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1 BINARY CHOICE MODELS: PROBIT ANALYSIS In the case of probit analysis, the sigmoid function F(Z) giving the probability is the cumulative standardized.

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Presentation on theme: "1 BINARY CHOICE MODELS: PROBIT ANALYSIS In the case of probit analysis, the sigmoid function F(Z) giving the probability is the cumulative standardized."— Presentation transcript:

1 1 BINARY CHOICE MODELS: PROBIT ANALYSIS In the case of probit analysis, the sigmoid function F(Z) giving the probability is the cumulative standardized normal distribution, and the marginal effect f(Z) is given by the normal distribution itself.

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

3 . probit GRAD ASVABC SM SF MALE Iteration 0: log likelihood = -118.67769 Iteration 1: log likelihood = -98.195303 Iteration 2: log likelihood = -96.666096 Iteration 3: log likelihood = -96.624979 Iteration 4: log likelihood = -96.624926 Probit estimates Number of obs = 540 LR chi2(4) = 44.11 Prob > chi2 = 0.0000 Log likelihood = -96.624926 Pseudo R2 = 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 ------------------------------------------------------------------------------ 3 Here is the result of the probit regression using the example of graduating from high school. BINARY CHOICE MODELS: PROBIT ANALYSIS

4 4 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. BINARY CHOICE MODELS: PROBIT ANALYSIS. probit GRAD ASVABC SM SF MALE Iteration 0: log likelihood = -118.67769 Iteration 1: log likelihood = -98.195303 Iteration 2: log likelihood = -96.666096 Iteration 3: log likelihood = -96.624979 Iteration 4: log likelihood = -96.624926 Probit estimates Number of obs = 540 LR chi2(4) = 44.11 Prob > chi2 = 0.0000 Log likelihood = -96.624926 Pseudo R2 = 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 ------------------------------------------------------------------------------

5 5 As with logit analysis, 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: PROBIT ANALYSIS

6 6 The marginal effect of Z on p is given by the standardized normal distribution. The marginal effect of X i on Z is given by  i. BINARY CHOICE MODELS: PROBIT ANALYSIS

7 7 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. BINARY CHOICE MODELS: PROBIT ANALYSIS

8 8 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.. 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 BINARY CHOICE MODELS: PROBIT ANALYSIS

9 9 In this case Z is equal to 1.881 when the X variables are equal to their sample means. BINARY CHOICE MODELS: PROBIT ANALYSIS Probit: Marginal Effects mean b product ASVABC51.360.0653.328 SM11.58–0.008–0.094 SF11.840.0060.066 MALE0.500.0630.032 constant1.00–1.451–1.451 Total1.881

10 10 We then calculate f(Z). BINARY CHOICE MODELS: PROBIT ANALYSIS Probit: Marginal Effects mean b product ASVABC51.360.0653.328 SM11.58–0.008–0.094 SF11.840.0060.066 MALE0.500.0630.032 constant1.00–1.451–1.451 Total1.881

11 11 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. BINARY CHOICE MODELS: PROBIT ANALYSIS Probit: Marginal Effects mean b product f(Z) f(Z)b ASVABC51.360.0653.3280.0680.004 SM11.58–0.008–0.0940.068–0.001 SF11.840.0060.0660.0680.000 MALE0.500.0630.0320.0680.004 constant1.00–1.451–1.451 Total1.881

12 Probit: Marginal Effects mean b product f(Z) f(Z)b ASVABC51.360.0653.3280.0680.004 SM11.58–0.008–0.0940.068–0.001 SF11.840.0060.0660.0680.000 MALE0.500.0630.0320.0680.004 constant1.00–1.451–1.451 Total1.881 12 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. BINARY CHOICE MODELS: PROBIT ANALYSIS

13 13 The logit and probit results are displayed for comparison. The coefficients in the regressions are very different because different mathematical functions are being fitted. BINARY CHOICE MODELS: PROBIT ANALYSIS Logit Probit f(Z)b f(Z)b ASVABC0.0040.004 SM–0.001–0.001 SF0.0000.000 MALE0.0040.004

14 14 Nevertheless the estimates of the marginal effects are usually similar. BINARY CHOICE MODELS: PROBIT ANALYSIS Logit Probit f(Z)b f(Z)b ASVABC0.0040.004 SM–0.001–0.001 SF0.0000.000 MALE0.0040.004

15 15 However, if the outcomes in the sample are divided between a large majority and a small minority, they can differ. BINARY CHOICE MODELS: PROBIT ANALYSIS Logit Probit f(Z)b f(Z)b ASVABC0.0040.004 SM–0.001–0.001 SF0.0000.000 MALE0.0040.004

16 16 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. BINARY CHOICE MODELS: PROBIT ANALYSIS Logit Probit f(Z)b f(Z)b ASVABC0.0040.004 SM–0.001–0.001 SF0.0000.000 MALE0.0040.004

17 17 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. BINARY CHOICE MODELS: PROBIT ANALYSIS Logit Probit f(Z)b f(Z)b ASVABC0.0040.004 SM–0.001–0.001 SF0.0000.000 MALE0.0040.004

18 18 Finally, for comparison, the estimates for the corresponding regression using the linear probability model are displayed. BINARY CHOICE MODELS: PROBIT ANALYSIS Logit Probit Linear f(Z)b f(Z)b b ASVABC0.0040.0040.007 SM–0.001–0.001–0.002 SF0.0000.0000.001 MALE0.0040.004–0.007

19 Logit Probit Linear f(Z)b f(Z)b b ASVABC0.0040.0040.007 SM–0.001–0.001–0.002 SF0.0000.0000.001 MALE0.0040.004–0.007 19 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. BINARY CHOICE MODELS: PROBIT ANALYSIS

20 Copyright Christopher Dougherty 2012. These slideshows may be downloaded by anyone, anywhere for personal use. 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. Additional (free) resources for both students and instructors may be downloaded from the OUP Online Resource Centre http://www.oup.com/uk/orc/bin/9780199567089/http://www.oup.com/uk/orc/bin/9780199567089/. Individuals studying econometrics on their own 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 http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx http://www2.lse.ac.uk/study/summerSchools/summerSchool/Home.aspx or the University of London International Programmes distance learning course 20 Elements of Econometrics www.londoninternational.ac.uk/lsewww.londoninternational.ac.uk/lse. 2012.12.08

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