Lecture 12

# Lecture 12 - LECTURE 12 REGRESSION WITH A BINARY DEPENDENT...

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LECTURE 12: REGRESSION WITH A BINARY DEPENDENT VARIABLE So far the dependent variable (Y) has been continuous: Wages GDP, Investment, etc. What if Y is binary? Y = get into college, or not; X = years of education Y = person smokes, or not; X = income Y = mortgage application is accepted, or not; X = income, house characteristics, marital status, race Example: Mortgage Denial and Race: The Boston Fed HMDA data set Individual applications for single-family mortgages made in 1990 in the greater Boston area 2380 observations, collected under Home Mortgage Disclosure Act (HMDA) Variables Dependent variable: o Is the mortgage denied or accepted? Independent variables: o Income, wealth, employment status o Other loan, property characteristics o Race of applicant The Linear Probability Model A natural starting point is the linear regression model with a single regressor:

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Y i = β 0 + β 1 X i + u i But: What does β 1 mean when Y is binary? Is β 1 = X Δ Y Δ ? What does the line β 0 + β 1 X mean when Y is binary? What does the predicted value Y ˆ mean when Y is binary? For example, what does Y ˆ = 0.26 mean? Y i = β 0 + β 1 X i + u i Recall assumption #1: E(u i ) = 0, so E(Y i |X i ) = E( β 0 + β 1 X i + u i |X i ) = β 0 + β 1 X i When Y is binary, E(Y) = 1*Pr(Y = 1) + 0*Pr(Y = 0) = Pr(Y = 1) So E(Y|X) = Pr(Y = 1|X) When Y is binary, the linear regression model Y i = β 0 + β 1 X i + u i Is called the linear probability model . The predicted value is a probability : o E(Y|X = x) = Pr(Y = 1|X = x) = probability that Y = 1 given s o Y ˆ = the predicted probability that Y i = 1, given X β 1 = change in probability that Y = 1 for a given Δ x: x Δ ) x X | 1 Y Pr( ) x Δ x X | 1 Y Pr( β 1
Mortgage denial v. ratio of debt payments to income (P/I ratio) in the HMDA data set (subset) Deny = -.080 + .604P/I ratio (n=2380) (.032) (.098) What is the predicted value for P/I ratio = .3? Pr(deny = 1|P/I ratio = .3) = -.080 + .604*.3 = .151 Calculating “effects”: increase P/I ratio from .3 to .4: Pr(deny = 1|P/I ratio = .4) = -.080 + .604*.4 = .212 The effect on the probability of denial of an increase in P/I ratio from .3 to .4 to increase the probability by .061, that is, by 6.1 percentage points . Next include black as a regressor: Deny = -.091 + .559P/I ratio + .177black (.032) (.098) (.025) Predicted probability of denial:

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For black applicant with P/I ratio = .3: Pr(deny = 1) = -.091 + .559*.3 + .177*1 = .254 For white applicant, P/I ratio = .3: Pr(deny = 1) = -.091 + .559*.3 + .177*0 = .077 Difference = .177 = 17.7 percentage points Coefficient on black is significant at the 5% level (Still possibility of omitted variable bias) Models Pr(Y =1|X) as a linear function of X Advantages: o Simple to estimate and to interpret o Inference is the same as for multiple regression Disadvantages: o Does it make sense that the probability should be linear in X? o
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Lecture 12 - LECTURE 12 REGRESSION WITH A BINARY DEPENDENT...

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