Consider again the example introduced in Section 4.5 of a credit card company that has a database of information provided by its customers when the customers apply for credit cards. An analyst has created a multiple regression model for which the dependent variable in the model is credit card charges accrued by a customer in the data set over the past year (y), and the independent variables are the customer’s annual household income (x1), number of members of the household (x2), and number of years of post–high school education (x3). Figure 4.23 provides Excel output for a multiple regression model estimated using a data set the company created.
a. Estimate the corresponding simple linear regression with the customer’s annual household income as the independent variable and credit card charges accrued by a customer over the past year as the dependent variable. Interpret the estimated relationship between the customer’s annual household income and credit card charges accrued over the past year. How much variation in credit card charges accrued by a customer over the past year does this simple linear regression model explain?
b. Estimate the corresponding simple linear regression with the number of members in the customer’s household as the independent variable and credit card charges accrued by a customer over the past year as the dependent variable. Interpret the estimated relationship between the number of members in the customer’s household and credit card charges accrued over the past year. How much variation in credit card charges accrued by a customer over the past year does this simple linear regression model explain?
c. Estimate the corresponding simple linear regression with the customer’s number of years of post–high school education as the independent variable and credit card charges accrued by a customer over the past year as the dependent variable. Interpret the estimated relationship between the customer’s number of years of post–high school education and credit card charges accrued over the past year. How much variation in credit card charges accrued by a customer over the past year does this simple linear regression model explain?
d. Recall the multiple regression in Figure 4.23 with credit card charges accrued by a customer over the past year as the dependent variable and customer’s annual household income (x1), number of members of the household (x2), and number of years of post–high school education (x3) as the independent variables. Do the estimated slopes differ substantially from the corresponding slopes that were estimated using simple linear regression in parts a, b, and c? What does this tell you about multicollinearity in the multiple regression model in Figure 4.23?
e. Add the coefficients of determination for the simple linear regression in parts a, b, and c, and compare the result to the coefficient of determination for the multiple regression model in Figure 4.23. What does this tell you about multicollinearity in the multiple regression model in Figure 4.23?
f. Add age, a dummy variable for gender, and a dummy variable for whether a customer has exceeded his or her credit limit in the past 12 months as independent variables to the multiple regression model in Figure 4.23. Code the dummy variable for gender as
1 if the customer’s gender is female and 0 if male, and code the dummy variable for whether a customer has exceeded his or her credit limit in the past 12 months as 1 if the customer has exceeded his or her credit limit in the past 12 months and 0 otherwise.
Do these variables substantially improve the fit of your model?