Chapter 11 - Regression with a Binary Dependent Variable

# Chapter 11 - Regression with a Binary Dependent Variable -...

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9-1 Regression with a Binary Dependent Variable (SW Ch. 11) So far the dependent variable ( Y ) has been continuous: district-wide average test score traffic fatality rate But we might want to understand the effect of X on a binary variable: Y = get into college, or not Y = person smokes, or not Y = mortgage application is accepted, or not

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9-2 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
9-3 The Linear Probability Model (SW Section 11.1) A natural starting point is the linear regression model with a single regressor: Y i = β 0 + 1 X i + u i But: What does 1 mean when Y is binary? Is 1 = Y X Δ Δ ? What does the line 0 + 1 X mean when Y is binary? What does the predicted value ˆ Y mean when Y is binary? For example, what does ˆ Y = 0.26 mean?

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9-4 The linear probability model, ctd. Y i = β 0 + 1 X i + u i Recall assumption #1: E ( u i | X i ) = 0, so E ( Y i | X i ) = E ( 0 + 1 X i + u i | X i ) = 0 + 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 )
9-5 The linear probability model, ctd. When Y is binary, the linear regression model Y i = β 0 + 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 ) = prob. that Y = 1 given x o ˆ Y = the predicted probability that Y i = 1, given X 1 = change in probability that Y = 1 for a given Δ x : 1 = P r (1 | ) P r | ) YX x x x x == + Δ Δ Example: linear probability model, HMDA data

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9-6 Mortgage denial v. ratio of debt payments to income (P/I ratio) in the HMDA data set (subset)
9-7 Linear probability model: HMDA data n deny = -.080 + .604 P/I ratio ( n = 2380) (.032) (.098) What is the predicted value for P/I ratio = .3? Pr( 1| / .3) deny P Iratio == = -.080 + .604 × .3 = .151 Calculating “effects:” increase P/I ratio from .3 to .4: Pr( / .4) deny P Iratio = -.080 + .604 × .4 = .212 The effect on the probability of denial of an increase in P/I ratio from .3 to .4 is to increase the probability by .061, that is, by 6.1 percentage points .

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9-8 Next include black as a regressor: n deny = -.091 + .559 P/I ratio + .177 black (.032) (.098) (.025) Predicted probability of denial: for black applicant with P/I ratio = .3: n Pr( 1) deny = = -.091 + .559 × .3 + .177 × 1 = .254 for white applicant, P/I ratio = .3: n Pr( deny = = -.091 + .559 × .3 + .177 × 0 = .077 difference = .177 = 17.7 percentage points Coefficient on black is significant at the 5% level Still plenty of room for omitted variable bias…
9-9 The linear probability model: Summary Models probability as a linear function of X Advantages: o simple to estimate and to interpret o inference is the same as for multiple regression (need heteroskedasticity-robust standard errors) Disadvantages: o Does it make sense that the probability should be linear in X?

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## This note was uploaded on 09/13/2009 for the course ECONOMICS ECON 435 taught by Professor Pascal during the Spring '07 term at Simon Fraser.

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Chapter 11 - Regression with a Binary Dependent Variable -...

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