Question

A quant jock from your firm used a linear demand specification to estimate the demand for its product and sent you a hard copy of the results. Unfortunately, some entries are missing because the toner was low in her printer. Use the information presented below to find the missing values labelled ‘$\mathbf{1}$throughrole="math" localid="1657353618073" $\mathbf{7}$’ (round your answer to the nearest hundredth). Then, answer the accompanying questions.

a. Based on these estimates, write an equation that summarizes the demand for the firm’s product.

b. Which regression coefficients are statistically significant at the$\mathbf{5}$percent level?

c. Comment on how well the regression line fits the data.

Short answer

a. $\begin{array}{c}{\mathrm{Q}}^{\mathrm{d}}=\widehat{\mathrm{a}}+\widehat{\mathrm{b}}{\mathrm{P}}_{\mathrm{x}}+\mathrm{cM}\\ =58.87-1.64+7\end{array}$

b. The t-statistics coefficient intercept and income are statistically significant at a 5% level.

c. The regression line does not fit the data.

Step-by-step solution

Step 1: Given information

Based on the estimation table,the coefficient’s intercept is$58.87$ , the coefficient’s price is $-1.64$,and the coefficient’s income is $7$.

${\mathrm{S}}_{\mathrm{Total}}=73,807.49\mathrm{and}{\mathrm{SS}}_{\mathrm{Residual}}=63,408.62$

Step 2: To find the demand equation

a.

Demand for the firm’s product is as follows:

$\begin{array}{c}{\mathrm{Q}}^{\mathrm{d}}=\widehat{\mathrm{a}}+\widehat{\mathrm{b}}{\mathrm{P}}_{\mathrm{x}}+\mathrm{cM}\\ =58.87-1.64+7\end{array}$

Here, ${\mathrm{Q}}^{\mathrm{d}}$represents the demand firm’s product,$\widehat{\mathrm{a}}$ represents the coefficient’s intercept, $\widehat{\mathrm{b}}$represents the coefficient of price $\mathrm{X}$, and $\mathrm{C}$represents the coefficient’s income.

To find the statistically significant regression coefficients 

b.

Based on the table,the coefficient’s intercept is$58.87$ , and the coefficient’s income is$7$ .

Therefore, the t-statistics coefficient intercept and income are statistically significant.

Checking the fit of a regression line 

c.

Check if the regression line fits the data. We identify the value for $\mathrm{R}-\mathrm{square}$and the adjusted $\mathrm{R}-\mathrm{square}$—the greater F-statistic better fit into the regression line.

Usethe equationto identify the$\mathrm{R}-\mathrm{square}$.

${\mathrm{R}}^2=\frac{\mathrm{ExplainedVariation}}{\mathrm{TotalVariation}}=\frac{{\mathrm{SS}}_{\mathrm{Regression}}}{{\mathrm{SS}}_{\mathrm{Total}}}$

First, the value of the sum of squares is found using the equation asfollows:

${\mathrm{SS}}_{\mathrm{Total}}={\mathrm{SS}}_{\mathrm{Regression}}+{\mathrm{SS}}_{\mathrm{Residual}}$

Put the valuein the given equation:

$\begin{array}{c}{\mathrm{SS}}_{\mathrm{Total}}={\mathrm{SS}}_{\mathrm{Regression}}+{\mathrm{SS}}_{\mathrm{Residual}}\\ 73,807.49={\mathrm{SS}}_{\mathrm{Regression}}+63,408.62\\ {\mathrm{SS}}_{\mathrm{Regression}}=73,807.49-63,408.62\\ =10,398.87\end{array}$

Thus, the explained variation is$10,398.87$.

It is given that the explained variation is $10,398.87$, and the total variation is $73,807.49$.

$\begin{array}{c}{\mathrm{R}}^2=\frac{\mathrm{ExplainedVariation}}{\mathrm{TotalVariation}}=\frac{{\mathrm{SS}}_{\mathrm{Regression}}}{{\mathrm{SS}}_{\mathrm{Total}}}\\ =\frac{10,398.87}{73,807.49}\\ {\mathrm{R}}^2=0.141\end{array}$

Therefore, the value for ${\mathrm{R}}^2=0.141$.

We use the following equation to find the adjusted ${\mathrm{R}}^2$:

$\overline{{\mathrm{R}}^2}=1-(1-{\mathrm{R}}^2)\frac{(\mathrm{n}-1)}{(\mathrm{n}-\mathrm{k})}$

$\begin{array}{c}\begin{array}{l}\overline{{\mathrm{R}}^2}=1-(1-{\mathrm{R}}^2)\frac{(\mathrm{n}-1)}{(\mathrm{n}-\mathrm{k})}\\ \end{array}\\ =1-(1-0.141)\frac{(150-1)}{(150-3)}\\ =1-(0.859)\frac{149}{147}\\ =1-(0.859\times 1.014)\\ =1-0.871\\ \overline{{\mathrm{R}}^2}=0.129\end{array}$

Thus,based on the calculation,the adjusted value for ${\mathrm{R}}^2\mathrm{is}0.141$, andthe adjusted value for$\mathrm{R}-\mathrm{squareis}0.129.$.

The difference between the ${\mathrm{R}}^2$and the adjusted value for$\mathrm{R}-\mathrm{square}$ indicates that the regression line does not fit the data.