# 1 Regression with a Binary Dependent Variable (SW Chapter 11)

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1 Regression with a Binary Dependent Variable (SW Chapter 11)

2 Example: Mortgage denial and race The Boston Fed HMDA data set

3 The Linear Probability Model

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5 Example: Linear Prob Model

6 Linear probability model: HMDA data

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8 Linear probability model: Application Cattaneo, Galiani, Gertler, Martinez, and Titiunik (2009). “Housing, Health, and Happiness.” American Economic Journal: Economic Policy 1(1): 75 - 105 What was the impact of Piso Firme, a large-scale Mexican program to help families replace dirt floors with cement floors? A pledge by governor Enrique Martinez y Martinez led to State of Coahuila offering free 50m 2 of cement flooring (\$150 value), starting in 2000, for homeowners with dirt floors

9 Cattaneo et al. (AEJ:Economic Policy 2009) “Housing, Health, & Happiness” X 1 = “Program dummy” = 1 if offered Piso Firme.

10 Cattaneo et al. (AEJ:Economic Policy 2009) “Housing, Health, & Happiness” Interpretations? X 1 = “Program dummy” = 1 if offered Piso Firme

11 Probit and Logit Regression

12 Probit Regression

13 STATA Example: HMDA data

14 STATA Example: HMDA data, ctd.

15 Probit regression with multiple regressors

16 STATA Example: HMDA data

17 STATA Example: HMDA data

18 STATA Example: HMDA data

19 Probit Regression Marginal Effects

20 Probit Regression Marginal Effects. sum pratio; Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- pratio | 1140 1.027249.286608.497207 2.324675. scalar meanpratio = r(mean);. sum disp_pepsi; Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- disp_pepsi | 1140.3640351.4813697 0 1. scalar meandisp_pepsi = r(mean);. sum disp_coke; Variable | Obs Mean Std. Dev. Min Max -------------+-------------------------------------------------------- disp_coke | 1140.3789474.4853379 0 1. scalar meandisp_coke = r(mean);. probit coke pratio disp_coke disp_pepsi; Iteration 0: log likelihood = -783.86028 Iteration 1: log likelihood = -711.02196 Iteration 2: log likelihood = -710.94858 Iteration 3: log likelihood = -710.94858 Probit regression Number of obs = 1140 LR chi2(3) = 145.82 Prob > chi2 = 0.0000 Log likelihood = -710.94858 Pseudo R2 = 0.0930 ------------------------------------------------------------------------------ coke | Coef. Std. Err. z P>|z| [95% Conf. Interval] -------------+---------------------------------------------------------------- pratio | -1.145963.1808833 -6.34 0.000 -1.500487 -.791438 disp_coke |.217187.0966084 2.25 0.025.027838.4065359 disp_pepsi | -.447297.1014033 -4.41 0.000 -.6460439 -.2485502 _cons | 1.10806.1899592 5.83 0.000.7357465 1.480373 ------------------------------------------------------------------------------

21 Probit Regression Marginal Effects

22 Logit Regression

23 STATA Example: HMDA data

24 Predicted probabilities from estimated probit and logit models usually are (as usual) very close in this application.

25 Logit Regression Marginal Effects

26 Comparison of Marginal Effects LPMProbitLogit Marginal Effect at Means for Price Ratio -.4008 (.0613) -.4520 (.0712) via nlcom -.4905 (.0773) via nlcom Average Marginal Effect of Price Ratio -.4008 (.0613) -.4096 (beyond eco205) -.4332 (beyond eco205) Marginal Effect at Means for Coke display dummy.0771 (.0343).0856 (.0381) via nlcom.0864 (.0390) via nlcom Average Marginal Effect For Coke display dummy.0771 (.0343).0776 (beyond eco205).0763 (beyond eco205)

27 Probit model: Application Arcidiacono and Vigdor (2010). “Does the River Spill Over? Estimating the Economic Returns to Attending a Racially Diverse College.” Economic Inquiry 48(3): 537 – 557. Does “diversity capital” matter and does minority representation increase it? Does diversity improve post- graduate outcomes of non-minority students? College & Beyond survey, starting college in 1976

28 Arcidiacono & Vigdor (EI, 2010)

29 Arcidiacono & Vigdor (EI, 2010)

30 Arcidiacono & Vigdor (EI, 2010)

31 Logit model: Application Bodvarsson & Walker (2004). “Do Parental Cash Transfers Weaken Performance in College?” Economics of Education Review 23: 483 – 495. When parents pay for tuition & books does this undermine the incentive to do well? Univ of Nebraska @ Lincoln & Washburn Univ in Topeka, KS, 2001-02 academic year

34 Estimation and Inference in Probit (and Logit) Models

35 Probit estimation by maximum likelihood

36 Special case: probit MLE with no X

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38

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40 The MLE in the “no-X” case (Bernoulli distribution), ctd.:

41 The MLE in the “no-X” case (Bernoulli distribution), ctd:

42 The probit likelihood with one X

43 The probit likelihood function:

44 The Probit MLE, ctd.

45 The logit likelihood with one X

46 Measures of fit for logit and probit

47 Application to the Boston HMDA Data (SW Section 11.4)

48 The HMDA Data Set

49 The loan officer’s decision

50 Regression specifications

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52

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54 Table 11.2, ctd.

55 Table 11.2, ctd.

56 Summary of Empirical Results

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