Question

Question: Impact of race on football card values. Refer to the Electronic Journal of Sociology (2007) study of the Impact of race on the value of professional football players’ “rookie” cards, Exercise 12.72 (p. 756). Recall that the sample consisted of 148 rookie cards of NFL players who were inducted into the Football Hall of Fame (HOF). The researchers modelled the natural logarithm of card price (y) as a function of the following independent variables:

$$\begin{gathered} \mathbf{\text{Race:}}{\mathit{x}}_{\mathbf{1}}\mathbf{=}\mathbf{1}\mathit{i}\mathit{f}\mathit{b}\mathit{l}\mathit{a}\mathit{c}\mathit{k}\mathbf{,}\mathbf{0}\mathit{i}\mathit{f}\mathit{w}\mathit{h}\mathit{i}\mathit{t}\mathit{e} \\ \mathbf{\text{Cardavailability:}}{\mathit{x}}_{\mathbf{2}}\mathbf{=}\mathbf{1}\mathit{i}\mathit{f}\mathit{h}\mathit{i}\mathit{g}\mathit{h}\mathbf{,}\mathbf{0}\mathit{i}\mathit{f}\mathit{l}\mathit{o}\mathit{w} \\ \mathbf{\text{Cardvintage:}}{\mathit{x}}_{\mathbf{3}}\mathbf{=}\mathit{y}\mathit{e}\mathit{a}\mathit{r}\mathit{c}\mathit{a}\mathit{r}\mathit{d}\mathit{p}\mathit{r}\mathbf{int}\mathit{e}\mathit{d} \\ \mathbf{\text{Finalist:}}{\mathbf{x}}_{\mathbf{4}}\mathbf{=}\mathbf{naturallogarithmof}\mathbf{ }\mathbf{numberoftimesplayeron}\mathbf{ }\mathbf{finalHOFballot} \\ \mathbf{Position}\mathbf{-}\mathbf{QB}\mathbf{:}\mathbf{\text{:}}{\mathit{x}}_{\mathbf{5}}\mathbf{=}\mathbf{1}\mathit{i}\mathit{f}\mathit{q}\mathit{u}\mathit{a}\mathit{r}\mathit{t}\mathit{e}\mathit{r}\mathit{b}\mathit{a}\mathit{c}\mathit{k}\mathbf{,}\mathbf{0}\mathit{i}\mathit{f}\mathit{n}\mathit{o}\mathit{t} \\ \mathbf{\text{Position-RB:}}{\mathit{x}}_{\mathbf{7}}\mathbf{=}\mathbf{1}\mathit{i}\mathit{f}\mathit{r}\mathit{u}\mathit{n}\mathit{n}\mathit{i}\mathit{n}\mathit{g}\mathit{b}\mathit{a}\mathit{c}\mathit{k}\mathbf{,}\mathbf{0}\mathit{i}\mathit{f}\mathit{n}\mathit{o}\mathit{t} \\ \mathbf{Position}\mathbf{-}\mathbf{WR}\mathbf{\text{:}}{\mathit{x}}_{\mathbf{8}}\mathbf{=}\mathbf{1}\mathit{i}\mathit{f}\mathit{w}\mathit{i}\mathit{d}\mathit{e}\mathit{r}\mathit{e}\mathit{c}\mathit{e}\mathit{i}\mathit{v}\mathit{e}\mathit{r}\mathbf{,}\mathbf{0}\mathit{i}\mathit{f}\mathit{n}\mathit{o}\mathit{t}\mathbf{} \\ \mathbf{Position}\mathbf{-}{\mathbf{TEx}}_{\mathbf{9}}\mathbf{=}\mathbf{1}\mathbf{iftightend}\mathbf{,}\mathbf{0}\mathbf{ifnot} \\ \mathbf{\text{Position-DL:}}{\mathit{x}}_{\mathbf{10}}\mathbf{=}\mathbf{1}\mathit{i}\mathit{f}\mathit{d}\mathit{e}\mathit{f}\mathit{e}\mathit{n}\mathit{s}\mathit{i}\mathit{v}\mathit{e}\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathit{m}\mathit{a}\mathit{n}\mathbf{,}\mathbf{0}\mathit{i}\mathit{f}\mathit{n}\mathit{o}\mathit{t} \\ \mathbf{\text{Position-LB:}}{\mathit{x}}_{\mathbf{11}}\mathbf{=}\mathbf{1}\mathit{i}\mathit{f}\mathit{l}\mathit{i}\mathit{n}\mathit{e}\mathit{b}\mathit{a}\mathit{c}\mathbf{ker}\mathbf{,}\mathbf{0}\mathit{i}\mathit{f}\mathit{n}\mathit{o}\mathit{t}\mathbf{} \\ \mathbf{\text{Position-DB:}}{\mathit{x}}_{\mathbf{12}}\mathbf{=}\mathbf{1}\mathit{i}\mathit{f}\mathit{d}\mathit{e}\mathit{f}\mathit{e}\mathit{n}\mathit{s}\mathit{i}\mathit{v}\mathit{e}\mathit{b}\mathit{a}\mathit{c}\mathit{k}\mathbf{,}\mathbf{0}\mathit{i}\mathit{f}\mathit{n}\mathit{o}\mathit{t} \end{gathered}$$

[Note: For position, offensive lineman is the base level.]

1. The model $\mathit{E}\left(y\right)\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{3}}\mathbf{+}{\mathit{\beta }}_{\mathbf{4}}{\mathit{x}}_{\mathbf{4}}\mathbf{+}{\mathit{\beta }}_{\mathbf{5}}{\mathit{x}}_{\mathbf{5}}\mathbf{+}{\mathit{\beta }}_{\mathbf{6}}{\mathit{x}}_{\mathbf{6}}\mathbf{+}{\mathit{\beta }}_{\mathbf{7}}{\mathit{x}}_{\mathbf{7}}\mathbf{+}{\mathit{\beta }}_{\mathbf{8}}{\mathit{x}}_{\mathbf{8}}\mathbf{+}{\mathit{\beta }}_{\mathbf{9}}{\mathit{x}}_{\mathbf{9}}\mathbf{+}{\mathit{\beta }}_{\mathbf{10}}{\mathit{x}}_{\mathbf{10}}\mathbf{+}{\mathit{\beta }}_{\mathbf{11}}{\mathit{x}}_{\mathbf{11}}\mathbf{+}{\mathit{\beta }}_{\mathbf{12}}{\mathit{x}}_{\mathbf{12}}$ was fit to the data with the following results:${\mathbf{R}}^{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{705}\mathbf{,}{{\mathbf{R}}_{\mathbf{a}}}^{\mathbf{2}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{681}\mathbf{,}\mathit{F}\mathbf{=}\mathbf{26}\mathbf{.}\mathbf{9}\mathbf{.}$Interpret the results, practically. Make an inference about the overall adequacy of the model.
2. Refer to part a. Statistics for the race variable were reported as follows:${\hat{\mathbf{\beta }}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{147}\mathbf{,}\mathit{s}{\hat{\mathbf{\beta }}}_{\mathbf{1}}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{145}\mathbf{,}\mathit{t}\mathbf{=}\mathbf{-}\mathbf{1}\mathbf{.}\mathbf{014}\mathbf{,}\mathit{p}\mathbf{-}\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{312}$ .Use this information to make an inference about the impact of race on the value of professional football players’ rookie cards.
3. Refer to part a. Statistics for the card vintage variable were reported as follows:${\hat{\mathbf{\beta }}}_{\mathbf{3}}\mathbf{=}\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{074}\mathbf{,}\mathit{s}{\hat{\mathbf{\beta }}}_{\mathbf{3}}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{007}\mathbf{,}\mathit{t}\mathbf{=}\mathbf{-}\mathbf{10}\mathbf{.}\mathbf{92}\mathbf{,}\mathit{p}\mathbf{-}\mathit{v}\mathit{a}\mathit{l}\mathit{u}\mathit{e}\mathbf{=}\mathbf{.}\mathbf{000}$.Use this information to make an inference about the impact of card vintage on the value of professional football players’ rookie cards.
4. Write a first-order model for E(y) as a function of card vintage $\left({\mathbf{x}}_{\mathbf{3}}\right)$and position $\left({\mathbf{x}}_{\mathbf{5}}\mathbf{-}{\mathbf{x}}_{\mathbf{12}}\right)$that allows for the relationship between price and vintage to vary depending on position.

Short answer

1. The value of $R_{}^2$ is 0.705 indicating that nearly 70% of variation in the data is explained by the model. The value of 68.1% for $R_a^2$ indicates that the variables are explaining the model to a higher degree. At 95% significance level, it can be concluded thatthe model is not a good fit for the data.
2. At95% significance level,${\beta }_1=0$ . Hence it can be concluded with enough evidence that x₁ is a significance variable.
3. At 95% significance level, ${\beta }_3=0$. Hence it can be concluded with enough evidence that x₂ is a significance variable.
4. The model is:
5. $$\begin{gathered} E\left(y\right)={\beta }_0+{\beta }_1{x}_3+{\beta }_2{x}_5+{\beta }_3{x}_6+{\beta }_4{x}_7+{\beta }_5{x}_8+{\beta }_6{x}_9 \\ +{\beta }_7{x}_{10}+{\beta }_8{x}_{11}+{\beta }_9{x}_{12}+{\beta }_{10}{x}_3{x}_5+{\beta }_{11}{x}_3{x}_6+{\beta }_{12}{x}_3{x}_7 \\ +{\beta }_{13}{x}_3{x}_8+{\beta }_{14}{x}_3{x}_9+{\beta }_{15}{x}_3{x}_{10}+{\beta }_{16}{x}_3{x}_{11}+{\beta }_{17}{x}_3{x}_{12} \end{gathered}$$

.

Step-by-step solution

Given Information

The model is given as-$E\left(y\right)={\beta }_0+{\beta }_1{x}_1+{\beta }_2{x}_2+{\beta }_3{x}_3+{\beta }_4{x}_4+{\beta }_5{x}_5+{\beta }_6{x}_6+{\beta }_7{x}_7+{\beta }_8{x}_8+{\beta }_9{x}_9+{\beta }_{10}{x}_{10}+{\beta }_{11}{x}_{11}+{\beta }_{12}{x}_{12}$

Where , ${R_{}}^2=0.705and{R_a}^2=0.681$ .

The statistics for the race variable are given as${\hat{\mathrm{\beta }}}_1=-0.147,s{\hat{\mathrm{\beta }}}_1=-0.145,t=-1.014,p-value=0.312$ whereas the statistics for the card vintage variable are given as ${\hat{\mathrm{\beta }}}_3=-0.074,s{\hat{\mathrm{\beta }}}_3=0.007,t=-10.92,p-value=.000$ .

Interpretation of results

The results got from the model were:${\mathrm{R}}^2=0.705,{{\mathrm{R}}_{\mathrm{a}}}^2=0.681,F=26.9.$ .

Here, the value of R² is 0.705 indicating that nearly 70% of variation in the data is explained by the model which is very good number. This denotes that the model is a good fit for the data.

Value of $,{{\mathrm{R}}_{\mathrm{a}}}^2=0.681,$ which adjusts for the added variables and checks if the added variables is explaining the variation in the model or not. The value of 68.1% indicates that the variables are explaining the model to a higher degree.

F-test statistic value is 26.9, to check the overall adequacy of the model, we conduct the f-test.

$H_0\text{:}{\beta }_1={\beta }_2={\beta }_3=0$

H_a :At least one of the parameters ${\beta }_1,{\beta }_2,{\beta }_3$ is non zero.

H₀:is rejected if F statistic $>F_{0.025,147,147}$.

For $\alpha =0.025,F_{0.025,147,147}=1.311$. Since , there is sufficient evidence to reject

Hence the model is not a good fit for the data.

 Step 3: Significance of  β1

b.

Here,

$$\begin{gathered} H_0\text{:}{\beta }_1=0 \\ {H}_a\text{:}{\beta }_1\ne 0 \end{gathered}$$

$$\begin{gathered} t-\text{test statistic}=\frac{{\hat{\beta }}_1}{s{\hat{\beta }}_1} \\ =\frac{-0.147}{0.145} \\ =-1.0137 \end{gathered}$$

Value of $t_{0.025,147}$is 1.98

H_o: is rejected if t statistic$>t_{0.05,24,24}$ .

For $\alpha =0.025$, since $t<t_{0.025,147}$

Not sufficient evidence to reject at 95% confidence interval.

Therefore, ${\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}\mathbf{0}$. Hence it can be concluded with enough evidence that x₁ is a significant variable.

Significance of  β2 

c.

$$\begin{gathered} H_0\text{:}{\beta }_3=0 \\ {H}_a\text{:}{\beta }_3\ne 0 \end{gathered}$$

Here,

$$\begin{gathered} t-\text{test} \text{statistic}=\frac{{\hat{\beta }}_3}{s{\hat{\beta }}_3} \\ =\frac{-0.074}{0.007} \\ =-10.57 \end{gathered}$$

Value of $t_{0.025,147}$ is 1.98

H₀ is rejected if t statistic $>t_{0.05,24,24}$. For $\alpha =0.025$, since $t<t_{0.025,147}$

Not sufficient evidence to reject H₀ at 95% confidence interval.

Therefore,${\mathit{\beta }}_{\mathbf{3}}\mathbf{=}\mathbf{0}$. Hence it can be concluded with enough evidence that x₃ is a significant variable.

Model equation

d.

To write a first-order model for E(y) as a function of card vintage $\left(x_3\right)$ and position $\left(x_5-x_{12}\right)$ that allows the relationship between price and vintage to vary depending on the position can be expressed by an interaction model.

The equation can be written as:

.$$\begin{gathered} E\left(y\right)={\beta }_0+{\beta }_1{x}_3+{\beta }_2{x}_5+{\beta }_3{x}_6+{\beta }_4{x}_7+{\beta }_5{x}_8+{\beta }_6{x}_9 \\ +{\beta }_7{x}_{10}+{\beta }_8{x}_{11}+{\beta }_9{x}_{12}+{\beta }_{10}{x}_3{x}_5+{\beta }_{11}{x}_3{x}_6+{\beta }_{12}{x}_3{x}_7 \\ +{\beta }_{13}{x}_3{x}_8+{\beta }_{14}{x}_3{x}_9+{\beta }_{15}{x}_3{x}_{10}+{\beta }_{16}{x}_3{x}_{11}+{\beta }_{17}{x}_3{x}_{12} \end{gathered}$$