Question

This case study concerns a bank's efforts to calculate credit risk scores (These are opposite of credit scores. Higher the value the riskier the customer) A loan officer at a bank needs to be able to identify characteristics that are indicative of people who are likely to default on loans and use those characteristics to identify good and bad credit risks. The loan officer also needs to be able to better quantify an individual’s credit risk level.

Information on 700 past customers is given in the file along with data for the following variables:

Age: customer age in years.

Employment: years that the customer has been with his/her current employer

Address: years that the customer has lived at his/her current address

Income: household annual income (in $1,000)

Debt _to _Income: debt to income ratio (x100)

Risk _Score: Credit risk score (the higher the score, the more risky)

Default _Indicator: an indicator of whether the customer had previously defaulted

Test the hypothesis that the average credit risk score of customers who have previously defaulted on their loans is higher than those who haven’t defaulted on their loans. (Divide the credit risk score into two groups those who previously defaulted and those who didn’t and then compare the two groups.)

You would like to build a regression model to predict the credit risk score based on all the other variables. What is the regression equation?

What is the predicted credit risk score of a customer aged 40, who has been with their employer for 10 years, has lived at the same address for 3 years, has an income of $200,000, debt to income ratio of 2.5 and has previously defaulted on a loan. Will the prediction be accurate? Explain

Compute the coefficient of determination (R2) and fully interpret its meaning.

Interpret the meaning of the coefficients of the independent variables income and address.

Which independent variables significant? Defend your answer.

Age in years	Years with current employer	Years at current address	Household income in thousands	Debt to income ratio (x100)	Previously Defaulted	Credit Risk Score
41	17	12	176	9.3	1	808.3943274
27	10	6	31	17.3	0	198.2974762
40	15	14	55	5.5	0	10.0361081
41	15	14	120	2.9	0	22.13828376
24	2	0	28	17.3	1	781.5883142
41	5	5	25	10.2	0	216.7089415
39	20	9	67	30.6	0	185.9601084
43	12	11	38	3.6	0	14.70865349
24	3	4	19	24.4	1	748.0412036
36	0	13	25	19.7	0	815.0570131
27	0	1	16	1.7	0	350.309226
25	4	0	23	5.2	0	239.0539023
52	24	14	64	10	0	9.790173473
37	6	9	29	16.3	0	364.4940475
48	22	15	100	9.1	0	11.87390385
36	9	6	49	8.6	1	96.70407786
36	13	6	41	16.4	1	212.0503906
43	23	19	72	7.6	0	1.404870603
39	6	9	61	5.7	0	104.1453903
41	0	21	26	1.7	0	91.9180135
39	22	3	52	3.2	0	4.373536462
47	17	21	43	5.6	0	3.047352362
28	3	6	26	10	0	293.9321797
29	8	6	27	9.8	0	106.7996198
21	1	2	16	18	1	629.7774553
25	0	2	32	17.6	0	861.3134014
45	9	26	69	6.7	0	16.46115799
43	25	21	64	16.7	0	1.437993467
33	12	8	58	18.4	0	276.7066755
26	2	1	37	14.2	0	503.3218674
45	3	15	20	2.1	0	76.41958523
30	1	10	22	10.5	0	433.6994251
27	2	7	26	6	0	288.7388759
25	8	4	27	14.4	0	231.1006843
25	8	1	35	2.9	0	74.95719559
26	6	7	45	26	0	950.0535168
30	10	4	22	16.1	0	211.9564036
32	12	1	54	14.4	0	335.9969153
28	1	8	24	17.1	1	643.9032953
45	23	5	50	4.2	0	2.268753579
23	7	2	31	6.6	0	132.8782071
34	17	3	59	8	0	31.76854323
42	7	23	41	4.6	0	31.90347933
39	19	5	48	13.1	0	28.07933138
26	0	0	14	7.5	1	511.04996
21	0	1	16	6.8	0	453.6168743
35	13	15	35	4.5	0	10.78188877
47	4	2	26	10.4	0	281.6573725
23	0	2	21	11.4	1	621.7847698

Answer 1

(first part)  Test the hypothesis that the average credit risk score of customers who have previously defaulted on their loans is higher than those who haven’t defaulted on their loans. (Divide the credit risk score into two groups those who previously defaulted and those who didn’t and then compare the two groups.)

here we used t-test and following information has been generated using ms-excel and found that there is average credit risk is more for previously defaulted(1) as the one-tailed p-value is less than typical level of significance alpha=0.05

t-Test: Two-Sample Assuming Equal Variances
	previously defaulted(0)	previously defaulted(1)
Mean	206.9791737	561.4770882
Variance	58383.0311	62496.89295
Observations	40	9
Pooled Variance	59083.26291
Hypothesized Mean Difference	0
df	47
t Stat	-3.953071406
P(T<=t) one-tail	0.000129087
t Critical one-tail(0.05)	1.677926722
P(T<=t) two-tail	0.000258174
t Critical two-tail(0.05)	2.01174048

(second part) What is the regression equation?

credit_risk_score=216.46-2.06x1-25.25x2-5.39x3+3.87x4+19.86x5+103.74x6

following regression analysis information has been generated using ms-excel

Regression Statistics
Multiple R	0.891660903
R Square	0.795059166
Adjusted R Square	0.765781904
Standard Error	134.3698582
Observations	49
ANOVA
	df	SS	MS	F	Significance F
Regression	6	2941873.257	490312.2	27.1562	5.73E-13
Residual	42	758320.8694	18055.26
Total	48	3700194.126
	Coefficients	Standard Error	t Stat	P-value	Lower 95%	Upper 95%
Intercept	216.4603585	121.8743419	1.776095	0.082962	-29.492	462.4127
Age in years(x1)	-2.061511871	4.069594181	-0.50656	0.61511	-10.2743	6.151262
Years with current employer(x2)	-25.24501541	3.941245778	-6.40534	1.04E-07	-33.1988	-17.2913
Years at current address(x3)	-5.391063761	4.066615102	-1.32569	0.192109	-13.5978	2.815698
Household income(x4)	3.865940338	0.921752112	4.194121	0.000138	2.005769	5.726111
Debt to income ratio (x5)	19.86416094	3.148379715	6.309328	1.43E-07	13.51047	26.21785
Previously Defaulted(x6)	103.7410698	55.6767152	1.863276	0.069426	-8.61909	216.1012

(third part) What is the predicted credit risk score of a customer aged x1= 40, who has been with their employer for x2=10 years, has lived at the same address for x3=3 years, has an income of x4=$200,000, debt to income ratio of x5=2.5 and has previously defaulted (x6=1) on a loan.

credit_risk_score=216.46-2.06*40-25.25*10-5.39*3+3.87*200+19.86*2.5+103.74*1=792.98

(fourth part) Compute the coefficient of determination (R2) and fully interpret its meaning

coefficient of determination (R2) =0.795

(fifth part)

Interpret the meaning of the coefficients of the independent variables income and address.

Which independent variables significant? Defend your answer.

Income is significant(p-value < 0.05) as its p-value is less than typical level of significance alpha=0.05 and address is not significant(p-value>0.05) p-value is greater than typical level of significance alpha=0.05