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library(distill)
library(dplyr)
library(readr)
library(tidyverse)
library(knitr)
library(ggplot2)
library("readxl")

BACKGROUND

One of the federal loan programs available to help finance college education is the parent PLUS program. This college financing program has increased popularity through the years. It is a loan program for parents of undergraduates to borrow money from the federal government to bridge the gap between the cost of attendance and available financial resources such as student loans, grants, and scholarships. According to the data from over 800 school institutions, there are currently over 273,000 parent borrowers comprising the $ 84.3B outstanding loan balance in this program. There are minimal requirements to qualify for parent PLUS loans, namely being a parent of a dependent student and not have adverse credit history (with exemptions allowed). There is no debt-to-income ratio requirement to secure any loan amount. These minimal requirements have made the loans and college education more accessible to parents, notwithstanding their ability to repay the debt. Parent debt is an inter-generational social science problem. Both parents and children are saddled with loans that threaten their current and future financial security.

RESEARCH AND METHODS

This multifaceted study uses quantitative methods to examine the characteristics of parent PLUS loans including demographic profiles, loan amounts, type of borrowers, and analyze trends on repayment and default rates from over 800 post-secondary schools in the US.

        METHODS:
        -Data is sourced from March 2022  US Department of Education
        College Scorecard 
        -Missing Not At Random (MNAR) data (e.g. NULL, Privacy
        Suppressed) was handled using listwise deletion. Imputation
        of MNAR is not preferred as it will give misleading results
        -Exploratory data analysis, data visualization and
        descriptive statistics.
        -Correlation Matrix to evaluate strongly correlated variables
        -Simple and multiple regression analyses 
        -Post hoc test using EMM (Estimated Marginal Means)
        -Diagnostic Plots evaluation of regression model

Data

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loans<- read_excel("LOANS.xlsx",)
Eloans <-na.omit(loans)

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Eloans<-rename(Eloans, State=STABBR)

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Eloans<-rename(Eloans,SAT=SAT_AVG)

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Eloans<-  rename(Eloans,COST=COSTT4_A)

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Eloans <-rename(Eloans,Students=LPPPLUS_CNT)

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Eloans<-  rename(Eloans,Repay5yr=DBRR5_PP_UG_RT)

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Eloans<-  rename(Eloans,Repay10yr=DBRR10_PP_UG_RT)

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Eloans<-  rename(Eloans,Repay20yr=DBRR20_PP_UG_RT)

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Eloans<-  rename(Eloans,ParentLoan=PLUS_DEBT_INST_MD)

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Eloans<-  rename(Eloans,StudentLoan=DEBT_MDN )

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Eloans<-rename(Eloans, Default3yr=BBRR3_PP_UG_DFLT)

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Eloans<-  rename(Eloans,NoDegree=OMENRUP_ALL_POOLED_SUPP)

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Eloans<-rename(Eloans, PellBorrowerLoan=PLUS_DEBT_INST_PELL_MD)

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Eloans<-  rename(Eloans,NoPellBorrowerLoan=PLUS_DEBT_INST_NOPELL_MD)

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Eloans<-rename(Eloans,Parents=BBRR1_PP_UG_N)

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dim(Eloans)

[1] 865  35

Sum of Total outstanding Parent PLUS Loan balance

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TPL<-sum(Eloans$LPPPLUS_AMT)
print(paste("Total outstanding Parent PLUS Loan balance is",TPL))

[1] "Total outstanding Parent PLUS Loan balance is 84303788470"

Sum of Number of students associated with outstanding Parent PLUS Loan balances

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TSL<-sum(Eloans$Students)
print(paste("Total Number of students associated with outstanding Parent PLUS Loan balances is", TSL))

[1] "Total Number of students associated with outstanding Parent PLUS Loan balances is 2730197"

LOAN DISTRIBUTION ACROSS THE US

Fig 1. Who is borrowing how much?

Method: Descriptive Statistics

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ggplot(Eloans, aes(x = State , y= Parents, fill = COST)) +
  geom_bar(position="dodge",stat = "identity") +
  labs(x = "State",
       y = "Parents with Loans",
       title = "Number of Parents who took Loans vs Cost to attend
       (One Academic Year Data)") +
  theme(axis.text.x = element_text(angle = 90, vjust = 0.5, hjust=1,size=8),
        axis.text.y = element_text(size=12),
        axis.title.y = element_text(size=12),
        axis.title.x = element_text(size=12),
        plot.title = element_text(size=12,hjust=0.5))

Observation

    -One Year Data Distribution
    -Pennsylvania had highest number of parent PLUS borrowers followed by 
    Arizona,Texas and Michigan, ave cost of attendance <$40,000
    -New York has highest ave cost of attendance > $40,000
    -California has a mix of high and mid-level ave cost of attendance.
    -Mean parent PLUS loans =$15,551
    -Ave cost of attendance =$27,548

LOAN AMOUNTS AND SCHOOL SELECTIVITY

Fig 2. Where are the loans disbursed?

Method: Regression Analysis

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library(hexbin)
ggplot(Eloans, aes(SAT, ParentLoan)) +
  geom_hex(bins = 20, color = "white")+
  scale_fill_gradient(low =  "#00AFBB", high = "#FC4E07")+
  theme_minimal() +
  geom_smooth(method = "lm") +
ggtitle( "Loans Distribution Across Selective Universities (p-value<0.05)" )+
  theme(axis.text=element_text(size=14),
        axis.title=element_text(size=14,face="bold"))

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SAT_lm<-lm(ParentLoan~SAT,Eloans)
summary(SAT_lm)


Call:
lm(formula = ParentLoan ~ SAT, data = Eloans)

Residuals:
   Min     1Q Median     3Q    Max 
-21865  -6885  -1560   4992  66762 

Coefficients:
              Estimate Std. Error t value Pr(>|t|)    
(Intercept) -42600.486   3577.839  -11.91   <2e-16 ***
SAT             58.120      3.101   18.74   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 10420 on 863 degrees of freedom
Multiple R-squared:  0.2893,    Adjusted R-squared:  0.2885 
F-statistic: 351.3 on 1 and 863 DF,  p-value: < 2.2e-16

Observations

      -Figure 2 shows positive correlation between parent PLUS loan amounts and average SAT scores at the institutions. 
      -Higher loan amounts are disbursed at more selective schools with higher SAT score averages 
      -Universities with mid-range SAT scores disbursed more loans < $25,000

BORROWER BEHAVIOR

Fig 3. Does Income Influence Borrower Behavior?

Method: Regression Analysis

The maximum 4-year undergraduate student loan is $27,000 at an interest rate of 3.73% while parent loans have no limit and a higher interest rate of 6.28%. This study used regression analysis to compare borrowing behaviors between 2 groups namely Pell and Non-Pell Grant recipients. Pell grants are need based awards that does not have to be repaid and are available only to undergraduates who display exceptional financial need. Most Pell grants are awarded to 90.6% of parent of households with income < $20,000 and 86.7% of parents with income 20K-40K. The mean debt of parent of Pell grant recipients is $18,412 vs non-Pell grant recipients with $30,821.

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PPL2<-lm(ParentLoan~PellBorrowerLoan, Eloans)
summary(PPL2)


Call:
lm(formula = ParentLoan ~ PellBorrowerLoan, data = Eloans)

Residuals:
   Min     1Q Median     3Q    Max 
-20307  -2563  -1041   1338  32771 

Coefficients:
                  Estimate Std. Error t value Pr(>|t|)    
(Intercept)      2.962e+03  3.358e+02   8.819   <2e-16 ***
PellBorrowerLoan 1.150e+00  1.605e-02  71.623   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 4689 on 863 degrees of freedom
Multiple R-squared:  0.856, Adjusted R-squared:  0.8558 
F-statistic:  5130 on 1 and 863 DF,  p-value: < 2.2e-16

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PPL1<-lm(ParentLoan~NoPellBorrowerLoan, Eloans)
summary(PPL1)


Call:
lm(formula = ParentLoan ~ NoPellBorrowerLoan, data = Eloans)

Residuals:
     Min       1Q   Median       3Q      Max 
-14541.6  -1441.7    303.1   1717.2  21088.1 

Coefficients:
                     Estimate Std. Error t value Pr(>|t|)    
(Intercept)        -7.704e+02  2.656e+02    -2.9  0.00382 ** 
NoPellBorrowerLoan  8.079e-01  7.779e-03   103.9  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 3363 on 863 degrees of freedom
Multiple R-squared:  0.9259,    Adjusted R-squared:  0.9258 
F-statistic: 1.079e+04 on 1 and 863 DF,  p-value: < 2.2e-16

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L1<-ggplot(Eloans, aes(x = StudentLoan,y = PellBorrowerLoan)) +
    geom_point(color=2) + 
    geom_smooth(method = "lm") +
    labs(x="Student Loan Amounts", y="Parent Loan Amounts", title = "Pell Grant Recipients
         (𝑅²:  0.856,p-value: < 0.05)")

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L2<-ggplot(Eloans, aes(x = StudentLoan,y = NoPellBorrowerLoan)) +
    geom_point(color=2) + 
    geom_smooth(method = "lm") +
    labs(x="Student Loan Amounts", y="Parent Loan Amounts", title = "Non-Pell Grant Recipents
         (𝑅²:0.9259,p-value: < 0.05)")

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library(see)
plots(L1, L2, n_rows = 1)

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Default_Cor <- select(Eloans,c("Default3yr","ParentLoan","COST",  "Repay5yr", "Repay10yr" , "Repay20yr",
                          "PellBorrowerLoan","NoPellBorrowerLoan","NoDegree"))

Observations

      -Student maximum loan amounts for both groups are $27,000
      -Low- and moderate-income parents borrowed less amount compared
      to moderate-and high-income households. 
      -However, considering the median household income, the 
      debt-to-income ratio of low- and moderate-income parents of
      Pell grant recipients is much higher than parents of Non-Pell
      grant recipients. Parents with higher debt-to-income ratio are
      more likely to have difficulty repaying loans.
      -Median 3-year default rate of parents loans with Pell grants
      is higher at 14% compared to 10% for parents Non-Pell grant
      recipients.

EXAMINING DEFAULT RATES

Fig 4. What influences default rates?

Method: Correlation Matrix

The College Scorecard has a rich and very wide data on aid, repayment, income, completion, etc on an institutional level. This study is limited to exploring some variables that may be correlated to default rates on parent loans. The correlation matrix to understand default rates factors the following variables: Loan Amounts, Cost of Attendance, 5-,10-,and 20-year Repayment Rates, Pell Grant and Non-Pell Grant Loan Amounts and Degree Completion.

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library(GGally)
ggcorr(Default_Cor, nbreaks = 4,low = "steelblue", mid = "white", high = "red", label = TRUE, label_size = 5, hjust = 0.85, layout.exp = 3)

Observation

        The two variableswith highest correlation coefficients (positively 
        correlation) to default rates are:
        1.***Repay5yr, (cor 0.6)*** - Undergraduate Parent PLUS Loan 
        dollar-based 5-year repayment rate
        2.***No Degree, (cor 0.5)***- Percentage of all students that did 
        not receive an award and whose enrollment status is unknown after 
        leaving this institution within 8 years of entry

PREDICTING DEFAULT RATES

Fig 5. Can default rates be predicted?

Method: Regression Analysis, Model Evaluation, Estimated Marginal Means (EMM) Post Hoc Test lm(formula = Default3yr ~ Repay5yr + NoDegree, data = Eloans)

Predicted values of a 3-year default rate using EMM (estimated marginal means) fitted with multiple linear regression model.

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Default1 <-lm(Default3yr~Repay5yr+NoDegree, data = Eloans)

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summary(Default1)


Call:
lm(formula = Default3yr ~ Repay5yr + NoDegree, data = Eloans)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.102440 -0.021622 -0.005443  0.016983  0.131494 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.041400   0.005007  -8.268 5.12e-16 ***
Repay5yr     0.139053   0.006954  19.995  < 2e-16 ***
NoDegree     0.111405   0.010970  10.155  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.03111 on 862 degrees of freedom
Multiple R-squared:  0.4659,    Adjusted R-squared:  0.4646 
F-statistic: 375.9 on 2 and 862 DF,  p-value: < 2.2e-16

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library(ggeffects)
lm(Default3yr~Repay5yr+NoDegree, data = Eloans) %>% 
  ggemmeans(terms = c("Repay5yr", "NoDegree")) %>% 
  plot () +
  ggtitle( "EMMEANS:Predicted Values of 3-year Default Rate" )+
  theme(axis.text=element_text(size=14),
        axis.title=element_text(size=14,face="bold"))

Observation

The mean percentage of undergraduate student Parent PLUS Loan borrowers in default after 3 years is 14.7%, with a max of 59% (Data from 2,766 institutions)

    HIGHER DEFAULT RATE FOR: 
    -Parents  who entered a repayment plan within 5 years of loan disbursement AND
    -Parents whose students did not complete a degree

MODEL EVALUATION -Diagnostic Plots

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Default_lm<- lm(Default3yr~Repay5yr+NoDegree, data=Eloans)
summary(Default_lm)


Call:
lm(formula = Default3yr ~ Repay5yr + NoDegree, data = Eloans)

Residuals:
      Min        1Q    Median        3Q       Max 
-0.102440 -0.021622 -0.005443  0.016983  0.131494 

Coefficients:
             Estimate Std. Error t value Pr(>|t|)    
(Intercept) -0.041400   0.005007  -8.268 5.12e-16 ***
Repay5yr     0.139053   0.006954  19.995  < 2e-16 ***
NoDegree     0.111405   0.010970  10.155  < 2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.03111 on 862 degrees of freedom
Multiple R-squared:  0.4659,    Adjusted R-squared:  0.4646 
F-statistic: 375.9 on 2 and 862 DF,  p-value: < 2.2e-16

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par(mfrow=c(2,2))
plot(Default_lm)

Assumptions: Multicollinearity and Heteroskedascity

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performance::check_collinearity(Default_lm)

# Check for Multicollinearity

Low Correlation

     Term  VIF Increased SE Tolerance
 Repay5yr 1.15         1.07      0.87
 NoDegree 1.15         1.07      0.87

Low Correlation, No Multicollinearity, regression model is good.

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library(lmtest)
library(caret)

Assumption: Heteroskesdacity, use Breusch-Pagan test

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lmtest::bptest(lm(Default3yr~NoDegree+Repay5yr, data=Eloans))


    studentized Breusch-Pagan test

data:  lm(Default3yr ~ NoDegree + Repay5yr, data = Eloans)
BP = 20.909, df = 2, p-value = 2.882e-05

Result: At p-value <0.05, homoscedasticity rejected in favor of heteroskedasticity

Transformation:

Transformed dependent variable using Box Cox Transformation and applied Breusch-Pagan test

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Default_BCMod <- caret::BoxCoxTrans(Eloans$Default3yr)
print(Default_BCMod)

Box-Cox Transformation

865 data points used to estimate Lambda

Input data summary:
   Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
0.02000 0.05000 0.08000 0.08081 0.10000 0.30000 

Largest/Smallest: 15 
Sample Skewness: 1.18 

Estimated Lambda: 0.1 
With fudge factor, Lambda = 0 will be used for transformations

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NEloans <- cbind(Eloans, dist_new=predict(Default_BCMod, Eloans$Default3yr)) # append the transformed variable 
head(NEloans) # view the top 6 rows

                               INSTNM State  UGDS  SAT  COST PCTPELL
1            Alabama A & M University    AL  5271  939 23053  0.7019
2 University of Alabama at Birmingham    AL 13328 1234 24495  0.3512
3 University of Alabama in Huntsville    AL  7785 1319 23917  0.2536
4            Alabama State University    AL   375  946 21866  0.7627
5           The University of Alabama    AL   319 1261 29872  0.1772
6     Auburn University at Montgomery    AL   447 1082 19849  0.4644
  PCTFLOAN LPSTAFFORD_CNT LPSTAFFORD_AMT Students LPPPLUS_AMT Parents
1   0.7361          32214      952539931     5398   124751228    1062
2   0.4798          59241     1805698825     3903    95560297     806
3   0.3976          20387      423083320     1444    32335435     289
4   0.8232          34809      968253338     4638    98316982    1135
5   0.3802          77636     1837713282    12086   676026693    2568
6   0.5391          23837      511627872     1369    18241142     300
  BBRR1_PP_NOPELL_N BBRR1_PP_PELL_N  CDR3 DBRR1_PP_UG_RT  Repay5yr
1               150             912 0.176       1.155044 1.2040422
2               322             484 0.063       1.068891 0.9748241
3               112             177 0.065       1.040121 0.8645943
4               147             988 0.180       1.140748 1.2583712
5              1377            1191 0.054       1.065396 0.9187667
6                73             227 0.109       1.097795 1.1208157
  Repay10yr Default3yr ParentLoan StudentLoan
1 0.9109411       0.20      16552       15250
2 0.6820359       0.08      18586       15085
3 0.4597370       0.09      15354       14000
4 0.8876563       0.20      18289       17500
5 0.6487990       0.07      44851       17671
6 0.5972791       0.14       8000       12000
  PPLUS_PCT_HIGH_POOLED_SUPP Repay20yr BBRR3_PP_PELL_N
1                         30 0.3771945             683
2                         10 0.4306524             353
3                         10 0.2390932             155
4                         25 0.3885780             711
5                         15 0.3586384            1062
6                         10 0.4559485             139
  BBRR3_PP_PELL_DFLT BBRR3_PP_NOPELL_DFLT BBRR3_PP_NOPELL_N
1               0.29                 0.10               116
2               0.14                 0.05               249
3               0.20                 0.20                84
4               0.39                 0.20                87
5               0.10                 0.03              1082
6               0.20                 0.20                64
  PellBorrowerLoan NoPellBorrowerLoan PELL_RPY_3YR_RT_SUPP
1            15944              19004            0.2390405
2            15304              22178            0.4716981
3            13300              16956            0.4832808
4            16882              25046            0.2079327
5            38868              48411            0.5159860
6             7507              10000            0.3853631
  NOPELL_RPY_3YR_RT_SUPP OMENRUP_ALL NoDegree OMENRUP_PELL_ALL
1              0.3492958      0.3016   0.2789           0.3271
2              0.5892070      0.1832   0.2017           0.1915
3              0.6783920      0.1716   0.1730           0.2063
4              0.3333333      0.2262   0.2481           0.2543
5              0.7126540      0.0820   0.0813           0.1448
6              0.5775034      0.2807   0.2993           0.2945
  OMENRUP8_FTFT  dist_new
1        0.3243 -1.609438
2        0.1549 -2.525729
3        0.1450 -2.407946
4        0.2229 -1.609438
5        0.0623 -2.659260
6        0.2898 -1.966113

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lmMod_def <- lm(dist_new ~ NoDegree, data=NEloans)
bptest(lmMod_def)


    studentized Breusch-Pagan test

data:  lmMod_def
BP = 0.047752, df = 1, p-value = 0.827

Result: New p-value = 0.827 after transformation: homoscedasticity not rejected

Evaluate New Diagnostic Plots using Box Cox Transformed Variable

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Default_lm2<- lm(dist_new~Repay5yr+NoDegree, data=NEloans)
summary(Default_lm2)


Call:
lm(formula = dist_new ~ Repay5yr + NoDegree, data = NEloans)

Residuals:
     Min       1Q   Median       3Q      Max 
-1.20735 -0.25334 -0.01247  0.29037  1.17292 

Coefficients:
            Estimate Std. Error t value Pr(>|t|)    
(Intercept) -4.06186    0.06509 -62.402   <2e-16 ***
Repay5yr     1.58272    0.09041  17.507   <2e-16 ***
NoDegree     1.40450    0.14261   9.849   <2e-16 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Residual standard error: 0.4045 on 862 degrees of freedom
Multiple R-squared:  0.4149,    Adjusted R-squared:  0.4135 
F-statistic: 305.6 on 2 and 862 DF,  p-value: < 2.2e-16

Show code

par(mfrow=c(2,2))
plot(Default_lm2)

CONCLUSION

With rising costs of education, parent PLUS loans have become increasing popular for families. The availability of parent PLUS loans have made post-secondary education accessible to more low- and moderate-income families. The mean cost of attendance per year is $38,144 and the mean parent PLUS loan is $24,131, with a max of $91,987 at an interest rate of 6.28%. According to the US Census Bureau, the median household income in the US is $67,521 The lack of limit of loan amounts and absence of debt-to-income ratio requirement have allowed parents to borrow more than their ability to repay. Comparison of parents of Pell grant recipients and non-Pell grant recipient borrower behavior is alarming. Most Pell grant recipient parents have income <$40,000 and median loan amount of $16,000, with a max of $95,089. Parents who entered repayment with 5 years of loan disbursement with students who did not complete the degree are most likely to default on the loan. This is likely due to personal circumstances such as family obligations or sought employment.

At a default rate of 14.7%, further analysis on this federal loan program is needed to better understand and prevent families from being entrapped in debt and exacerbate wealth disparities across races.

LIMITATIONS OF THE STUDY

  -The College Scorecard has data on 6,694 institutions all over the US but 
  this study is limited to only 865 schools due to missing data (e.g NULL and Privacy Suppressed).
  -The most recent data also discontinued collecting information relating to 
  income, an important variable that could increase statistical power of this
  model.
  -The researcher used listwise deletion to handle the missing data. Due to 
  the large size of the missing not at random (MNAR)  data, multiple 
  imputation is not preferred as it may provide misleading results.
  -Regression model used to analyze default rates was limited to a couple of
  the variables with high correlation coefficients. The College Scorecard 
  data is a very rich source of information. Unfortunately, the most recent
  data have discontinued releasing information on earnings and income. 
  Further examination and analysis of underlying factors (e.g. race, 
  first-generation students, gender, family characteristics) could probably 
  increase the explanatory power of the model.

FUTURE WORK: Loan Default Prediction using Decision Tree Classifier

Model currently has low accuracy, possible due to variance of continuous variables.Further work is needed.

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Eloans_train_test <- function(Eloans, size = 0.8, train = TRUE) {
    n_row = nrow(Eloans)
    total_row = size * n_row
    train_sample <- 1:total_row
    if (train == TRUE) {
        return (Eloans[train_sample, ])
    } else {
        return (Eloans[-train_sample, ])
    }
}

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Eloans_train <-Eloans_train_test(Eloans, 0.8, train = TRUE)
Eloans_test <- Eloans_train_test(Eloans, 0.8, train = FALSE)
dim(Eloans_train)

[1] 692  35

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dim(Eloans_test)

[1] 173  35

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prop.table(table(Eloans_train$Default3yr))


       0.02        0.03        0.04        0.05        0.06 
0.031791908 0.044797688 0.075144509 0.235549133 0.062138728 
       0.07        0.08        0.09         0.1        0.11 
0.010115607 0.036127168 0.174855491 0.187861272 0.001445087 
       0.12        0.14        0.16        0.17        0.18 
0.011560694 0.069364162 0.007225434 0.002890173 0.002890173 
       0.19         0.2        0.22        0.24         0.3 
0.031791908 0.005780347 0.004335260 0.002890173 0.001445087

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prop.table(table(Eloans_test$Default3yr))


       0.02        0.03        0.04        0.05        0.06 
0.046242775 0.075144509 0.075144509 0.260115607 0.069364162 
       0.07        0.08        0.09         0.1        0.12 
0.023121387 0.011560694 0.104046243 0.173410405 0.023121387 
       0.14        0.16        0.18        0.19         0.2 
0.057803468 0.017341040 0.005780347 0.040462428 0.005780347 
       0.22        0.24 
0.005780347 0.005780347

Show code

library(rpart)
library(rpart.plot)
Eloansfit <- rpart(Default3yr~NoDegree+Repay5yr, data = Eloans_train, method = 'anova')
rpart.plot(Eloansfit, extra = 101)

Show code

predict(Eloansfit, Eloans_train, type = 'vector')

         1          2          3          4          5          6 
0.19294118 0.13200000 0.09466667 0.19294118 0.10161290 0.19294118 
         7          8          9         10         11         12 
0.07046512 0.07745342 0.07745342 0.14884615 0.19294118 0.09466667 
        13         14         15         16         17         18 
0.07745342 0.07745342 0.19294118 0.05383886 0.09466667 0.07046512 
        19         20         21         22         23         24 
0.13200000 0.19294118 0.09466667 0.07046512 0.05383886 0.13200000 
        25         26         27         28         29         30 
0.07745342 0.13200000 0.09466667 0.07745342 0.13200000 0.05383886 
        31         32         33         34         35         36 
0.05383886 0.05383886 0.07046512 0.09466667 0.07745342 0.05383886 
        37         38         39         40         41         42 
0.09466667 0.05383886 0.05383886 0.09466667 0.05383886 0.05383886 
        43         44         45         46         47         48 
0.09466667 0.05383886 0.05383886 0.05383886 0.05383886 0.05383886 
        49         50         51         52         53         54 
0.05383886 0.05383886 0.05383886 0.07745342 0.05383886 0.07046512 
        55         56         57         58         59         60 
0.07745342 0.07745342 0.05383886 0.09466667 0.13200000 0.05383886 
        61         62         63         64         65         66 
0.05383886 0.07046512 0.09466667 0.09466667 0.07046512 0.05383886 
        67         68         69         70         71         72 
0.05383886 0.05383886 0.07745342 0.05383886 0.05383886 0.05383886 
        73         74         75         76         77         78 
0.07046512 0.05383886 0.05383886 0.07046512 0.05383886 0.07046512 
        79         80         81         82         83         84 
0.05383886 0.07046512 0.07745342 0.09466667 0.07745342 0.05383886 
        85         86         87         88         89         90 
0.05383886 0.07745342 0.05383886 0.07745342 0.07745342 0.07745342 
        91         92         93         94         95         96 
0.07745342 0.05383886 0.09466667 0.07745342 0.14884615 0.07745342 
        97         98         99        100        101        102 
0.05383886 0.07046512 0.10161290 0.10161290 0.07046512 0.09466667 
       103        104        105        106        107        108 
0.19294118 0.07745342 0.14884615 0.07046512 0.05383886 0.05383886 
       109        110        111        112        113        114 
0.14884615 0.10161290 0.19294118 0.09466667 0.07745342 0.07745342 
       115        116        117        118        119        120 
0.07046512 0.14884615 0.09466667 0.07745342 0.09466667 0.10161290 
       121        122        123        124        125        126 
0.09466667 0.07046512 0.07046512 0.13200000 0.07046512 0.05383886 
       127        128        129        130        131        132 
0.14884615 0.09466667 0.09466667 0.09466667 0.07046512 0.13200000 
       133        134        135        136        137        138 
0.07745342 0.07745342 0.14884615 0.14884615 0.05383886 0.13200000 
       139        140        141        142        143        144 
0.13200000 0.05383886 0.14884615 0.05383886 0.07046512 0.09466667 
       145        146        147        148        149        150 
0.13200000 0.05383886 0.07046512 0.09466667 0.14884615 0.05383886 
       151        152        153        154        155        156 
0.09466667 0.10161290 0.14884615 0.09466667 0.14884615 0.09466667 
       157        158        159        160        161        162 
0.10161290 0.09466667 0.09466667 0.09466667 0.07046512 0.09466667 
       163        164        165        166        167        168 
0.07745342 0.07745342 0.09466667 0.07745342 0.07745342 0.09466667 
       169        170        171        172        173        174 
0.07046512 0.05383886 0.07745342 0.05383886 0.09466667 0.05383886 
       175        176        177        178        179        180 
0.07745342 0.05383886 0.05383886 0.05383886 0.07046512 0.07046512 
       181        182        183        184        185        186 
0.07745342 0.07745342 0.07745342 0.05383886 0.09466667 0.07046512 
       187        188        189        190        191        192 
0.05383886 0.05383886 0.07745342 0.09466667 0.10161290 0.07046512 
       193        194        195        196        197        198 
0.07745342 0.10161290 0.07745342 0.05383886 0.07046512 0.05383886 
       199        200        201        202        203        204 
0.07745342 0.05383886 0.05383886 0.05383886 0.07046512 0.07046512 
       205        206        207        208        209        210 
0.09466667 0.09466667 0.09466667 0.09466667 0.13200000 0.07745342 
       211        212        213        214        215        216 
0.09466667 0.05383886 0.09466667 0.07745342 0.07745342 0.05383886 
       217        218        219        220        221        222 
0.05383886 0.05383886 0.05383886 0.07745342 0.07046512 0.07745342 
       223        224        225        226        227        228 
0.05383886 0.07745342 0.07046512 0.07745342 0.05383886 0.09466667 
       229        230        231        232        233        234 
0.09466667 0.07745342 0.05383886 0.05383886 0.05383886 0.05383886 
       235        236        237        238        239        240 
0.09466667 0.07745342 0.05383886 0.05383886 0.05383886 0.07745342 
       241        242        243        244        245        246 
0.07745342 0.07745342 0.05383886 0.07745342 0.07745342 0.07745342 
       247        248        249        250        251        252 
0.07745342 0.05383886 0.07745342 0.07745342 0.07745342 0.07046512 
       253        254        255        256        257        258 
0.09466667 0.09466667 0.09466667 0.19294118 0.07046512 0.09466667 
       259        260        261        262        263        264 
0.07745342 0.09466667 0.07745342 0.09466667 0.09466667 0.19294118 
       265        266        267        268        269        270 
0.07046512 0.09466667 0.13200000 0.09466667 0.13200000 0.09466667 
       271        272        273        274        275        276 
0.19294118 0.09466667 0.07745342 0.14884615 0.07745342 0.07745342 
       277        278        279        280        281        282 
0.07046512 0.07745342 0.07745342 0.14884615 0.07046512 0.07745342 
       283        284        285        286        287        288 
0.05383886 0.07745342 0.05383886 0.07046512 0.19294118 0.14884615 
       289        290        291        292        293        294 
0.07046512 0.05383886 0.05383886 0.07046512 0.10161290 0.05383886 
       295        296        297        298        299        300 
0.05383886 0.05383886 0.05383886 0.07745342 0.09466667 0.14884615 
       301        302        303        304        305        306 
0.07046512 0.07046512 0.07745342 0.07745342 0.05383886 0.05383886 
       307        308        309        310        311        312 
0.07046512 0.07046512 0.07745342 0.05383886 0.07046512 0.07745342 
       313        314        315        316        317        318 
0.05383886 0.07046512 0.09466667 0.07046512 0.05383886 0.07745342 
       319        320        321        322        323        324 
0.07046512 0.05383886 0.07046512 0.05383886 0.07745342 0.05383886 
       325        326        327        328        329        330 
0.05383886 0.13200000 0.05383886 0.05383886 0.05383886 0.07745342 
       331        332        333        334        335        336 
0.07046512 0.09466667 0.09466667 0.07745342 0.07745342 0.07745342 
       337        338        339        340        341        342 
0.05383886 0.07745342 0.09466667 0.07745342 0.05383886 0.05383886 
       343        344        345        346        347        348 
0.05383886 0.07745342 0.09466667 0.07745342 0.09466667 0.09466667 
       349        350        351        352        353        354 
0.09466667 0.07745342 0.09466667 0.07745342 0.09466667 0.07745342 
       355        356        357        358        359        360 
0.05383886 0.05383886 0.05383886 0.07745342 0.05383886 0.05383886 
       361        362        363        364        365        366 
0.07745342 0.07745342 0.07745342 0.07046512 0.07745342 0.05383886 
       367        368        369        370        371        372 
0.07745342 0.05383886 0.19294118 0.09466667 0.10161290 0.13200000 
       373        374        375        376        377        378 
0.14884615 0.09466667 0.05383886 0.07745342 0.07745342 0.07046512 
       379        380        381        382        383        384 
0.07745342 0.09466667 0.07745342 0.07745342 0.09466667 0.05383886 
       385        386        387        388        389        390 
0.07745342 0.05383886 0.09466667 0.05383886 0.07745342 0.07046512 
       391        392        393        394        395        396 
0.05383886 0.07745342 0.07745342 0.07745342 0.05383886 0.05383886 
       397        398        399        400        401        402 
0.07745342 0.07745342 0.07745342 0.07745342 0.05383886 0.07745342 
       403        404        405        406        407        408 
0.05383886 0.05383886 0.07745342 0.07745342 0.05383886 0.05383886 
       409        410        411        412        413        414 
0.09466667 0.05383886 0.05383886 0.07745342 0.05383886 0.10161290 
       415        416        417        418        419        420 
0.13200000 0.10161290 0.05383886 0.10161290 0.07046512 0.07046512 
       421        422        423        424        425        426 
0.07046512 0.05383886 0.10161290 0.09466667 0.07046512 0.07046512 
       427        428        429        430        431        432 
0.14884615 0.10161290 0.05383886 0.07046512 0.05383886 0.09466667 
       433        434        435        436        437        438 
0.09466667 0.10161290 0.05383886 0.05383886 0.05383886 0.05383886 
       439        440        441        442        443        444 
0.05383886 0.05383886 0.09466667 0.07046512 0.07046512 0.07745342 
       445        446        447        448        449        450 
0.05383886 0.10161290 0.05383886 0.07046512 0.13200000 0.05383886 
       451        452        453        454        455        456 
0.09466667 0.07745342 0.09466667 0.14884615 0.07046512 0.05383886 
       457        458        459        460        461        462 
0.10161290 0.10161290 0.05383886 0.07046512 0.10161290 0.19294118 
       463        464        465        466        467        468 
0.09466667 0.09466667 0.07745342 0.07745342 0.05383886 0.05383886 
       469        470        471        472        473        474 
0.07046512 0.09466667 0.05383886 0.09466667 0.05383886 0.07745342 
       475        476        477        478        479        480 
0.05383886 0.10161290 0.07745342 0.07745342 0.07745342 0.07745342 
       481        482        483        484        485        486 
0.09466667 0.09466667 0.05383886 0.05383886 0.05383886 0.05383886 
       487        488        489        490        491        492 
0.05383886 0.07745342 0.05383886 0.09466667 0.05383886 0.05383886 
       493        494        495        496        497        498 
0.05383886 0.05383886 0.05383886 0.05383886 0.05383886 0.09466667 
       499        500        501        502        503        504 
0.05383886 0.05383886 0.07046512 0.10161290 0.07745342 0.05383886 
       505        506        507        508        509        510 
0.10161290 0.09466667 0.07046512 0.07745342 0.14884615 0.05383886 
       511        512        513        514        515        516 
0.05383886 0.05383886 0.10161290 0.10161290 0.09466667 0.07046512 
       517        518        519        520        521        522 
0.19294118 0.09466667 0.13200000 0.07745342 0.07046512 0.09466667 
       523        524        525        526        527        528 
0.14884615 0.05383886 0.07745342 0.07745342 0.14884615 0.05383886 
       529        530        531        532        533        534 
0.14884615 0.05383886 0.07046512 0.10161290 0.10161290 0.13200000 
       535        536        537        538        539        540 
0.07745342 0.07046512 0.14884615 0.05383886 0.07745342 0.07745342 
       541        542        543        544        545        546 
0.09466667 0.05383886 0.05383886 0.09466667 0.09466667 0.07046512 
       547        548        549        550        551        552 
0.05383886 0.05383886 0.05383886 0.09466667 0.09466667 0.05383886 
       553        554        555        556        557        558 
0.09466667 0.09466667 0.09466667 0.09466667 0.05383886 0.09466667 
       559        560        561        562        563        564 
0.07745342 0.07745342 0.05383886 0.05383886 0.07046512 0.07745342 
       565        566        567        568        569        570 
0.07745342 0.07745342 0.09466667 0.09466667 0.05383886 0.05383886 
       571        572        573        574        575        576 
0.09466667 0.07046512 0.05383886 0.09466667 0.09466667 0.10161290 
       577        578        579        580        581        582 
0.07046512 0.07745342 0.07745342 0.07745342 0.05383886 0.09466667 
       583        584        585        586        587        588 
0.09466667 0.09466667 0.07745342 0.07745342 0.07745342 0.09466667 
       589        590        591        592        593        594 
0.05383886 0.05383886 0.13200000 0.07745342 0.07745342 0.09466667 
       595        596        597        598        599        600 
0.05383886 0.05383886 0.05383886 0.07745342 0.07745342 0.05383886 
       601        602        603        604        605        606 
0.05383886 0.09466667 0.05383886 0.07745342 0.07745342 0.07745342 
       607        608        609        610        611        612 
0.05383886 0.07745342 0.07046512 0.07046512 0.05383886 0.05383886 
       613        614        615        616        617        618 
0.09466667 0.09466667 0.09466667 0.05383886 0.09466667 0.09466667 
       619        620        621        622        623        624 
0.09466667 0.09466667 0.05383886 0.09466667 0.07745342 0.07745342 
       625        626        627        628        629        630 
0.05383886 0.07745342 0.05383886 0.09466667 0.09466667 0.09466667 
       631        632        633        634        635        636 
0.09466667 0.09466667 0.09466667 0.09466667 0.07745342 0.05383886 
       637        638        639        640        641        642 
0.14884615 0.09466667 0.05383886 0.09466667 0.07046512 0.05383886 
       643        644        645        646        647        648 
0.07745342 0.07046512 0.14884615 0.09466667 0.07046512 0.05383886 
       649        650        651        652        653        654 
0.05383886 0.07745342 0.07745342 0.07745342 0.05383886 0.07745342 
       655        656        657        658        659        660 
0.09466667 0.09466667 0.07745342 0.07745342 0.05383886 0.05383886 
       661        662        663        664        665        666 
0.07745342 0.05383886 0.13200000 0.07046512 0.05383886 0.05383886 
       667        668        669        670        671        672 
0.07046512 0.05383886 0.07745342 0.09466667 0.07046512 0.05383886 
       673        674        675        676        677        678 
0.05383886 0.09466667 0.05383886 0.07745342 0.05383886 0.05383886 
       679        680        681        682        683        684 
0.07745342 0.10161290 0.19294118 0.05383886 0.07046512 0.19294118 
       685        686        687        688        689        690 
0.05383886 0.13200000 0.07046512 0.13200000 0.13200000 0.09466667 
       691        692 
0.07046512 0.10161290

Show code

predict_unseen <-predict(Eloansfit, Eloans_test, type = 'vector')

Show code

Eloans_mat <- table(Eloans_test$Default3yr, predict_unseen)
Eloans_mat

      predict_unseen
       0.0538388625592417 0.0704651162790697 0.0774534161490683
  0.02                  5                  0                  3
  0.03                 10                  0                  3
  0.04                  3                  4                  4
  0.05                 15                  8                 17
  0.06                  1                  2                  5
  0.07                  0                  0                  1
  0.08                  0                  0                  0
  0.09                  1                  5                  3
  0.1                   4                  4                 11
  0.12                  0                  1                  0
  0.14                  0                  0                  0
  0.16                  0                  1                  0
  0.18                  0                  0                  0
  0.19                  0                  1                  3
  0.2                   0                  0                  0
  0.22                  0                  0                  0
  0.24                  0                  0                  0
      predict_unseen
       0.0946666666666665 0.101612903225806 0.132 0.148846153846154
  0.02                  0                 0     0                 0
  0.03                  0                 0     0                 0
  0.04                  2                 0     0                 0
  0.05                  4                 1     0                 0
  0.06                  3                 0     1                 0
  0.07                  3                 0     0                 0
  0.08                  2                 0     0                 0
  0.09                  6                 3     0                 0
  0.1                   9                 1     0                 1
  0.12                  2                 0     0                 0
  0.14                  6                 1     1                 1
  0.16                  0                 0     0                 1
  0.18                  0                 0     0                 1
  0.19                  1                 1     0                 0
  0.2                   0                 0     0                 0
  0.22                  0                 0     0                 1
  0.24                  0                 0     0                 0
      predict_unseen
       0.192941176470588
  0.02                 0
  0.03                 0
  0.04                 0
  0.05                 0
  0.06                 0
  0.07                 0
  0.08                 0
  0.09                 0
  0.1                  0
  0.12                 1
  0.14                 1
  0.16                 1
  0.18                 0
  0.19                 1
  0.2                  1
  0.22                 0
  0.24                 1

Show code

accuracy_Test <- sum(diag(Eloans_mat)) / sum(Eloans_mat)
print(paste('Accuracy for test', accuracy_Test))

[1] "Accuracy for test 0.0751445086705202"

FUTURE WORK: Loan Default Prediction using Random Forest

Random Forest using mtry=3 and 500 trees has been initiated Need to continue work on running and testing the model, confusionMatrix and importance plots

Show code

Eloans <-select(Eloans,-c(INSTNM))

Show code

library(randomForest)
library(caret)
library(e1071)

Show code

Eloans.rf <- randomForest(Default3yr~ ., data = Eloans, mtry = 3,
                         importance = TRUE, na.action = na.omit)
print(Eloans.rf)


Call:
 randomForest(formula = Default3yr ~ ., data = Eloans, mtry = 3,      importance = TRUE, na.action = na.omit) 
               Type of random forest: regression
                     Number of trees: 500
No. of variables tried at each split: 3

          Mean of squared residuals: 0.0003730025
                    % Var explained: 79.35

Show code

Etrain_test <- function(Eloans, size = 0.8, train = TRUE) {
    n_row = nrow(Eloans)
    total_row = size * n_row
    train_sample <- 1:total_row
    if (train == TRUE) {
        return (Eloans[train_sample, ])
    } else {
        return (Eloans[-train_sample, ])
    }
}

Show code

Etrain <-Etrain_test(Eloans, 0.8, train = TRUE)
Etest <- Etrain_test(Eloans, 0.8, train = FALSE)
dim(Etrain)

[1] 692  34

Show code

dim(Etest)

[1] 173  34

Show code

set.seed(120)  # Setting seed
classifier_RF <- randomForest(x = Etrain[-5],
                             y = Etrain$STABBR,
                             ntree = 500)
  
classifier_RF


Call:
 randomForest(x = Etrain[-5], y = Etrain$STABBR, ntree = 500) 
               Type of random forest: unsupervised
                     Number of trees: 500
No. of variables tried at each split: 5

REFERENCES

Data Home: College Scorecard. Data Home | College Scorecard. (n.d.). Retrieved May 11, 2022, from https://collegescorecard.ed.gov/data/
2015–16 National Postsecondary Student Aid Study (npsas:16) student … (n.d.). Retrieved May 12, 2022, from https://nces.ed.gov/pubs2018/2018466.pdf
Federal Student Aid. (n.d.). Retrieved May 11, 2022, from https://studentaid.gov/plus-app/parent/landing
Wickham, H., & Grolemund, G. (2016). R for data science: Visualize, model, transform, tidy, and import data. OReilly Media.
R Core Team (2017). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL https://www.R-project.org/.
[SMSS] Agresti, Alan. Statistical Methods for the Social Sciences. 5th Edition. Pearson, 2018. Wickham, H., & Grolemund, G. (2016). R for data science: Visualize, model, transform, tidy, and import data. OReilly Media

Financing College: Anatomy of Parent PLUS Loans

BACKGROUND

RESEARCH AND METHODS

Data

Sum of Total outstanding Parent PLUS Loan balance

Sum of Number of students associated with outstanding Parent PLUS Loan balances

LOAN DISTRIBUTION ACROSS THE US

Fig 1. Who is borrowing how much?

Observation

LOAN AMOUNTS AND SCHOOL SELECTIVITY

Fig 2. Where are the loans disbursed?

Observations

BORROWER BEHAVIOR

Fig 3. Does Income Influence Borrower Behavior?

Observations

EXAMINING DEFAULT RATES

Fig 4. What influences default rates?

Observation

PREDICTING DEFAULT RATES

Fig 5. Can default rates be predicted?

Observation

MODEL EVALUATION -Diagnostic Plots

Assumptions: Multicollinearity and Heteroskedascity

Low Correlation, No Multicollinearity, regression model is good.

Assumption: Heteroskesdacity, use Breusch-Pagan test

Evaluate New Diagnostic Plots using Box Cox Transformed Variable

CONCLUSION

LIMITATIONS OF THE STUDY

FUTURE WORK: Loan Default Prediction using Decision Tree Classifier

FUTURE WORK: Loan Default Prediction using Random Forest

REFERENCES