1. 1 BINARY CHOICE MODELS: LOGIT ANALYSIS The linear probability model may make the nonsense predictions that an event will occur with probability greater than 1 or less than 0. XXiXi 1 0  1 +  2 X i Y, p 11 A 1 -  1 -  2 X i B  1 +  2 X i

2. 2 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. BINARY CHOICE MODELS: LOGIT ANALYSIS

3. 3 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). BINARY CHOICE MODELS: LOGIT ANALYSIS

4. 4 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. BINARY CHOICE MODELS: LOGIT ANALYSIS

5. 5 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 and applies it to F(Z). BINARY CHOICE MODELS: LOGIT ANALYSIS

6. 6 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. 7 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. BINARY CHOICE MODELS: LOGIT ANALYSIS

8. 8 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. BINARY CHOICE MODELS: LOGIT ANALYSIS

9. . logit GRAD ASVABC Iteration 0: Log Likelihood =-162.29468 Iteration 1: Log Likelihood =-132.97646 Iteration 2: Log Likelihood =-117.99291 Iteration 3: Log Likelihood =-117.36084 Iteration 4: Log Likelihood =-117.35136 Iteration 5: Log Likelihood =-117.35135 Logit Estimates Number of obs = 570 chi2(1) = 89.89 Prob > chi2 = 0.0000 Log Likelihood = -117.35135 Pseudo R2 = 0.2769 ------------------------------------------------------------------------------ grad | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- asvabc |.1666022.0211265 7.886 0.000.1251951.2080094 _cons | -5.003779.8649213 -5.785 0.000 -6.698993 -3.308564 ------------------------------------------------------------------------------ BINARY CHOICE MODELS: LOGIT ANALYSIS 9 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. . logit GRAD ASVABC Iteration 0: Log Likelihood =-162.29468 Iteration 1: Log Likelihood =-132.97646 Iteration 2: Log Likelihood =-117.99291 Iteration 3: Log Likelihood =-117.36084 Iteration 4: Log Likelihood =-117.35136 Iteration 5: Log Likelihood =-117.35135 Logit Estimates Number of obs = 570 chi2(1) = 89.89 Prob > chi2 = 0.0000 Log Likelihood = -117.35135 Pseudo R2 = 0.2769 ------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- ASVABC |.1666022.0211265 7.886 0.000.1251951.2080094 _cons | -5.003779.8649213 -5.785 0.000 -6.698993 -3.308564 ------------------------------------------------------------------------------ 10 In this case the coefficients of the Z function are as shown. BINARY CHOICE MODELS: LOGIT ANALYSIS

11. 11 Since there is only one explanatory variable, we can draw the probability function and marginal effect function as functions of ASVABC. BINARY CHOICE MODELS: LOGIT ANALYSIS

12. 12 We see that ASVABC has its greatest effect on graduating when it is in the range 20-40, that is, in the lower ability range. Any individual with a score above the average (50) is almost certain to graduate.

13. 13 The t statistic indicates that the effect of variations in ASVABC on the probability of graduating from high school is highly significant. BINARY CHOICE MODELS: LOGIT ANALYSIS. logit GRAD ASVABC Iteration 0: Log Likelihood =-162.29468 Iteration 1: Log Likelihood =-132.97646 Iteration 2: Log Likelihood =-117.99291 Iteration 3: Log Likelihood =-117.36084 Iteration 4: Log Likelihood =-117.35136 Iteration 5: Log Likelihood =-117.35135 Logit Estimates Number of obs = 570 chi2(1) = 89.89 Prob > chi2 = 0.0000 Log Likelihood = -117.35135 Pseudo R2 = 0.2769 ------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- ASVABC |.1666022.0211265 7.886 0.000.1251951.2080094 _cons | -5.003779.8649213 -5.785 0.000 -6.698993 -3.308564 ------------------------------------------------------------------------------

14. BINARY CHOICE MODELS: LOGIT ANALYSIS. logit GRAD ASVABC Iteration 0: Log Likelihood =-162.29468 Iteration 1: Log Likelihood =-132.97646 Iteration 2: Log Likelihood =-117.99291 Iteration 3: Log Likelihood =-117.36084 Iteration 4: Log Likelihood =-117.35136 Iteration 5: Log Likelihood =-117.35135 Logit Estimates Number of obs = 570 chi2(1) = 89.89 Prob > chi2 = 0.0000 Log Likelihood = -117.35135 Pseudo R2 = 0.2769 ------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- ASVABC |.1666022.0211265 7.886 0.000.1251951.2080094 _cons | -5.003779.8649213 -5.785 0.000 -6.698993 -3.308564 ------------------------------------------------------------------------------ 14 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. BINARY CHOICE MODELS: LOGIT ANALYSIS 15 The coefficients of the Z function do not have any direct intuitive interpretation.

16. 16 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. BINARY CHOICE MODELS: LOGIT ANALYSIS

17. 17 Since p is a function of Z, and Z is a function of the X variables, the marginal effect of X i on p can be written as the product of the marginal effect of Z on p and the marginal effect of X i on Z. BINARY CHOICE MODELS: LOGIT ANALYSIS

18. 18 We have already derived an expression for dp/dZ. The marginal effect of X i on Z is given by its  coefficient. BINARY CHOICE MODELS: LOGIT ANALYSIS

19. 19 Hence we obtain an expression for the marginal effect of X i on p. BINARY CHOICE MODELS: LOGIT ANALYSIS

20. 20 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. BINARY CHOICE MODELS: LOGIT ANALYSIS

21. 21 The sample mean of ASVABC was 50.15. BINARY CHOICE MODELS: LOGIT ANALYSIS. sum GRAD ASVABC Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- GRAD | 570.9175439.2753 0 1 ASVABC | 570 50.15088 9.214589 22 65 Logit Estimates Number of obs = 570 chi2(1) = 89.89 Prob > chi2 = 0.0000 Log Likelihood = -117.35135 Pseudo R2 = 0.2769 ------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- ASVABC |.1666022.0211265 7.886 0.000.1251951.2080094 _cons | -5.003779.8649213 -5.785 0.000 -6.698993 -3.308564 ------------------------------------------------------------------------------

22. . sum GRAD ASVABC Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- GRAD | 570.9175439.2753 0 1 ASVABC | 570 50.15088 9.214589 22 65 Logit Estimates Number of obs = 570 chi2(1) = 89.89 Prob > chi2 = 0.0000 Log Likelihood = -117.35135 Pseudo R2 = 0.2769 ------------------------------------------------------------------------------ GRAD | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- ASVABC |.1666022.0211265 7.886 0.000.1251951.2080094 _cons | -5.003779.8649213 -5.785 0.000 -6.698993 -3.308564 ------------------------------------------------------------------------------ 22 When evaluated at the mean, Z is equal to 3.371. BINARY CHOICE MODELS: LOGIT ANALYSIS

23. 23 e -Z is 0.034. Hence f(Z) is 0.032. BINARY CHOICE MODELS: LOGIT ANALYSIS. sum GRAD ASVABC Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- GRAD | 570.9175439.2753 0 1 ASVABC | 570 50.15088 9.214589 22 65

24. 24 The marginal effect, evaluated at the mean, is therefore 0.005. This implies that a one point increase in ASVABC would increase the probability of graduating from high school by 0.5 percent. BINARY CHOICE MODELS: LOGIT ANALYSIS. sum GRAD ASVABC Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- GRAD | 570.9175439.2753 0 1 ASVABC | 570 50.15088 9.214589 22 65

25. 25 In this example, the marginal effect at the mean of ASVABC is very low. The reason is that anyone with an average score is very likely to graduate anyway. So an increase in the score has little effect. BINARY CHOICE MODELS: LOGIT ANALYSIS 50.15

26. . sum COLLEGE ASVABC Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- COLLEGE | 570.4964912.5004269 0 1 ASVABC | 570 50.15088 9.214589 22 65 26 To show that the marginal effect varies, we will also calculate it for ASVABC equal to 30. A one point increase in ASVABC then increases the probability by 4.2 percent. BINARY CHOICE MODELS: LOGIT ANALYSIS

27. 27 An individual with a score of 30 has only a 50 percent probability of graduating, and an increase in the score has a relatively large impact. BINARY CHOICE MODELS: LOGIT ANALYSIS 0.890.042 30

28. . logit GRAD ASVABC SM SF MALE Iteration 0: Log Likelihood =-162.29468 Iteration 1: Log Likelihood =-133.21451 Iteration 2: Log Likelihood =-117.16714 Iteration 3: Log Likelihood =-116.51081 Iteration 4: Log Likelihood =-116.49969 Iteration 5: Log Likelihood =-116.49968 Logit Estimates Number of obs = 570 chi2(4) = 91.59 Prob > chi2 = 0.0000 Log Likelihood = -116.49968 Pseudo R2 = 0.2822 ------------------------------------------------------------------------------ grad | Coef. Std. Err. z P>|z| [95% Conf. Interval] ---------+-------------------------------------------------------------------- ASVABC |.1563271.0224382 6.967 0.000.1123491.2003051 SM |.0645542.0773804 0.834 0.404 -.0871086.216217 SF |.0054552.0616822 0.088 0.930 -.1154397.12635 MALE | -.2790915.3601689 -0.775 0.438 -.9850095.4268265 _cons | -5.15931.994783 -5.186 0.000 -7.109049 -3.209571 ------------------------------------------------------------------------------ 28 Here is the output for a model with a somewhat better specification. BINARY CHOICE MODELS: LOGIT ANALYSIS

29. . sum GRAD ASVABC SM SF MALE Variable | Obs Mean Std. Dev. Min Max ---------+----------------------------------------------------- GRAD | 570.9175439.2753 0 1 ASVABC | 570 50.15088 9.214589 22 65 SM | 570 11.65263 2.561449 0 20 SF | 570 11.81754 3.533178 0 20 MALE | 570.5701754.4954857 0 1 29 We will estimate the marginal effects, putting all the explanatory variables equal to their sample means. BINARY CHOICE MODELS: LOGIT ANALYSIS

30. Logit: Marginal Effects mean b product f(Z) f(Z)b ASVABC50.150.1567.8390.0330.005 SM11.650.0650.7530.0330.002 SF11.820.0060.0650.0330.000 MALE0.57-0.279-0.1590.033-0.009 Constant1.00-5.159-5.159 Total3.338 30 BINARY CHOICE MODELS: LOGIT ANALYSIS The first step is to calculate Z, when the X variables are equal to their sample means.

31. Logit: Marginal Effects mean b product f(Z) f(Z)b ASVABC50.150.1567.8390.0330.005 SM11.650.0650.7530.0330.002 SF11.820.0060.0650.0330.000 MALE0.57-0.279-0.1590.033-0.009 Constant1.00-5.159-5.159 Total3.338 31 BINARY CHOICE MODELS: LOGIT ANALYSIS We then calculate f(Z).

32. Logit: Marginal Effects mean b product f(Z) f(Z)b ASVABC50.150.1567.8390.0330.005 SM11.650.0650.7530.0330.002 SF11.820.0060.0650.0330.000 MALE0.57-0.279-0.1590.033-0.009 Constant1.00-5.159-5.159 Total-3.338 32 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. Every extra year of schooling of the mother increases the probability of graduating by 0.2 percent. BINARY CHOICE MODELS: LOGIT ANALYSIS

33. Logit: Marginal Effects mean b product f(Z) f(Z)b ASVABC50.150.1567.8390.0330.005 SM11.650.0650.7530.0330.002 SF11.820.0060.0650.0330.000 MALE0.57-0.279-0.1590.033-0.009 Constant1.00-5.159-5.159 Total-3.338 33 BINARY CHOICE MODELS: LOGIT ANALYSIS Father's schooling has no discernible effect. Males have 0.9 percent lower probability of graduating than females. These effects would all have been larger if they had been evaluated at a lower ASVABC score.

34. Individuals who graduated: outcome probability is 34 This sequence will conclude with an outline explanation of how the model is fitted using maximum likelihood estimation. BINARY CHOICE MODELS: LOGIT ANALYSIS

35. 35 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. BINARY CHOICE MODELS: LOGIT ANALYSIS Individuals who graduated: outcome probability is

36. 36 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. BINARY CHOICE MODELS: LOGIT ANALYSIS Maximize F(Z 1 ) x... x F(Z s ) x [1 - F(Z s+1 )] x... x [1 - F(Z n )] Individuals who graduated: outcome probability is Individuals did not graduate: outcome probability is

37. Maximize F(Z 1 ) x... x F(Z s ) x [1 - F(Z s+1 )] x... x [1 - F(Z n )] Did graduate Did not graduate 37 We choose b 1 and b 2 so as to maximize the joint probability of the outcomes, that is, F(Z 1 ) x... x F(Z s ) x [1 - F(Z s+1 )] x... x [1 - F(Z n )]. There are no mathematical formulae for b 1 and b 2. They have to be determined iteratively by a trial-and-error process. BINARY CHOICE MODELS: LOGIT ANALYSIS