

40. BINARY CHOICE MODELS: LOGIT ANALYSIS
Maximize
Did graduate
Did not graduate
We choose b1 and b2 so as to maximize the joint probability of the outcomes, that is, F(Z1) x ... x F(Zs) x [1 – F(Zs+1)] x ... x [1 – F(Zn)]. There are no mathematical formulae for b1 and b2. They have to be determined iteratively by a trial-and-error process. 40

41. Copyright Christopher Dougherty 2012.
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