Predictive Analytics & The U.S.SBA (Small Business Administration) - Statistics Assignment Help

Assignment Task -   

Predictive Analytics

The U.S. SBA (Small Business Administration) was founded in 1953 on the principle of promoting and assisting small enterprises in the U.S. credit market. Small businesses have been a primary source of job creation in the United States; therefore, fostering small business formation and growth has social benefits by creating job opportunities and reducing unemployment.

One way SBA assists these small business enterprises is through a loan guarantee program which is designed to encourage banks to grant loans to small businesses. SBA acts much like an insurance provider to reduce the risk for a bank by taking on some of the risk through guaranteeing a portion of the loan. In the case that a loan goes into default, SBA then covers the amount they guaranteed.

There have been many success stories of start-ups receiving SBA loan guarantees such as FedEx and Apple. However, there have also been stories of small businesses and/or start-ups that have defaulted on their SBA-guaranteed loans. 

Since SBA loans only guarantee a portion of the entire loan balance, banks will incur some losses if a small business defaults on its SBA-guaranteed loan. Therefore, banks are still faced with a difficult choice as to the amount they should grant such a loan because of the high risk of default. One way to inform their decision making is through analyzing relevant historical data such as the data-set provided here. 

This data set contains: 

Variable Name Data Type Description of variable 

Name Text Borrower Name 

State Text Borrower State 

Term Number Loan term in months 

NoEmp Number Number of Business Employees 

NewExist Text 1 = Existing Business, 2 = New Business 

CreateJob Number Number of jobs created 

RetainedJob Number Number of jobs retained 

UrbanRural Text 1= Urban, 2= Rural, 0 = Undefined 

GrAppv Currency Gross Amount of Loan Approved by Bank 

SBA_Appv Currency SBA’s Guaranteed Amount of Approved Loan 

Regression Modelling 

Using R fit a multiple linear regression model to the data with log10(GrAppv) as the response variable and the variable chosen in the exploratory analysis section as the predictor variables. Define and describe the mathematical equation for the model (Also provide your R code) (5 marks). 

(a) Diagnostics tests on the regression model 

For each question in this section, please provide the lines of R code required to produce your results and any tables and/or figures produced by R. 

1. Check the linearity assumption of your model. Provide a reasonable justification as to whether your model satisfies the linearity assumption, and provide supporting evidence.

2. What are the consequences of the model not satisfying the linearity assumption?  

3. The linear regression model assumes that you have an independent and identically distributed random sample. Provide a reasonable justification as to whether your model satisfies this assumption, and provide supporting evidence.

4. What are the consequences of the data not being from an independent and identically distributed random sample? 

5. An assumption of the linear regression model is that you have no multicollinearity in your model. Provide a reasonable justification as to whether your model satisfies this assumption, and provide supporting evidence.

6. What are the consequences of having multicollinearity in your model .

7. The linear regression model assumes that the residuals have a zero conditional mean. Provide a reasonable justification as to whether your model satisfies this assumption, and provide supporting evidence .

8. An assumption of the linear regression model is that the residuals have constant variance. Provide a reasonable justification as to whether your model satisfies this assumption, and provide supporting evidence .

9. What are the consequences of having non-constant variance in your model (2 marks)? 

10. The linear regression model assumes that the residuals are normally distributed. Provide a reasonable justification as to whether your model satisfies this assumption, and provide supporting evidence 

11. What are the consequences of non-normal residuals in your model? 

(b) Interpret the regression model 

1. Interpret the estimate of the parameters β in your model 

2. Which values of the predictor variables will lead to the largest increase in log10(GrAppv)? 

3. Discuss the effect that one of the categorical variables in your model has on the E[log10(GrAppv)] 

4. Which of the parameters β in your model are significantly different from zero. Provide supporting evidence 

5. Interpret the F-statistic in the output in the summary of the regression model. Hint: State the hypothesis being tested, the test statistic and p-value and the conclusion in the context of the problem. 

6. Interpret the Adjusted R-squared value. 

7. What is the difference between the Multiple R-squared and the Adjusted R-squared value? 

(c) ANOVA

1. Compute and interpret the ANOVA Type I table for your model. Hint: State the hypothesis being tested, the test statistic and p-value and the conclusion in the context of the problem . 

2. Compute and interpret the ANOVA Type II table for your model. Hint: State the hypothesis being tested, the test statistic and p-value and the conclusion in the context of the problem.

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