CHAPTER 11 Regression with a Binary Dependent Variable. Kazu Matsuda IBEC PHBU 430 Econometrics

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1 CHAPTER 11 Regression with a Binary Dependent Variable Kazu Matsuda IBEC PHBU 430 Econometrics

2 Mortgage Application Example Two people, identical but for their race, walk into a bank and apply for a mortgage, a large loan so that each can buy an identical house. Does the bank treat them the same way? Are they both equally likely to have their mortgage application accepted? Buy law they must receive identical treatment.

3 Mortgage Application Example But how, precisely, should one check for statistical evidence of discrimination in the mortgage market? A start is to compare the fraction of minority and white applicants who were denied a mortgage. In our data, gathered from mortgage applications in 1990 in the Boston, MA, area, 28% of black applicants were denied mortgages but only 9% of white applicants were denied. But this comparison does not really answer the question that opened this chapter, because the black and white applicants were not necessarily identical but for their race.

4 11.1 Binary Dependent Variables & the Linear Probability Model Dependent variable: Deny = You have to create Deny from s7. One important piece of information is the size of the required loan payments relative to the applicant s income. Explanatory variable: The ratio of the applicant s anticipated total monthly loan payments to his/her monthly income. You have to create this by s46/100.

5 Sample average (deny) = About the data Sample average (black) = Sample average (P/I ratio) =

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7 The Linear Probability Model It is the name for the multiple regression model when the dependent variable is binary rather than continuous. Because the dependent variable Y is binary, the population regression function corresponds to the probability that the dependent variable equals one, given X. The OLS predicted value,, computed using the estimated regression function, is the predicted probability that the dependent variable equals 1.

8 The Linear Probability Model Deny R = P/ Iratio (0.032) (0.098) = SER= 2 Adj.039, 0.32 The estimated coefficient on P/I ratio is positive, and the population coefficient is statistically significant at the? % level. If the P/I ratio increases by?, then the probability of denial increases by?. If the P/I ratio increases by?, then the probability of denial increases by?.

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10 According to this linear probability model, an applicant whose projected debt payments are 30% of income has a probability of? that his/her application will be denied.

11 The Linear Probability Model Because the errors of the linear probability model are always heteroscedastic, it is essential that heteroscedasticity-robust standard errors be used for inference. One tool that does not carry over is the R 2. The R 2 is not a particularly useful statistic here.

12 Including the race dummy Black = 2 Adj.0752, if the applicant is black if the applicant is white Deny = P/ Iratio Black R (0.029) (0.089) (0.025) = SER= The coefficient on Black,?, indicates that an African American applicant has a? % higher probability of having a mortgage application denied than a white, holding constant their P/I ratio. This coefficient is significant at the? % level.

13 According to this linear probability model, an applicant who is black & whose projected debt payments are 30% of income has a probability of? that his/her application will be denied.

14 Shortcomings of the linear probability model

15 Probit & Logit Regression These are nonlinear regression models specifically designed for dummy dependent variables. These force the predicted values to be between 0 and 1. Because cumulative probability distribution functions produce probabilities between 0 and 1, they are used in lotgit and probit regressions. Probit regression uses the standard? c.d.f. Logit regression uses the? c.d.f.

16 Probit Regression Φ ( ) Where is the cumulative standard normal distribution function. Suppose β 0 = -2 and β 1 = 3. What is the probability of denial if P/I ratio = 0.4? In the probit model, the term, β 0 +β 1 X, plays the role of z in the cumulative standard normal distribution table.

17 Estimation result without race dummy ( Deny = P Iratio) =Φ( + P Iratio) Pr 1 / / (0.14) (0.39) What is the change in the predicted probability than an application will be denied when the P/I ratio increases from 0.3 to 0.4? The coefficients are best interpreted indirectly by computing probabilities and changes in probabilities.

18 Estimation result without race dummy

19 Estimation result with race dummy ( Deny = P Iratio Black ) =Φ( + P Iratio + Black ) Pr 1 /, / 0.71 (0.14) (0.38) (0.083) The coefficient on Black is positive, indicating that an African American applicant has a? probability of denial than a white applicant, holding constant their P/I ratio. This coefficient is statistically significant at the? % level. For a white applicant with P/I ratio = 0.3, the predicted denial probability is? %, while for a black applicant with P/I ratio = 0.3, the predicted denial probability is? %.

20 Estimation result with race dummy

21 Logistic Regression Logit regression is similar to probit regression, except that the cumulative distribution function is different. The population logit model of the binary dependent variable Y with multiple regressors is: ( Y = X X X ) = F( β + β X + β X + + β X ) Pr 1,,...,... = k k k 1 ( X X... X ) e β + β + β + + β k k 1

22 Estimation result with race dummy ( Deny = P Iratio Black ) Pr 1 /, ( / 1.27 ) = Logistic + P Iratio + Black (0.27) (0.73) (0.15) This coefficient is statistically significant at the? % level. For a white applicant with P/I ratio = 0.3, the predicted denial probability is? %, while for a black applicant with P/I ratio = 0.3, the predicted denial probability is? %.

23 Estimation result with race dummy

24 Comparing the LP, Probit,, & Logit Models Probability of Application Denial given P/I ratio = 0.3 LPM Probit Logit Black White