A bank finds that the estimated proportion of clients defaulting on a loan, give

emancipezN 2021-09-08 Answered
A bank finds that the estimated proportion of clients defaulting on a loan, given the interest rate is below \(\displaystyle{15}\%\), is 0.34, They also find that the estimated proportion of clients defaulting on a loan, given the interest is greater than or equal to \(\displaystyle{15}\%\), is 0.52.
a) What is the odds ratio of defaulting given the interest rate is greater than or equal to \(\displaystyle{15}\%\) relative to the interest rate is lower than \(\displaystyle{15}\%\)? Interpret this odds ratio.
b) If we were to analyze this data using the following logistic regression model, what are the estimates of \(\displaystyle\beta_{{{0}}}\) and \(\displaystyle\beta_{{{1}}}\)? Show your work.
\(\displaystyle{l}{g}{\left\lbrace{\left({\frac{{{p}}}{{{1}-{p}}}}\right)}\right\rbrace}=\beta_{{{0}}}+\beta_{{{1}}}{x}\)
Where p is the probability of defaulting on a loan and x is an indicator variable that is 1 when the interest rate is greater than or equal to \(\displaystyle{15}\%\) and 0 when the interest rate is less than \(\displaystyle{15}\%\).

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Expert Answer

Aubree Mcintyre
Answered 2021-09-09 Author has 8598 answers

SOLUTION: we translate data into tabular formate.
\(\begin{array}{|c|c|}\hline &&Cases&Contol\\ \hline &&Defaulting&non-Defaulting\\ \hline exposed&<15\%&0.34&0.66\\ \hline unexposed&\geq15\%&0.52&0.48\\ \hline\end{array}\)
a) What is the odds ratio of defaulting given the interest rate is greater than or equal to \(\displaystyle{15}\%\) relative to the interest rate is lower than \(\displaystyle{15}\%\)? Interpret this odds ratio.
ANS: odds of defaulting given interest rate \(\displaystyle{15}\geq{15}\%\) is \(\displaystyle{o}{d}{d}{1}={\left({\frac{{{0.52}}}{{{0.48}}}}\right)}={1.083}\)
odds of defaulting given interest rate \(\displaystyle{<}{15}\%\) is \(\displaystyle{o}{d}{d}{1}={\left({\frac{{{0.34}}}{{{0.66}}}}\right)}={0.5151}\)
SO odds ratio is given by,
addratio=\(\displaystyle{\frac{{{o}{d}{d}{s}{1}}}{{{o}{d}{d}{s}{20}}}}={\frac{{{1.083}}}{{{0.5151}}}}={2.1025}\)
Interpretation: odds of defaulting given interest rate \(\displaystyle{15}\geq{15}\%\) is greater than odds of defaulting given interest rate \(\displaystyle{<}{15}\%\), that is high intereset rate \(\displaystyle{15}\geq{15}\%\) may be a risk factor for defualting a loan.
b) If we were to analyze this data using the following logistic regression model, what are the estimates of \(\displaystyle\beta_{{{0}}}\) and \(\displaystyle\beta_{{{1}}}\)? Show your work.
\(\displaystyle{l}{g}{\left\lbrace{\left({\frac{{{p}}}{{{1}-{p}}}}\right)}\right\rbrace}=\beta_{{{0}}}+\beta_{{{1}}}{x}\)
ANS:
where bl and bo are \(\displaystyle{b}{1}={\log{}}\) (base catergory) and \(\displaystyle{b}{1}={\log{}}\) (odd ratio)
\(\displaystyle{b}{o}={\log{}}\) (odds of defaulting given interest rate \(\displaystyle{15}\geq{15}\%={\log{{\left({1.083}\right)}}}={0.03462}\)
\(\displaystyle{b}{1}={\log{}}\) (oddratio of defaulting given interest rate \(\displaystyle{15}\geq{15}\%\) relative to the defaulting given interest rate \(\displaystyle{<}{15}\%={\log{{\left({2.1025}\right)}}}\)

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