Handout
Regression with a Binary Dependent Variable
Binary Dependent Variables and the Linear Probability Model
• • •
Many of the decisions made by people are binary. What factors drive a person's decision? This question leads to regression with a binary dependent
variable.
The binary choice problem is an example of models with limited dependent variables (see Appendix 9.3 for details). Note that the multiple regression model discussed earlier does not preclude a dependent variable from being binary.
Boston HMDA data set
•
Under Home Mortgage Disclosure Act (HMDA), researchers at Federal Reserve Bank of Boston collected information about mortgage applicants and lending institutions (banks and others) in the greater Boston metropolitan area in 1990.
• • •
The full data set contains 2,925 observations, consisting of all mortgage applications by blacks and Hispanics plus a random sample of mortgage applications by whites. We here use a subsample only containing the applications for single-family residences. The sample size of this subset is 2,380. In this data set, 28% of black applicants were denied mortgages, while only 9% of white applicants were denied. Does this indicate some discriminatory treatment of applications?
Key variables in our example:
deny :
A binary variable that is one if and only if the loan application is denied.
P/I ratio: The applicant's anticipated monthly loan payment divided by his/her monthly income.
Q: What relationship would you expect between A:
deny
and the P/I ratio?
76
Note: This graph was created by using only 127 observations out of 2,380. Q: What can we learn from the OLS regression result? A:
Provided that the linear regression of
deny
on the P/I ratio is the correct speci cation, we have that
The OLS regression of
deny
on the P/I ratio estimates
β0
and
β1
in this equality, so that we can learn the
probability of application denial conditional on the P/I ratio.
Binary Dependent Variables and the Linear Probability Model
• • •
Many of the decisions made by people are binary. What factors drive a person's decision? This question leads to regression with a binary dependent
variable.
The binary choice problem is an example of models with limited dependent variables (see Appendix 9.3 for details). Note that the multiple regression model discussed earlier does not preclude a dependent variable from being binary.
Boston HMDA data set
•
Under Home Mortgage Disclosure Act (HMDA), researchers at Federal Reserve Bank of Boston collected information about mortgage applicants and lending institutions (banks and others) in the greater Boston metropolitan area in 1990.
• • •
The full data set contains 2,925 observations, consisting of all mortgage applications by blacks and Hispanics plus a random sample of mortgage applications by whites. We here use a subsample only containing the applications for single-family residences. The sample size of this subset is 2,380. In this data set, 28% of black applicants were denied mortgages, while only 9% of white applicants were denied. Does this indicate some discriminatory treatment of applications?
Key variables in our example:
deny :
A binary variable that is one if and only if the loan application is denied.
P/I ratio: The applicant's anticipated monthly loan payment divided by his/her monthly income.
Q: What relationship would you expect between A:
deny
and the P/I ratio?
76
Note: This graph was created by using only 127 observations out of 2,380. Q: What can we learn from the OLS regression result? A:
Provided that the linear regression of
deny
on the P/I ratio is the correct speci cation, we have that
The OLS regression of
deny
on the P/I ratio estimates
β0
and
β1
in this equality, so that we can learn the
probability of application denial conditional on the P/I ratio.