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Regression with a Binary Dependent Variable (SW Chapter 11)

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Example: Mortgage denial and raceThe Boston Fed HMDA data set

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The Linear Probability Model

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The Linear Probability Model

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Example: Linear Prob Model

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Linear probability model: HMDA data

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Linear probability model: HMDA data

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Linear probability model: ApplicationCattaneo, 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 50m2 of cement flooring ($150 value), starting in 2000, for homeowners with dirt floors

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Cattaneo et al. (AEJ:Economic Policy 2009) “Housing, Health, & Happiness” X1 = “Program dummy” = 1 if offered Piso Firme.

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Cattaneo et al. (AEJ:Economic Policy 2009) “Housing, Health, & Happiness”

Interpretations?

X1 = “Program dummy” = 1 if offered Piso Firme

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Probit and Logit Regression

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Probit Regression

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STATA Example: HMDA data

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STATA Example: HMDA data, ctd.

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Probit regression with multiple regressors

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STATA Example: HMDA data

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STATA Example: HMDA data

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STATA Example: HMDA data

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Probit Regression Marginal Effects

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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 = 0.0000Log 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------------------------------------------------------------------------------

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Probit Regression Marginal Effects

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Logit Regression

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STATA Example: HMDA data

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

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Logit Regression Marginal Effects

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Comparison of Marginal Effects

LPM Probit Logit

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)

. probit coke pratio disp_coke disp_pepsi /* should request robust std errors, I didn’t when making original slide */

. margins , dydx(*) /* Average marginal effect */

. margins , dydx(*) atmeans /* Marginal effect at means of X’s */

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Probit model: ApplicationArcidiacono 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

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Arcidiacono & Vigdor (EI, 2010)

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Arcidiacono & Vigdor (EI, 2010)

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Arcidiacono & Vigdor (EI, 2010)

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Logit model: ApplicationBodvarsson & 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

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Bodvarsson & Walker (EconEduR,2004)

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Bodvarsson & Walker (EconEduR,2004)

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Estimation and Inference in Probit (and Logit) Models

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Probit estimation by maximum likelihood

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Special case: probit MLE with no X

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

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

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The probit likelihood with one X

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The probit likelihood function:

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The Probit MLE, ctd.

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The logit likelihood with one X

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Measures of fit for logit and probit

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Measures of fit for logit and probit

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. lsens /* graphs sensitivity & specificity vs. probability cutoff */

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Application to the Boston HMDA Data (SW Section 11.4)

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The HMDA Data Set

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The loan officer’s decision

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Regression specifications

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Table 11.2, ctd.

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Table 11.2, ctd.

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Summary of Empirical Results