Chapter 16, Problem 29SE

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, ŷ = b₀ + b₁ln(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 29SE

The scatter diagram of the data using $x=\text{years to maturity}$ as the independent variable is:

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

Explanation of Solution

Calculation:

The data gives information on yield (%) of 40 corporate bonds and the respective years to maturity.

Scatterplot:

Software procedure:

Step by step procedure to draw scatter diagram using MINITAB software is given below:

• Choose Graph > Scatterplot.
• Choose Simple, and then click OK.
• In Y–variables, enter the column of Yield.
• In X–variables enter the column of Years.
• Click OK.

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. Thus, 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 29SE

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

$\underset{\_}{\text{Yield}=1.01695+\text{0}\text{.460636 Years}-0.0102532\text{ YearsSq}}$.

Explanation of Solution

Calculation:

Square transformation:

Software procedure:

Step by step procedure to make square transformation using MINITAB software is given as,

• Choose Calc > Calculator.
• In Store result in variable, enter YearsSq.
• In Expression, enter ‘Years’^2.
• Click OK.

The squared variable is stored in the column of ‘YearsSq’.

Regression:

Software procedure:

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

• Choose Stat > Regression > General Regression.
• Under Responses, enter the column of Yield.
• Under Model, enter the columns of Years, YearsSq.
• Click OK.

Output using MINITAB 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:

$\underset{\_}{\text{Yield}=1.01695+\text{0}\text{.460636 Years}-0.0102532\text{ YearsSq}}$.

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 29SE

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

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

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 MINITAB software is given as,

• Choose Calc > Calculator.
• In Store result in variable, enter Years.
• In Expression, enter ln(‘Years’).
• Click OK.

The logarithm of the variable is stored in the column of ‘ln(‘Years’)’.

Regression:

Software procedure:

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

• Choose Stat > Regression > General Regression.
• Under Responses, enter the column of Yield.
• Under Model, enter the columns of ln(Years).
• Click OK.

Output using MINITAB software is given below:

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

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

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 ‘YearsSq’ 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 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.