Mortgage lenders are interested in determining borrower and loan characteristics that may lead to delinquency or foreclosure. In the data file lasvegas are 1000 observations on mortgages for single family homes in Las Vegas, Nevada during 2008. The variable of interest is DELINQUENT, an indicator variable \(=1\) if the borrower missed at least three payments \((90+\) days late \()\), but 0 otherwise. Explanatory variables are \(L V R=\) the ratio of the loan amount to the value of the property; \(R E F=1\) if purpose of the loan was a "refinance" and \(=0\) if loan was for a purchase; \(I N S U R=1\) if mortgage carries mortgage insurance, 0 otherwise; \(R A T E=\) initial interest rate of the mortgage; AMOUNT \(=\) dollar value of mortgage (in \(\$ 100,000\) ); CREDIT \(=\) credit score, TERM \(=\) number of years between disbursement of the loan and the date it is expected to be fully repaid, \(A R M=1\) if mortgage has an adjustable rate, and \(=0\) if mortgage has a fixed rate.

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

b. Use logit 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 DELINQENT for the 500th and 1000th observations using both the linear probability model and the logit model. Interpret the values.

d. Construct a histogram of CREDIT. Using both linear probability and logit models, calculate the probability of delinquency for \(C R E D I T=500,600\), and 700 for a loan of \(\$ 250,000\) \((A M O U N T=2.5)\). For the other variables, let the loan to value ratio \((L V R)\) be \(80 \%\), the initial interest rate is \(8 \%\), all indicator variables take the value 0 , and \(T E R M=30\). Discuss similarities and differences among the predicted probabilities from the two models.

e. Using both linear probability and logit models, compute the marginal effect of CREDIT on the probability of delinquency for \(C R E D I T=500,600\), 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 \(L V R\). Using both linear probability and logit models, calculate the probability of delinquency for \(L V R=20\) and \(L V R=80\), with \(C R E D I T=600\) 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 logit 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 logit 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?