Mortgage lenders are interested in determining borrower and loan factors that may lead to delinquency or foreclosure. In the data file vegas5_small are 1000 observations on mortgages for single family homes in Las Vegas, Nevada during 2010. (The data file vegas 5 contains 10,000 observations.) The variable of interest is DEFAULT, an indicator variable \(=1\) if the borrower's payment was 90 + days late, but 0 otherwise. Explanatory variables are \(A R M=1\) if it's an adjustable rate mortgage, 0 if fixed; REFINANCE \(=1\) if loan is for a refinance of any type, 0 if for purchase; LIEN2 = 1 if there is a second lien mortgage, 0 if it is the first lien; TERM30 = 1 if it is a 30 -year mortgage, 0 if 15 -year mortgage; \(U N D E R W A T E R ~=1\) if borrower estimated to owe more than the property is worth at time of origination, 0 otherwise; \(L T V=\) loan to value ratio of property at origination, percent; \(R A T E=\) current interest rate on loan, percent; \(A M O U N T=\) loan amount in \$10,000 units; and \(F I C O=\) borrower's credit score at origination.

a. Estimate the linear probability (regression) model explaining DEFAULT as a function of the remaining variables. Use White robust standard errors. Are the signs of the estimated coefficients reasonable?

b. Use probit to estimate the model in (a). Are the signs and significance of the estimated coefficients the same as for the linear probability model?

c. Compute the predicted value of DEFAULT for the 500th and 1000th observations using both the linear probability model and the probit model. Interpret the values.

d. Construct a histogram of FICO. Using both linear probability and probit models, calculate the probability of default for \(F I C O=500,600\), and 700 for a loan of \(\$ 250,000\) (AMOUNT =25). For the other variables, the loan to value ratio ( \(L T V)\) is \(80 \%\), initial interest rate is \(8 \%\), indicator variables take the value 0 except for \(T E R M 30=1\). Discuss similarities and differences among the predicted probabilities from the two models.

e. Using both linear probability and probit models, compute the marginal effect of FICO on the probability of delinquency for \(F I C O=500,600\), and 700 , given that the other explanatory variables take the values in (d). Discuss the interpretation of the marginal effect.

f. Construct a histogram of \(L T V\). Using both linear probability and probit models, calculate the probability of delinquency for \(L V R=20\) and \(L V R=80\), with \(F I C O=600\) and other variables set as they are in (d). Compare and contrast the results.

g. Compare the percentage of correct predictions from the linear probability model and the probit model using a predicted probability of 0.5 as the threshold.

h. As a loan officer, you wish to provide loans to customers who repay on schedule and are not delinquent. Suppose you have available to you the first 500 observations in the data on which to base your loan decision on the second 500 applications \((501-1,000)\). Is using the probit model with a threshold of 0.5 for the predicted probability the best decision rule for deciding on loan applications? If not, what is a better rule?