Final Project Questions

Data Loading and Create necessary variables (5 points)

# Load libraries 
library(dplyr)

## 
## Attaching package: 'dplyr'

## The following objects are masked from 'package:stats':
## 
##     filter, lag

## The following objects are masked from 'package:base':
## 
##     intersect, setdiff, setequal, union

library(moments)
library(car)

## Loading required package: carData

## 
## Attaching package: 'car'

## The following object is masked from 'package:dplyr':
## 
##     recode

library(caret)

## Loading required package: ggplot2

## Loading required package: lattice

library(pROC)

## Type 'citation("pROC")' for a citation.

## 
## Attaching package: 'pROC'

## The following objects are masked from 'package:stats':
## 
##     cov, smooth, var

library(ggplot2)
# Load Data
hmda <- read.csv("~/Downloads/HMDA2023_MT.csv")
set.seed(38503)
indata <- sample(nrow(hmda),10000)
indata <- hmda %>% 
  mutate(race=case_when(
    derived_race == "Asian" ~ "Asian",
    derived_race == "White" ~ "White",
    derived_race == "Black or African American" ~ "Black",
    TRUE ~ "Others"),
  sex = as.factor(derived_sex),
  age = as.factor(applicant_age),
  denial = ifelse(action_taken == 3,1,0),
  loan_type = as.factor(loan_type),
  purchaser_type = as.factor(purchaser_type),
  loan_amount = log(as.numeric(loan_amount)),
  loan_term = as.numeric(loan_term),
  property_value = log(as.numeric(property_value)),
  occupancy_type = as.factor(occupancy_type),
  area_income = log(ffiec_msa_md_median_family_income),
  area_minority = tract_minority_population_percent,
  income = log(income)
) %>% 
  filter(action_taken==1 | action_taken ==3,
         loan_type == 1,
         loan_purpose== 1,
         property_value > 0,
        loan_amount > 0,
        property_value > loan_amount,
        area_income > 0,
        total_units == "1",
        income > 0) %>% 
        select(denial, race, sex, age, purchaser_type, income, loan_amount,
               loan_term, property_value, occupancy_type, area_minority, area_income)

## Warning: There were 3 warnings in `mutate()`.
## The first warning was:
## ℹ In argument: `loan_term = as.numeric(loan_term)`.
## Caused by warning:
## ! NAs introduced by coercion
## ℹ Run `dplyr::last_dplyr_warnings()` to see the 2 remaining warnings.

# Check any missing cases by column
colSums(is.na(indata))

##         denial           race            sex            age purchaser_type 
##              0              0              0              0              0 
##         income    loan_amount      loan_term property_value occupancy_type 
##              0              0             10              0              0 
##  area_minority    area_income 
##              0              0

indata <- indata[complete.cases(indata),]
table(indata$denial)

## 
##    0    1 
## 6896  701

summary(indata)

##      denial            race                          sex            age      
##  Min.   :0.00000   Length:7597        Female           :1463   25-34  :2015  
##  1st Qu.:0.00000   Class :character   Joint            :3742   35-44  :1982  
##  Median :0.00000   Mode  :character   Male             :1976   45-54  :1372  
##  Mean   :0.09227                      Sex Not Available: 416   55-64  :1094  
##  3rd Qu.:0.00000                                               65-74  : 581  
##  Max.   :1.00000                                               <25    : 367  
##                                                                (Other): 186  
##  purchaser_type     income        loan_amount       loan_term    
##  0      :2347   Min.   : 1.099   Min.   : 8.517   Min.   :  4.0  
##  1      :1983   1st Qu.: 4.369   1st Qu.:12.231   1st Qu.:360.0  
##  3      :1443   Median : 4.804   Median :12.692   Median :360.0  
##  71     : 897   Mean   : 4.885   Mean   :12.575   Mean   :333.5  
##  9      : 488   3rd Qu.: 5.278   3rd Qu.:13.071   3rd Qu.:360.0  
##  6      : 337   Max.   :10.228   Max.   :16.487   Max.   :480.0  
##  (Other): 102                                                    
##  property_value   occupancy_type area_minority    area_income   
##  Min.   : 9.616   1:6498         Min.   : 0.00   Min.   :11.35  
##  1st Qu.:12.692   2: 555         1st Qu.: 9.58   1st Qu.:11.35  
##  Median :13.071   3: 544         Median :11.80   Median :11.35  
##  Mean   :13.075                  Mean   :13.28   Mean   :11.39  
##  3rd Qu.:13.452                  3rd Qu.:14.08   3rd Qu.:11.46  
##  Max.   :17.148                  Max.   :89.73   Max.   :11.52  
## 

Descriptive Analytics (10 points)

1. Provide a summary of the dataset and describe key patterns observed in the data.
   • Discuss how these patterns may relate to business decisions in the mortgage industry.

summary_statistics <-
  summary(indata)
print(summary_statistics)

##      denial            race                          sex            age      
##  Min.   :0.00000   Length:7597        Female           :1463   25-34  :2015  
##  1st Qu.:0.00000   Class :character   Joint            :3742   35-44  :1982  
##  Median :0.00000   Mode  :character   Male             :1976   45-54  :1372  
##  Mean   :0.09227                      Sex Not Available: 416   55-64  :1094  
##  3rd Qu.:0.00000                                               65-74  : 581  
##  Max.   :1.00000                                               <25    : 367  
##                                                                (Other): 186  
##  purchaser_type     income        loan_amount       loan_term    
##  0      :2347   Min.   : 1.099   Min.   : 8.517   Min.   :  4.0  
##  1      :1983   1st Qu.: 4.369   1st Qu.:12.231   1st Qu.:360.0  
##  3      :1443   Median : 4.804   Median :12.692   Median :360.0  
##  71     : 897   Mean   : 4.885   Mean   :12.575   Mean   :333.5  
##  9      : 488   3rd Qu.: 5.278   3rd Qu.:13.071   3rd Qu.:360.0  
##  6      : 337   Max.   :10.228   Max.   :16.487   Max.   :480.0  
##  (Other): 102                                                    
##  property_value   occupancy_type area_minority    area_income   
##  Min.   : 9.616   1:6498         Min.   : 0.00   Min.   :11.35  
##  1st Qu.:12.692   2: 555         1st Qu.: 9.58   1st Qu.:11.35  
##  Median :13.071   3: 544         Median :11.80   Median :11.35  
##  Mean   :13.075                  Mean   :13.28   Mean   :11.39  
##  3rd Qu.:13.452                  3rd Qu.:14.08   3rd Qu.:11.46  
##  Max.   :17.148                  Max.   :89.73   Max.   :11.52  
## 

# Summary of data- grouped by occupancy type, comparing denial and income
summary_occupancy <- indata %>% 
  group_by(occupancy_type) %>% 
  summarize(
    Denial_mean = mean(denial, na.rm = TRUE),
    Denial_median = median(denial, na.rm = TRUE),
    Denial_sd = sd(denial, na.rm = TRUE),
    Denial_skew = skewness(denial, na.rm = TRUE),
    Denial_kurt = kurtosis(denial, na.rm = TRUE),
    Denial_CV = Denial_sd / Denial_mean,
    Income_mean = mean(income, na.rm = TRUE),
    Income_median = median(income, na.rm = TRUE),
    Income_sd = sd(income, na.rm = TRUE),
    Income_skew = skewness(income, na.rm = TRUE),
    Income_kurt = kurtosis(income, na.rm = TRUE),
    Income_CV = Income_sd / Income_mean
  )
print(summary_occupancy)

## # A tibble: 3 × 13
##   occupancy_type Denial_mean Denial_median Denial_sd Denial_skew Denial_kurt
##   <fct>                <dbl>         <dbl>     <dbl>       <dbl>       <dbl>
## 1 1                   0.0877             0     0.283        2.91        9.50
## 2 2                   0.124              0     0.330        2.28        6.19
## 3 3                   0.114              0     0.318        2.43        6.90
## # ℹ 7 more variables: Denial_CV <dbl>, Income_mean <dbl>, Income_median <dbl>,
## #   Income_sd <dbl>, Income_skew <dbl>, Income_kurt <dbl>, Income_CV <dbl>

Key Patterns & Business Implications

1. Type of Occupancy and Loan Denial

• The denial rate is highest for Occupancy Type 2 (12.4%), followed by Type 3 (11.4%) and Type 1 (8.8%).

• Implication: Depending on the type of occupancy, lenders may modify risk assessments or pricing.

2. Demographics and Refusal of Loans

• The majority of applicants are between the ages of 25 and 44, and 49% of them are joint borrowers, suggesting that their households have two incomes.

• Implication: Age-specific requirements and affordability can be met by customized mortgage packages.

3. Trends in Loan and Property Values

• 30-year periods and high loan-to-value ratios point to financial limitations.

• Implication: Lenders might provide flexible repayment plans or tighten loan-to-income ratios.

4. Considerations for Minority and Geographic Areas

• Disparities in loan availability may exist in some minority-heavy areas (up to 89.7%).

• Implication: Inclusivity can be enhanced by bolstering fair lending policies and financial literacy initiatives.

• Financial institutions can optimize loan offerings, improve risk models, and advance financial inclusion while maintaining profitability by taking these aspects into consideration.

2. Create graphical representations (using ggplot2) to explore relationships between loan denial vs. gender, race, and income area (high, mid, and lower-income areas).
   • Define income area by high, mid, and low income areas using area_income.
   • Clearly interpret and discuss the findings from these visualizations.
   • Explain how financial institutions might use these insights in their lending policies and risk assessments.

# Loan Denial by Sex
ggplot(indata, aes(x = sex, fill = as.factor(denial))) +
  geom_bar(position = "fill") +
  scale_y_continuous(labels = scales::percent_format()) +  
  labs(title = "Loan Denial by Gender", x = "Gender", y = "Proportion") +
  theme_minimal()

# Loan Denial by Race
ggplot(indata, aes(x = race, fill = as.factor(denial))) +
  geom_bar(position = "fill") +
  scale_y_continuous(labels = scales::percent_format()) +  
  labs(title = "Loan Denial by Race", x = "Race", y = "Proportion") +
  theme_minimal()

# Loan Denial by Income Area
indata <- indata %>%
  mutate(income_area = factor(area_income, levels = c("Low", "Mid", "High")))

ggplot(indata, aes(x = area_income, fill = as.factor(denial))) +
  geom_bar(position = "fill") +
  scale_y_continuous(labels = scales::percent_format()) +  
  labs(title = "Loan Denial by Income Area",
       x = "Income Area",
       y = "Proportion",
       fill = "Denial Status") +
  scale_fill_manual(values = c("0" = "blue", "1" = "red"),
                    labels = c("Approved", "Denied")) +
  theme_minimal()

### Intreprtation of Findings 1. Loan Denial by Gender

• The loan denial rates for various gender categories are examined in the first visualization.

• The denial percentages for male and female applications are comparable, however the “Sex Not Available” category seems to have a little greater percentage of denials.

• It’s possible that combined incomes increase creditworthiness, which explains why joint applications have the lowest denial rates.

2. Loan Denial by Race

• Differences in loan denial rates between various racial groupings are shown in the second visualization.

• Some racial groupings seem to have a larger percentage of loan denials (blue) than others.

• White candidates appear to have the lowest rejection rate, while Black applicants appear to have the highest. Due to institutional biases, variations in creditworthiness, or other causes, this may indicate racial discrepancies in loan approval procedures.

3. Loan Denial by Income Area

• The third graphic displays the percentage of loans denied in various income brackets.

• The majority of applications were accepted (blue), while a tiny portion were rejected (red).

• The fact that the rejection rate is largely constant across income levels suggests that income level might not be the main determinant in loan denials.

Implications for Financial Institutions

• Fair Lending Practices: Racial differences in loan rejections point to possible prejudices, necessitating a review of policies to guarantee equity and adherence to anti-discrimination legislation.

• Enhancements to Risk Assessment: In order to avoid inadvertent disadvantages for specific groups, lenders should examine credit models. Machine learning methods and alternative credit data may lessen biases.

• Financial Education & Outreach: Programs that teach financial literacy about credit building and loan applications are necessary, as evidenced by higher refusal rates in particular populations.

• Regulatory Compliance & Transparency: While increased loan process transparency fosters accountability and confidence, internal audits can guarantee equitable lending.

3. Conduct hypothesis tests to analyze differences in loan denial rates across groups in 2.
   • State the null and alternative hypotheses.
   • Report the results.
   • Explain the implications of the findings in the context of mortgage lending fairness, regulatory compliance, and business strategy.

# Chi-Square Test (Denial by Race)
chisq_test_race <- chisq.test(table(indata$denial, indata$race))

## Warning in chisq.test(table(indata$denial, indata$race)): Chi-squared
## approximation may be incorrect

print(chisq_test_race)

## 
##  Pearson's Chi-squared test
## 
## data:  table(indata$denial, indata$race)
## X-squared = 18.675, df = 3, p-value = 0.0003192

Null: There is no association between race and loan denial rates. In other words, race does not influence the likehold of a loan being denied.

Alternative: There is an association between race and loan denial rates, meaning race does influence the likelilhood of loan being derived.

Report:

X^2 = 18.675
Degree of Freedom = 3
P-value = 0.0003192
- Since the P value is lower than the conventional threshold of 0.05, we reject the null hypothesis. This means that there is a significant relationship between race and loan denial rates.

Implications

• For Fair Lending and Mortgage Lending may raise concerns for disparities, and assurance of fair lending practices are implemented and needed to avoid discrimination. If the disparities are found, an investigation may be underway in order to see if there are biases with the loan approval decisions.
• In terms of business strategy and risk assessment, understanding the disparities can help with can help with evaluating and ensuring a fair, and compliant model according to regulations. Specifically with financial institutions, they may add a blind lending assessment to have unbiased loan decisions looking at financial criteria rather than demographic factors. Some institutions might use this data to provide lending opportunities to marginalized communities.

4. Perform correlation analysis:
   • Analyze the correlation between loan amounts and other continuous variables.
   • Analyze the correlation between loan denial and other relevant variables.

# Loan Amounts and Continuous Variables
cor(indata[c("loan_amount", "income", "property_value", "loan_term")])

##                loan_amount     income property_value  loan_term
## loan_amount      1.0000000 0.56829929      0.7153256 0.41251086
## income           0.5682993 1.00000000      0.6438250 0.08230416
## property_value   0.7153256 0.64382501      1.0000000 0.13467394
## loan_term        0.4125109 0.08230416      0.1346739 1.00000000

# Loan Denial and Other Relevant Variables
chisq.test(indata$denial, indata$sex)

## 
##  Pearson's Chi-squared test
## 
## data:  indata$denial and indata$sex
## X-squared = 35.367, df = 3, p-value = 1.019e-07

chisq.test(indata$denial, indata$race)

## Warning in chisq.test(indata$denial, indata$race): Chi-squared approximation
## may be incorrect

## 
##  Pearson's Chi-squared test
## 
## data:  indata$denial and indata$race
## X-squared = 18.675, df = 3, p-value = 0.0003192

Correlation between loan amounts and other continuous variables

• The correlation between loan amount and property value is 0.715. This is a strong positive correlation indicating higher the property values, the larger the loans.
• The correlation between loan amount and income is 0.568. This is a moderate correlation suggesting those who have higher-incomes will typically borrow larger loans.
• The correlation between loan amount and loan term is 0.412. This is a weak correlation meaning the loan size is not severely impacted by the loan term.

Correlation between loan denial and other relevant variables

***Chi-square test reveal: - For loan denial and sex there is (p < 0.001) statistically significant association, a need for further investigation for fairness - For loan denial and race there is (p < 0.0003) significant disparity, there is a concern for potential regulatory issues.

Implications

• Lenders need to ensure that fair lending practices are being held to regulations
• A data driven risk assessment can help improve loan models that will help maintain fairness

Regression Analysis on the Log of Loan Amount (15 points)

1. Develop and analyze three different regression models using the log of loan amount as the dependent variable.
   • Clearly define the independent variables used in each model.
   • Justify their inclusion in terms of business decision-making in the mortgage industry.

# Model 1 - Income and Property Value
model1 <- lm(loan_amount ~ income + property_value, data = indata)
summary(model1)

## 
## Call:
## lm(formula = loan_amount ~ income + property_value, data = indata)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8939 -0.0931  0.1888  0.3384  1.1267 
## 
## Coefficients:
##                Estimate Std. Error t value Pr(>|t|)    
## (Intercept)     1.00938    0.15036   6.713 2.05e-11 ***
## income          0.21389    0.01193  17.929  < 2e-16 ***
## property_value  0.80461    0.01384  58.144  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6173 on 7594 degrees of freedom
## Multiple R-squared:  0.5315, Adjusted R-squared:  0.5314 
## F-statistic:  4308 on 2 and 7594 DF,  p-value: < 2.2e-16

# Model 2
model2 <- lm(loan_amount ~ income + property_value + sex + race, data = indata)
summary(model2)

## 
## Call:
## lm(formula = loan_amount ~ income + property_value + sex + race, 
##     data = indata)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.8858 -0.0926  0.1885  0.3377  1.1360 
## 
## Coefficients:
##                       Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           1.035797   0.165541   6.257 4.14e-10 ***
## income                0.214328   0.012086  17.733  < 2e-16 ***
## property_value        0.804314   0.013875  57.968  < 2e-16 ***
## sexJoint              0.004271   0.019672   0.217   0.8281    
## sexMale               0.025808   0.021504   1.200   0.2301    
## sexSex Not Available  0.074381   0.040282   1.847   0.0649 .  
## raceBlack            -0.466741   0.197995  -2.357   0.0184 *  
## raceOthers           -0.064040   0.070844  -0.904   0.3660    
## raceWhite            -0.032619   0.067605  -0.482   0.6295    
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.6171 on 7588 degrees of freedom
## Multiple R-squared:  0.5322, Adjusted R-squared:  0.5317 
## F-statistic:  1079 on 8 and 7588 DF,  p-value: < 2.2e-16

# Model 3
model3 <- lm(loan_amount ~ income + property_value + sex + race + loan_term, data = indata)
summary(model3)

## 
## Call:
## lm(formula = loan_amount ~ income + property_value + sex + race + 
##     loan_term, data = indata)
## 
## Residuals:
##     Min      1Q  Median      3Q     Max 
## -3.6083 -0.1207  0.1236  0.2626  1.8277 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           4.136e-01  1.469e-01   2.815  0.00489 ** 
## income                2.169e-01  1.068e-02  20.306  < 2e-16 ***
## property_value        7.431e-01  1.233e-02  60.250  < 2e-16 ***
## sexJoint              1.472e-02  1.739e-02   0.847  0.39723    
## sexMale               2.839e-02  1.900e-02   1.494  0.13525    
## sexSex Not Available  9.705e-02  3.560e-02   2.726  0.00642 ** 
## raceBlack            -3.822e-01  1.750e-01  -2.184  0.02897 *  
## raceOthers           -9.657e-02  6.261e-02  -1.543  0.12299    
## raceWhite            -2.938e-02  5.974e-02  -0.492  0.62284    
## loan_term             4.214e-03  9.131e-05  46.147  < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.5454 on 7587 degrees of freedom
## Multiple R-squared:  0.6347, Adjusted R-squared:  0.6343 
## F-statistic:  1465 on 9 and 7587 DF,  p-value: < 2.2e-16

Independent Variables and Justification

• Income: Higher income generally increases borrowing capacity, making it a critical determinant of loan approval and loan amount.

• Property Values: Loan amounts are directly influenced by property values, this is due to loans being typically built around what the property values are.

• Sex: Included to assess whether disparities exist in loan amounts based on gender, ensuring compliance with fair lending regulations.

• Race: Identifying potential racial disparities, it is relevant for regulatory fair lending practices.

• Loan Term: Longer loan terms can effect loan amounts by impacting the monthly payment affordability.

Model Insights

• Base Model: Strong predictors (p < 0.001), R^2 is 0.531, this explains that the variables are roughly 53% of the loan amount variation.

• Expanded Model: The R^2 increases slightly to 0.532, sex and most of the categories are not significant, suggesting potential disparities.

• Full Model: R^2 improves to 0.635 with the Loan term being (p < 0.001) being a significant factors. Black or African American borrows receive lower loans (p = 0.029), and “Sex Not Available” becomes significant (p = 0.006).

Business Implications

• Lenders must ensure that income and property value remain the primary factors in the decision making process while also monitoring the disparities in the loan allocation.

• Regulators must look into the differences in lending practices to make sure that no one is being discriminated against because of their race or other factors.

• Financial institutions should consider loan term adjustments to help with balancing affordability and risk.

2. Identify significant variables in each model by evaluating:
   • P-values and confidence intervals.
   • Economic and statistical significance in influencing loan amounts and how these insights could inform pricing strategies and capital allocation.

Model 1: Income and Property Value

• Income: Estimate = 0.213889, p-value < 2e-16 (highly significant)

• Property Value: Estimate = 0.80461, p-value < 2e-16 (higly significant)

Interpretation: Both income and property value are statistically significant, meaning that they have a strong influence over loan amount.

Economic Significance: Income has a positive relationship with loan amount, this suggests that higher income can lead to to higher loans. This is important to lenders when creating loans based on income. Property value also highly significant implying that its also important to loan size. Lenders could use this in pricing the loans and assessing collateral.

Statistical Significance: There’s a strong relationship due to both variables being highly significant.

Model 2: Income, Property Value, Sex, and Race

• Income: Estimate = 0.214328, p-value < 2e-16 (highly significant)

• Property Value: Estimate = 0.804314, p-value < 2e-16 (highly significant)

• Sex (Male, Joint, Not Available): All p-values > 0.05, indicating not significant.

• Race (Black): Estimate = -0.466741, p-value = 0.0184 (significant)

• Race (Others, White): p-values > 0.05, indicating not significant.

Interpretation: Income and Property Value remains significant still, none of sex related variables are significant meaning they don’t really influence loan amount. Only Race(Black) is significant, suggesting that there are racial disparities in loan amounts.

Economic Significance: Sex doesn’t have an impact, however Race(Black) shows a negative coefficient, this can indicate that black borrowers may receive lower loans compared to others. This can suggest that there are potential inequities that may require addressing in business strategies.

Statistical Significance: The significance for Race(Black) means that addressing racial disparities could improve the fairness of loan amounts and allocation

Model 3: Income, Property Value, Sex, Race, and Loan Term

• Income: Estimate = 0.2169, p-value < 2e-16 (highly significant)

• Property Value: Estimate = 0.7431, p-value < 2e-16 (highly significant)

• Sex (Male, Joint, Not Available): Only Sex Not Available is significant (p-value = 0.00642), with a positive coefficient of 0.09705. This indicates a slightly higher loan amount for “Sex Not Available” compared to the baseline category.

• Race (Black): Estimate = -0.3822, p-value = 0.02897 (significant)

• Loan Term: Estimate = 0.004214, p-value < 2e-16 (highly significant)

Interpretations: Income and Property Value are still significant, the significance of “Sex Not Available” can mean that there’s a bit of an increase in loan amounts. There is still a negative coefficient for Race(Black) still indicating racial disparities. For Loan Term, it is a significant variable with a very low positive coefficient, the loan amount increases very slightly for each additional loan term.

Economic Significance: Loan Term as a signifcant positive effect on loan amounts, longer loan terms increases loan size. Race(Black) still indicates lower amounts of loan for black borrowers, meaning that there is still an unfairness and potential areas for intervention.

Statistical Significance: Income, Property Value, and Loan Term all are consistently significant, supporting the importance in pricing allocation strategies for loans.

3. Check for multicollinearity among independent variables using the Variance Inflation Factor (VIF).

# Variance Inflation Factor (VIF)
vif(model3)

##                    GVIF Df GVIF^(1/(2*Df))
## income         1.754149  1        1.324443
## property_value 1.738228  1        1.318419
## sex            1.545145  3        1.075214
## race           1.456778  3        1.064712
## loan_term      1.020338  1        1.010118

Intrepretation:

• GVIF (Generalized Variance Inflation Factor) values assess multicollinearity. Higher GVIF indicates stronger correlation with other variables.

• GVIF^(1/(2*Df)) provides a normalized measure of multicollinearity. Values above 1 suggest potential multicollinearity issues.

• Income & Property Value: GVIF close to 1, indicating minimal multicollinearity

• Sex & Race: GVIF is slightly higher GVIF, there is some degree of correlation but not a huge amount

• Loan Term: lowest GVIF, minimal correlation

Impact on Business Decision-Making

• The understanding of the importance of individual variables may be distorted by high multicollinearity, which could make loan terms and price-setting more difficult.

• Collinearity between property value and income, for example, may result in inaccurate conclusions about how much of an impact each factor has on loan amounts.

Recommendations:

• Remove or combine strongly correlated variables (such income and property value) to simplify the model.

• Refine correlated variables into a smaller, easier-to-manage set of uncorrelated components.

4. Perform F-tests and restricted F-tests to compare model performance.

anova(model1, model2, model3)

## Analysis of Variance Table
## 
## Model 1: loan_amount ~ income + property_value
## Model 2: loan_amount ~ income + property_value + sex + race
## Model 3: loan_amount ~ income + property_value + sex + race + loan_term
##   Res.Df    RSS Df Sum of Sq         F  Pr(>F)    
## 1   7594 2893.9                                   
## 2   7588 2889.8  6      4.04    2.2629 0.03484 *  
## 3   7587 2256.5  1    633.37 2129.5917 < 2e-16 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

a) Hypothesis, Test Statisitcs, and Correlation

1. Hypothesis

• Null Hypothesis (H₀): The model is not substantially enhanced by the addition of new variables (sex, race, and loan_term).

• Alternative Hypothesis (H₁): The model is much enhanced by the addition of new variables (sex, race, and loan_term).

2. Test Statistics:

• The base model, Model 1 (income + property_value), is not directly tested in this instance.

• With a p-value of 0.03484, Model 2 (income + property_value + sex + race) produces an F-statistic of 2.2629. This suggests that the model is considerably improved (at the 5% level) by including sex and race.

• With a p-value of less than 2e-16 and an exceptionally significant F-statistic of 2129.5917, Model 3 (income + property_value + sex + race + loan_term) demonstrates how significantly the model is improved by including loan_term.

3. Conclusion:

• Model 2: The model is greatly enhanced by the inclusion of sex and ethnicity, indicating that demographic variables are crucial in predicting loan amounts.

• Model 3: By adding loan_term, the model is further enhanced and yields a highly significant result, demonstrating the importance of the loan duration in explaining loan amounts.

b) Risk Management & Competitive Advantage

1. Risk Management

• Financial organizations can more precisely evaluate the risk associated with loan applications by identifying important criteria such as income, property_value, sex, race, and loan_term. For example, knowing how loan terms affect loan amounts enables institutions to more accurately evaluate long-term repayment risks.

• While demographic information can be used to predict default risks, race and sex factors can be used to inform credit scoring models in order to reduce the risk of discrimination and guarantee fair lending practices.

2. Competitive Advantage

• Based on the borrower’s demographics (e.g., loan terms catered to various income groups) and property value, institutions might use these results to create customized loan products.

• Banks can maintain an advantage over rivals by implementing more precise pricing strategies and focusing on underserved market groups by delivering competitive lending terms and products based on these findings. Both market share and client loyalty may rise as a result.

5. Identify the best-performing model based on adjusted R-squared values.

a) Model 3 is better than the others since it explains the most variation in loan amounts, as evidenced by its much lower RSS (2256.5). A strong model fit is indicated by Model 3’s big F-statistic and p-value < 2e-16, which imply that adding “loan_term” enhances predictive capacity and further lowers error in comparison to Models 1 and 2.

b) Potential Applicants:

• Business Forecasting: Model 3 is perfect for predicting loan amounts because of its extensive variables, which include loan length, which enable more precise financial estimates.

• Profitability Analysis: By identifying important variables affecting loan amounts, the model helps banks optimize loan pricing and boost profitability.

• Regulatory Compliance: Model 3 guarantees equity in lending practices by taking into account factors including sex, color, and loan length. This is essential for fulfilling legal obligations and preventing discriminatory practices.

6. Conduct cross-validation by splitting the dataset into 60% training and 40% testing subsets.

# Evaluate Adjusted R-squared for models
cat("Adjusted R-squared for Model 1: ", summary(model1)$adj.r.squared, "\n")

## Adjusted R-squared for Model 1:  0.5313987

cat("Adjusted R-squared for Model 2: ", summary(model2)$adj.r.squared, "\n")

## Adjusted R-squared for Model 2:  0.5316826

cat("Adjusted R-squared for Model 3: ", summary(model3)$adj.r.squared, "\n")

## Adjusted R-squared for Model 3:  0.6342758

# Cross-validation for all models
set.seed(38503)
trainIndex <- createDataPartition(indata$loan_amount, p = 0.6, list = FALSE)
trainData <- indata[trainIndex, ]
testData <- indata[-trainIndex, ]

# Model 1: Income and Property Value
predictions1 <- predict(lm(loan_amount ~ income + property_value, data = trainData), testData)
rmse1 <- sqrt(mean((testData$loan_amount - predictions1)^2))
mse1 <- mean((testData$loan_amount - predictions1)^2)
mpe1 <- mean(abs(testData$loan_amount - predictions1) / testData$loan_amount) * 100

# Model 2: Income, Property Value, Sex, Race
predictions2 <- predict(lm(loan_amount ~ income + property_value + sex + race, data = trainData), testData)
rmse2 <- sqrt(mean((testData$loan_amount - predictions2)^2))
mse2 <- mean((testData$loan_amount - predictions2)^2)
mpe2 <- mean(abs(testData$loan_amount - predictions2) / testData$loan_amount) * 100

# Model 3: Income, Property Value, Sex, Race, Loan Term
predictions3 <- predict(lm(loan_amount ~ income + property_value + sex + race + loan_term, data = trainData), testData)
rmse3 <- sqrt(mean((testData$loan_amount - predictions3)^2))
mse3 <- mean((testData$loan_amount - predictions3)^2)
mpe3 <- mean(abs(testData$loan_amount - predictions3) / testData$loan_amount) * 100

# Display Performance Metrics
cat("Performance Metrics for Model 1: RMSE =", rmse1, "MSE =", mse1, "MPE =", mpe1, "\n")

## Performance Metrics for Model 1: RMSE = 0.6351613 MSE = 0.4034299 MPE = 3.391547

cat("Performance Metrics for Model 2: RMSE =", rmse2, "MSE =", mse2, "MPE =", mpe2, "\n")

## Performance Metrics for Model 2: RMSE = 0.6348638 MSE = 0.4030521 MPE = 3.3915

cat("Performance Metrics for Model 3: RMSE =", rmse3, "MSE =", mse3, "MPE =", mpe3, "\n")

## Performance Metrics for Model 3: RMSE = 0.565539 MSE = 0.3198344 MPE = 2.982872

Questtion 6/7:

1. Evaluation:

1. Root Mean Squared Error

• Model 1: RMSE = 0.6352
• Model 2: RMSE = 0.6349
• Model 3: RMSE = 0.5655 Interpretation: Better prediction accuracy is indicated by a lower RMSE. With the lowest RMSE, Model 3 appears to be the best accurate model for loan amount prediction.

2. Mean Squared Error

• Model 1: MSE = 0.4034
• Model 2: MSE = 0.4031
• Model 3: MSE = 0.3198 Interpretation: Lower MSE is favored for improved model performance, much like RMSE. Once more demonstrating its greater predictive power, Model 3 has the lowest MSE.

3. Mean Percentage Error

• Model 1: MPE = 3.3915
• Model 2: MPE = 3.3915
• Model 3: MPE = 2.9829 Interpretation: The average percentage error of forecasts is assessed using MPE. With the lowest MPE, Model 3 outperforms the others once more, suggesting that its projections are more in line with actual loan amounts.

2. Comparison of Predictive Accuracy & Relevance

1. Predictive Accuracy: Model 3 is the best at forecasting loan amounts since it is the most accurate.
2. Financial Risk Assessment: Model 3’s consistent amount predictions aid in precisely evaluating loan risk.
3. Investment Strategies: Model 3’s accurate loan amount forecasting facilitates improved capital allocation.
4. Decision-Making in Operations: By helping businesses target the appropriate consumer categories, Model 3 increases operational efficiency.

Conclusion: Even though Model 2 strikes a good balance between fit and accuracy, Model 3 offers the best performance and is most helpful for financial decisions.

7. Evaluate model performance using Root Mean Squared Error (RMSE), Mean Squared Error (MSE), and Mean Percentage Error (MPE)

1. Evaluation:

1. Root Mean Squared Error

• Model 1: RMSE = 0.6352
• Model 2: RMSE = 0.6349
• Model 3: RMSE = 0.5655 Interpretation: Better prediction accuracy is indicated by a lower RMSE. With the lowest RMSE, Model 3 appears to be the best accurate model for loan amount prediction.

2. Mean Squared Error

• Model 1: MSE = 0.4034
• Model 2: MSE = 0.4031
• Model 3: MSE = 0.3198 Interpretation: Lower MSE is favored for improved model performance, much like RMSE. Once more demonstrating its greater predictive power, Model 3 has the lowest MSE.

3. Mean Percentage Error

• Model 1: MPE = 3.3915
• Model 2: MPE = 3.3915
• Model 3: MPE = 2.9829 Interpretation: The average percentage error of forecasts is assessed using MPE. With the lowest MPE, Model 3 outperforms the others once more, suggesting that its projections are more in line with actual loan amounts.

Binary Dependent Regression Analysis on Denial (10 points)

1. Develop three models to predict loan denial:
   • One Linear Probability Model (LPM).
   • Two best Logistic Regression Models with different sets of independent variables.

# Linear Probability Model
lpm <- lm(denial ~ income + property_value + sex + race, data = indata)
summary(lpm)

## 
## Call:
## lm(formula = denial ~ income + property_value + sex + race, data = indata)
## 
## Residuals:
##      Min       1Q   Median       3Q      Max 
## -0.22843 -0.11371 -0.08687 -0.05828  1.05350 
## 
## Coefficients:
##                        Estimate Std. Error t value Pr(>|t|)    
## (Intercept)           0.3680083  0.0770057   4.779 1.80e-06 ***
## income               -0.0408402  0.0056222  -7.264 4.13e-13 ***
## property_value        0.0006927  0.0064544   0.107  0.91453    
## sexJoint             -0.0207630  0.0091508  -2.269  0.02330 *  
## sexMale               0.0033860  0.0100033   0.338  0.73501    
## sexSex Not Available  0.0082914  0.0187382   0.442  0.65815    
## raceBlack            -0.1680222  0.0921029  -1.824  0.06815 .  
## raceOthers           -0.0521624  0.0329550  -1.583  0.11350    
## raceWhite            -0.0815270  0.0314483  -2.592  0.00955 ** 
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.2871 on 7588 degrees of freedom
## Multiple R-squared:  0.01727,    Adjusted R-squared:  0.01623 
## F-statistic: 16.66 on 8 and 7588 DF,  p-value: < 2.2e-16

# Logistic Regression Models
logit1 <- glm(denial ~ income + property_value + sex, data = indata, family = "binomial")
summary(logit1)

## 
## Call:
## glm(formula = denial ~ income + property_value + sex, family = "binomial", 
##     data = indata)
## 
## Coefficients:
##                       Estimate Std. Error z value Pr(>|z|)    
## (Intercept)           0.267237   0.861120   0.310   0.7563    
## income               -0.546798   0.073518  -7.438 1.03e-13 ***
## property_value        0.008956   0.078260   0.114   0.9089    
## sexJoint             -0.232650   0.108311  -2.148   0.0317 *  
## sexMale               0.040109   0.111786   0.359   0.7197    
## sexSex Not Available  0.370656   0.171970   2.155   0.0311 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 4676.2  on 7596  degrees of freedom
## Residual deviance: 4552.5  on 7591  degrees of freedom
## AIC: 4564.5
## 
## Number of Fisher Scoring iterations: 5

logit2 <- glm(denial ~ income + property_value + sex + race, data = indata, family = "binomial")
summary(logit2)

## 
## Call:
## glm(formula = denial ~ income + property_value + sex + race, 
##     family = "binomial", data = indata)
## 
## Coefficients:
##                        Estimate Std. Error z value Pr(>|z|)    
## (Intercept)            1.006444   0.912280   1.103   0.2699    
## income                -0.553767   0.073652  -7.519 5.53e-14 ***
## property_value         0.006839   0.078474   0.087   0.9305    
## sexJoint              -0.217015   0.108905  -1.993   0.0463 *  
## sexMale                0.055635   0.112121   0.496   0.6198    
## sexSex Not Available   0.102591   0.203010   0.505   0.6133    
## raceBlack            -12.968577 258.373450  -0.050   0.9600    
## raceOthers            -0.398929   0.312594  -1.276   0.2019    
## raceWhite             -0.738173   0.292781  -2.521   0.0117 *  
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## (Dispersion parameter for binomial family taken to be 1)
## 
##     Null deviance: 4676.2  on 7596  degrees of freedom
## Residual deviance: 4538.5  on 7588  degrees of freedom
## AIC: 4556.5
## 
## Number of Fisher Scoring iterations: 13

# AUC
roc_curve <- roc(testData$denial, predict(logit2, testData, type = "response"))

## Setting levels: control = 0, case = 1

## Setting direction: controls < cases

auc(roc_curve)

## Area under the curve: 0.629

2. Identify significant variables in each model by analyzing p-values, odds ratios (for logistic regression), and their economic and statistical implications. Discuss how lenders could use these variables to refine credit risk models, regulatory compliance, and decision-making processes.

Signficiant Variables

1. Linear Regression Model:

• Income has a negative impact on loan denial and is very significant (p = 4.13e-13).

• Property Value: p = 0.91453, not significant.

• Sex(Joint): Significant (p = 0.02330), has a negative impact on loan rejection.

• Race(White) applicants have a reduced denial rate, which is significant (p = 0.00955).

• Economic Implication: When evaluating loan risk, lenders should take sex and income into account. Although further research is required to determine its impact, race may possibly play a part.

2. Logistic Regression (Model 1)

• Income: Reduces the likelihood of denial; highly significant (p = 1.03e-13).

• Property Value: p = 0.9089, not significant.

• Denial likelihood is impacted by sex (joint, not available), which is significant (p = 0.0317, p = 0.0311).

• Economic Implication: Income can be used by lenders to determine the likelihood that a loan will be approved. The sex factor emphasizes how crucial it is to take into account many applications and a range of demographic information.

3. Logistic Regression (Model 2)

• Income: Reduces the likelihood of rejection; highly significant (p = 5.53e-14).

• Property Value: p = 0.9305, not significant.

• Sex(Joint): Significant (p = 0.0463), reduces the likelihood of denial.

• Race(White) applicants had a decreased denial rate, which was significant (p = 0.0117).

• Economic Significance: The findings support the necessity of taking race, sex, and income into account when determining loan eligibility. It might be necessary for lenders to create more inclusive policies in order to prevent racial and sexual prejudice.

Applications in Credit Risk Models, Regulatory Compliance, and Decision-Making

1. Credit Risk Models

• Race, sex (joint status), and income all increase the accuracy of loan denial predictions.

2. Regulatory Compliance

• Make sure models are equitable, refrain from racial and sexual discrimination, and abide by fair lending regulations.

3. Decision-Making

• When making judgments, prioritize income and joint sex status; take race into account to increase equity and inclusivity.

3. Generate confusion matrices using different probability thresholds (50%, 30%, and 10%).

threshold_50 <- ifelse(predict(logit2, type = "response") > 0.5, 1, 0)
confusionMatrix(as.factor(threshold_50), as.factor(indata$denial))

## Warning in confusionMatrix.default(as.factor(threshold_50),
## as.factor(indata$denial)): Levels are not in the same order for reference and
## data. Refactoring data to match.

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction    0    1
##          0 6896  701
##          1    0    0
##                                          
##                Accuracy : 0.9077         
##                  95% CI : (0.901, 0.9141)
##     No Information Rate : 0.9077         
##     P-Value [Acc > NIR] : 0.5101         
##                                          
##                   Kappa : 0              
##                                          
##  Mcnemar's Test P-Value : <2e-16         
##                                          
##             Sensitivity : 1.0000         
##             Specificity : 0.0000         
##          Pos Pred Value : 0.9077         
##          Neg Pred Value :    NaN         
##              Prevalence : 0.9077         
##          Detection Rate : 0.9077         
##    Detection Prevalence : 1.0000         
##       Balanced Accuracy : 0.5000         
##                                          
##        'Positive' Class : 0              
## 

threshold_30 <- ifelse(predict(logit2, type = "response") > 0.3, 1, 0)
confusionMatrix(as.factor(threshold_30), as.factor(indata$denial))

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction    0    1
##          0 6894  692
##          1    2    9
##                                          
##                Accuracy : 0.9086         
##                  95% CI : (0.9019, 0.915)
##     No Information Rate : 0.9077         
##     P-Value [Acc > NIR] : 0.4003         
##                                          
##                   Kappa : 0.0225         
##                                          
##  Mcnemar's Test P-Value : <2e-16         
##                                          
##             Sensitivity : 0.99971        
##             Specificity : 0.01284        
##          Pos Pred Value : 0.90878        
##          Neg Pred Value : 0.81818        
##              Prevalence : 0.90773        
##          Detection Rate : 0.90746        
##    Detection Prevalence : 0.99855        
##       Balanced Accuracy : 0.50627        
##                                          
##        'Positive' Class : 0              
## 

threshold_10 <- ifelse(predict(logit2, type = "response") > 0.1, 1, 0)
confusionMatrix(as.factor(threshold_10), as.factor(indata$denial))

## Confusion Matrix and Statistics
## 
##           Reference
## Prediction    0    1
##          0 4516  323
##          1 2380  378
##                                          
##                Accuracy : 0.6442         
##                  95% CI : (0.6333, 0.655)
##     No Information Rate : 0.9077         
##     P-Value [Acc > NIR] : 1              
##                                          
##                   Kappa : 0.0837         
##                                          
##  Mcnemar's Test P-Value : <2e-16         
##                                          
##             Sensitivity : 0.6549         
##             Specificity : 0.5392         
##          Pos Pred Value : 0.9333         
##          Neg Pred Value : 0.1371         
##              Prevalence : 0.9077         
##          Detection Rate : 0.5944         
##    Detection Prevalence : 0.6370         
##       Balanced Accuracy : 0.5971         
##                                          
##        'Positive' Class : 0              
## 

Confusion Matrix 1:

• Accuracy: 0.9077, High accuracy but skewed because of imbalance

• 1.0000 is the True Positive Rate (Sensitivity), which accurately indicates that the positive class is 0.

• 0.0000 is the False Positive Rate (1-Specificity) (no false positives).

Implications: - Risk Mitigation: This methodology may overlook positive cases (1), but it nearly always finds negatives (0).

• Customer satisfaction may suffer if customers who ought to be authorized are passed over.

• Business Profitability: This model may miss out on prospective business by failing to detect certain problematic clients, even while it is accurate in rejecting loans.

Confusion Matrix 2:

• Accuracy: 0.9086, which is marginally superior to Model 1.

• True Positive Rate (Sensitivity): 0.9997 (almost flawless positive detection)

• 0.01284 is the false positive rate (some false positives).

Implications: - Risk mitigation strikes a balance between minimizing false positives and discovering positives. could be helpful in locating and accepting applications who pose a risk.

• Customer satisfaction: There is a slight danger of mistakenly turning away loyal consumers, but there is a higher likelihood of accurate approval.

• Profitability for businesses: Assists in reducing risk while preserving client happiness.

Confusion Matrix 3:

• Accuracy: 0.6442 (poor accuracy)

• Sensitivity (True Positive Rate): 0.6549 (Sufficient positive detection)

• The rate of false positives is 0.5392, which is high.

Implications: - Risk Mitigation: This model is likely to approve high-risk applicants due to its high false positive rate, which increases financial exposure.

• Customer satisfaction: There may be a larger likelihood of loan defaults, but more customers may be approved.

• Business Profitability: As more clients are approved, there is a greater chance of defaults, which lowers profitability.

4. Compare model performance and justify the most effective model. Explain how these results could inform credit approval policies and financial service strategies.

Model 1:

Accuracy: 0.9077 Sensitivity: 1.0000 (perfect detection of negatives) Specificity: 0.0000 (no false positives) Implications: By identifying every loan denial, this methodology reduces risk, but it overlooks a large number of loan approvals, which could result in missed business possibilities.

Model 2:

Accuracy: 0.9086 Sensitivity: 0.9997 (near-perfect detection of positives) Specificity: 0.01284 (minimal false positives) Implications: This approach is perfect for granting legitimate loans while efficiently managing risk because it strikes a balance between identifying loan approvals and reducing false positives.

Model 3:

Accuracy: 0.6442 Sensitivity: 0.6549 (adequate detection of positives) Specificity: 0.5392 (higher false positives) Implications: Profitability may be harmed by this model’s decreased accuracy and increased chance of accepting unqualified applications.

Justification for the Most Effective Model

• Due to its balanced performance, which reliably detects both loan approvals and denials with few false positives, Model 2 is the most effective. It is perfect for reducing risk and seizing legitimate business chances.

Implications for Credit Approval Policies and Financial Services Strategies

1. Credit Approval Policies: Model 2 enables more informed decision-making, reducing defaults and sanctioning loans with assurance.

2. Financial Services Strategies: Targeting creditworthy clients, dynamically modifying loan terms, and increasing profitability while lowering risk are all made possible by this methodology.

3. Perform cross-validation using the Area Under the Curve (AUC) metric.

# Create a logistic regression model
log_model <- glm(denial ~ age + sex + income + loan_amount + property_value, 
                 data = indata, family = binomial())

# Create a decision tree model
library(rpart)
tree_model <- rpart(denial ~ age + sex + income + loan_amount + property_value, 
                    data = indata, method = "class")

# Predictions for Logistic Model
log_pred <- predict(log_model, type = "response")
log_roc <- roc(indata$denial, log_pred)

## Setting levels: control = 0, case = 1

## Setting direction: controls < cases

# Predictions for Decision Tree Model
tree_pred <- predict(tree_model, type = "prob")[,2]
tree_roc <- roc(indata$denial, tree_pred)

## Setting levels: control = 0, case = 1
## Setting direction: controls < cases

# Compare AUC for both models
log_auc <- auc(log_roc)
tree_auc <- auc(tree_roc)

# Print AUC values
cat("Logistic Regression AUC: ", log_auc, "\n")

## Logistic Regression AUC:  0.622226

cat("Decision Tree AUC: ", tree_auc, "\n")

## Decision Tree AUC:  0.5638311

# ROC curve plot comparison
plot.roc(log_roc, col = "blue", main = "Model Comparison: ROC Curves")
plot.roc(tree_roc, col = "red", add = TRUE)
legend("bottomright", legend = c("Logistic Regression", "Decision Tree"), 
       col = c("blue", "red"), lty = 1)

### Comparison of Model Performance - The Logistic Regression model (AUC = 0.622) performs better than the Decision Tree model (AUC = 0.564) in forecasting loan denial, according to the ROC curve and AUC values.

• The blue curve, or logistic regression, demonstrates a superior capacity to distinguish between loans that are granted and those that are denied. Its moderate predictive power is indicated by its AUC of 0.622.

• With a lower AUC (0.564), the Decision Tree (red curve) performs worse and is less reliable at differentiating loan denial cases.

• The model performs better the closer the AUC is to 1. In this instance, Logistic Regression is the superior model because to its higher AUC.

Business Implications for Financial Institutions

• These findings can be used by financial organizations to improve loan approval procedures and guarantee sustainability over the long run:

• Risk assessment and credit scoring: By providing a more accurate way to distinguish between loans that are authorized and those that are denied, logistic regression helps reduce defaults and guarantee eligible approvals.

• Operational Efficiency: Higher accuracy speeds up loan approvals and lowers expenses by reducing manual reviews.

• Fair & Responsible Lending: Data-driven methods encourage objective judgments, guaranteeing adherence to rules and lowering legal risks.

• Loan Portfolio Management: A strong predictive model adjusts to borrower profiles and economic conditions while striking a balance between approvals and financial stability.

Summary and Findings (5 points)

Summarize the findings from each section of the project:

1. Key insights from the descriptive analysis.
2. Findings from the regression analysis on loan amounts.
3. Findings from the binary dependent variable analysis.
4. Provide an overall assessment of the results, discussing limitations and real-world business applications.

Summary of Findings

1. Key Insights from Descriptive Study

• Important relationships in mortgage lending were found in the dataset. Loan amounts have a moderate correlation (0.568) with income and a substantial correlation (0.715) with property values. The association between loan length and loan amount is lower (0.412).

• Disparities are suggested by the examination of loan denial rates. According to chi-square tests, there is a statistically significant correlation between loan denial and race (p < 0.0003) and sex (p < 0.001). Concerns regarding equity and possible regulatory scrutiny are raised by these findings.

2. Findings from the Regression Analysis on Loan Amounts

• Model 3, which had the lowest Root Mean Squared Error (RMSE = 0.5655) and the greatest Adjusted R2 (0.634), outperformed the other two regression models in terms of predictive accuracy.

• Across all models, income and property worth were consistently significant predictors of loan amounts.

• Concerns over equity in lending were raised by the negative coefficient for race (Black), which suggested less loan amounts for Black customers.

• Longer loan durations are linked to somewhat larger loan amounts, according to the minor but substantial positive effect of loan length.

3. Findings from the Binary Dependent Variable Analysis

• To forecast loan denial, two logistic regression models and a linear probability model (LPM) were created.

• In both models, income significantly decreased the chance of loan denial (p < 0.0001).

• Perhaps as a result of the financial security that comes with having several sources of income, sex (combined applications) dramatically decreased loan denial rates.

• Racial differences in acceptance choices were evident in the much lower loan denial rate for White applicants.

• The model evaluation’s AUC values revealed that the logistic regression model (AUC = 0.622) outperformed the decision tree model (AUC = 0.564), suggesting a reasonable level of predictive potential.

4. Overall Evaluation and Implications

• Limitations: Although the models were able to identify important factors that affect loan amounts and denial rates, they could also improve predictive power by taking into account unobserved borrower characteristics like credit scores and debt-to-income ratios.

• Business and Regulatory Consequences: Fair lending issues: In order to prevent discrimination claims, there may be a need for compliance reviews and possible policy changes given the strong correlation between race and loan denial.

• Enhancements to risk assessment: While enhancing risk management, financial institutions should improve credit scoring models to integrate fair lending principles.

• Pricing strategies: By using the findings from the regression model, lenders can better target lucrative client segments and optimize loan pricing.

• Consumer protection: These results back up regulatory initiatives to keep an eye on mortgage lending practices and encourage openness.

Overall, the study shows that sex, ethnicity, and wealth are important considerations when it comes to mortgage lending decisions. To maintain good risk management and guarantee lending equity, financial institutions and regulators must collaborate.

Reminder: Each student must choose a different state for their HMDA dataset analysis. Duplications will not be accepted.