Chapter 16, Problem 26SE

A sample containing years to maturity and yield (%) for 40 corporate bonds is contained in the data file named CorporateBonds (Barron’s, April 2, 2012).

1. a. Develop a scatter diagram of the data using x = years to maturity as the independent variable. Does a simple linear regression model appear to be appropriate?
2. b. Develop an estimated regression equation with x = years to maturity and x² as the independent variables.
3. c. As an alternative to fitting a second-order model, fit a model using the natural logarithm of price as the independent variable; that is, $\hat{y}=b_0+b_1\text{ln}(x)$. Does the estimated regression using the natural logarithm of x provide a better fit than the estimated regression developed in part (b)? Explain.

To determine

Construct a scatter diagram of the data using $x=\text{years to maturity}$ as the independent variable.

Decide whether a simple linear regression model appears to be appropriate.

Answer to Problem 26SE

A simple linear regression model does not appear to be appropriate.

Explanation of Solution

Calculation:

The given information is about the yield (%) of 40 corporate bonds and their respective years to maturity.

Scatterplot:

Software procedure:

Step-by-step procedure to draw scatter diagram using EXCEL software:

• Open EXCEL file CorporateBonds.
• Select the two variables yield and years with labels.
• Go to Insert > Scatterplot.
• Click OK.

The scatter diagram of the data using EXCEL software is given below:

Observation:

The scatterplot shows a gradual increase in the yield at a decreasing rate, with increase in years up to 25. After this, there is a reduction in the values of yield. That is, the scatterplot shows a curvilinear relationship between the variables. Therefore, a simple linear regression model does not appear to be appropriate.

To determine

Develop an estimated multiple regression equation with $x=\text{years to maturity}$ and $x^2$ as the independent variables.

Answer to Problem 26SE

The estimated multiple regression equation with $x=\text{years to maturity}$ and $x^2$ as the independent variables is as follows:

$\underset{\_}{\text{Yield}=1.01695+\text{0}\text{.460636}\left(\text{Years}\right)-0.0102532{\left(\text{Years}\right)}^2}$.

Explanation of Solution

Calculation:

Square transformation:

Software procedure:

Step-by-step procedure to make square transformation using EXCEL software:

• Open EXCEL file CorporateBonds.
• Copy the variable Years in column D.
• Create a new variable using the command ‘=(Years)^2’ in column E.

The squared variable is stored in column E.

Regression:

Software procedure:

Step-by-step procedure to obtain the regression equation using EXCEL software:

• Select Data > Data Analysis > Regression.
• Click OK.
• Under Input Y Range, enter $C$1:$C$41.
• Under Input X Range, enter $D$1:$E$41.
• Click the box of Labels.
• Under Output Range, enter $J$1.
• Click OK.

The output obtained  using EXCEL software is given below:

From the output, the estimated multiple regression equation with $x=\text{years to maturity}$ and $x^2$ as the independent variables is as follows:

$\underset{\_}{\text{Yield}=1.01695+\text{0}\text{.460636}\left(\text{Years}\right)-0.0102532{\left(\text{Years}\right)}^2}$.

To determine

Develop an estimated multiple regression equation using the natural logarithm of years as the independent variable.

Explain whether the current regression provides a better fit than the estimated regression developed in Part b.

Answer to Problem 26SE

The estimated multiple regression equation using the natural logarithm of years as the independent variable is as follows:

$\underset{\_}{\text{Yield}=0.827876+\text{1}\text{.56261 }\ln \left(\text{Years}\right)}$.

The estimated regression using the natural logarithm of x provides a better fit than the estimated regression developed in Part (b).

Explanation of Solution

Calculation:

Logarithmic transformation:

Software procedure:

Step-by-step procedure to make logarithmic transformation using EXCEL software:

• Open EXCEL file CorporateBonds.
• Create a new variable using the command ‘=ln(Years)’ in column F.

The logarithm of the variable is stored in column F.

Regression:

Software procedure:

Step-by-step procedure to obtain the regression equation using EXCEL software:

• Select Data > Data Analysis > Regression.
• Click OK.
• Under Input Y Range, enter $C$1:$C$41.
• Under Input X Range, enter $F$1:$F$41.
• Click the box of Labels.
• Under Output Range, enter $J$1.
• Click OK.

The output obtained using EXCEL software is given below:

From the output, the estimated multiple regression equation using the natural logarithm of years as the independent variable is as follows:

$\underset{\_}{\text{Yield}=0.827876+\text{1}\text{.56261 }\ln \left(\text{Years}\right)}$.

Adjusted-$R^2$:

The adjusted $R^2$ is defined as the proportion of variation in the observed values of the response variable that is effectively explained by the regression. The squared correlation gives fraction of variability of response variable (y) accounted for by the linear regression model.

The value of adjusted $R^2$in the regression equation consisting of ‘Years’ and ‘Years^2’ is $\text{R-Sq}\left(\text{adj}\right)=64.99\%$.

The value of adjusted $R^2$in the regression equation consisting of ‘ln(Years)’ is $\text{R-Sq}\left(\text{adj}\right)=66.08\%$.

Evidently, the current regression equation effectively explains more of the variation of the response variable than the second regression equation.

Thus, the estimated regression using the natural logarithm of x provides a better fit than the estimated regression developed in Part (b).