Ch11binary - Regression with a Binary Dependent Variable...

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1 Regression with a Binary Dependent Variable (SW Chapter 11) So far the dependent variable ( Y ) has been continuous: district-wide average test score traffic fatality rate 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 2 Origins of Binary Choice Y = 1 if “patient” dies and =0 otherwise This is still the basic model in biostatistics and epidemiology A related model is Y is the time until the patient dies (or set to the maximum time if the patient lives) In these applications X = sex, race, age, health status, location, and sometimes income 3 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: Is the mortgage denied or accepted? Independent variables: income, wealth, employment status other loan, property characteristics race of applicant 4 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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5 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 ) 6 Interpretation Note that the only random variable we make assumptions about is u i Y i = 0 + 1 X i + u i E ( u i | X i ) = 0 means we are taking the expectation across a population of observationally identical people (those with the same value of X i ). Therefore E ( Y | X ) = Pr( Y =1| X ) can be interpreted as the proportion of these observationally identical people with Y=1 . 7 The linear probability model, ctd. When Y is binary, the linear regression model Y i = 0 + 1 X i + u i is called the linear probability model . The predicted value is a probability : E ( Y | X = x ) = Pr( Y =1| X = x ) = prob. that Y = 1 given x 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 8 Example : linear probability model, HMDA data Mortgage denial v. ratio of debt payments to income (P/I ratio) in the HMDA data set (subset)
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9 What are problems with the Linear probability model A. The errors are heteroskedastic B. Least Squares estimates are not efficient C. Predicted probabilities can be greater than 1 or less than 0 D. All of the above 10 Binary X variable If X is binary (or even discrete), an alternative model is a contingency table. Suppose that X = 1 if the debt/income ratio is greater than .4.
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Ch11binary - Regression with a Binary Dependent Variable...

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