Question

Impact of race on football card values. University of Colorado sociologists investigated the impact of race on the value of professional football players’ “rookie” cards (Electronic Journal of Sociology, 2007). The sample consisted of 148 rookie cards of National Football League (NFL) players who were inducted into the Football Hall of Fame. The price of the card (in dollars) was modeled as a function of several qualitative independent variables: race of player (black or white), card availability (high or low), and player position (quarterback, running back, wide receiver, tight end, defensive lineman, linebacker, defensive back, or offensive lineman).

1. Create the appropriate dummy variables for each of the qualitative independent variables.
2. Write a model for price (y) as a function of race. Interpret the$\mathbf{\beta }$’s in the model.
3. Write a model for price (y) as a function of card availability. Interpret the$\mathbf{\beta }$’s in the model.
4. Write a model for price (y) as a function of position. Interpret the$\mathbf{\beta }$’s in the model.

Short answer

1. To represent the 3 qualitative independent, 9 dummy variables will be created.
2. A model for price of the card (y) as a function of the race of the player can be written as $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$where ${\mathbf{x}}_{\mathbf{1}}$represents the player’s race.
3. A model for the price of the card(y) as a function of card availability can be written as $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_2$where ${\mathbf{x}}_2$represents card availability.
4. A model for wine quality (y) as a function of the grape-picking method can be written as$\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{5}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{6}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{7}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{6}}{\mathbf{x}}_{\mathbf{8}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{7}}{\mathbf{x}}_{\mathbf{9}}$ where ${\mathbf{x}}_{\mathbf{4}}\mathbf{,}{\mathbf{x}}_{\mathbf{5}}\mathbf{,}{\mathbf{x}}_{\mathbf{6}}\mathbf{,}{\mathbf{x}}_{\mathbf{7}}\mathbf{,}{\mathbf{x}}_{\mathbf{8}}$and${\mathbf{x}}_{\mathbf{9}}$ represent the player position.

Step-by-step solution

Creating dummy variables

The qualitative independent variables here are the grape-picking method, soil type, and slope orientation.

Let${\mathbf{x}}_{\mathbf{1}}$be the race of the player, where${\mathbf{x}}_{\mathbf{1}}$= 1if the player is white and${\mathbf{x}}_{\mathbf{1}}$= 0if he is black

${\mathbf{x}}_{\mathbf{2}}$= card availability where value of${\mathbf{x}}_{\mathbf{2}}$= 1if card availability is high;${\mathbf{x}}_{\mathbf{2}}$= 0if card availability is low

Since player position has 8 categories, (k-1) = 7dummy variables will be introduced

role="math" localid="1649842591035" ${\mathbf{x}}_{\mathbf{3}}$= player position;${\mathbf{x}}_{\mathbf{3}}$= 1if player position is quarterback;${\mathbf{x}}_{\mathbf{3}}$= 0otherwise

${\mathbf{x}}_{\mathbf{4}}$= player position;${\mathbf{x}}_{\mathbf{4}}$= 1if player position is running back,${\mathbf{x}}_{\mathbf{4}}$= 0otherwise

${\mathbf{x}}_{\mathbf{5}}$= player position;${\mathbf{x}}_{\mathbf{5}}$= 1if player position is wide receiver,${\mathbf{x}}_{\mathbf{5}}$= 0otherwise

${\mathbf{x}}_{\mathbf{6}}$= player position; role="math" localid="1649842839417" ${\mathbf{x}}_{\mathbf{6}}$= 1if player position is tight end,${\mathbf{x}}_{\mathbf{6}}$= 0otherwise

${\mathbf{x}}_{\mathbf{7}}$= player position;${\mathbf{x}}_{\mathbf{7}}$= 1if player position is defensive lineman,${\mathbf{x}}_{\mathbf{7}}$= 0otherwise

${\mathbf{x}}_{\mathbf{8}}$= player position;${\mathbf{x}}_{\mathbf{8}}$= 1if player position is linebacker,${\mathbf{x}}_{\mathbf{8}}$= 0otherwise

${\mathbf{x}}_{\mathbf{9}}$= player position;${\mathbf{x}}_{\mathbf{9}}$= 1if player position is defensive back,${\mathbf{x}}_{\mathbf{9}}$= 0otherwise

Therefore, to represent the 3 qualitative independents, 9 dummy variables will be created.

Dummy variable model

A model for the price of the card (y) as a function of the race of the player can be written as$\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$ where${\mathbf{x}}_{\mathbf{1}}$ represents the player’s race

${\mathbf{\beta }}_{\mathbf{0}}$represents the price of the card(y) at a base level (here base level means the level when ${\mathbf{x}}_{\mathbf{1}}$= 0, meaning the race of the player is black)

${\mathbf{\beta }}_{\mathbf{1}}$represents the changes in the price of the card (y) when the race of the player is white.

Dolt variable imitation

A model for the price of the card(y) as a function of card availability can be written as$\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}$ where${\mathbf{x}}_{\mathbf{2}}$ represents card availability.

${\mathbf{\beta }}_{\mathbf{0}}$represents the price of the card (y) at a base level (here base level means the level when ${\mathbf{x}}_{\mathbf{2}}$= 0, meaning the card availability is low)

${\mathbf{\beta }}_{\mathbf{1}}$represents the changes in the price of the card (y) when the card availability is high.

Dunce variable representation

A model for wine quality (y) as a function of the grape-picking method can be written as$\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{4}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{5}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{6}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{7}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{6}}{\mathbf{x}}_{\mathbf{8}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{7}}{\mathbf{x}}_{\mathbf{9}}$where${\mathbf{x}}_{\mathbf{4}}\mathbf{,}{\mathbf{x}}_{\mathbf{5}}\mathbf{,}{\mathbf{x}}_{\mathbf{6}}\mathbf{,}{\mathbf{x}}_{\mathbf{7}}\mathbf{,}{\mathbf{x}}_{\mathbf{8}}$and${\mathbf{x}}_{\mathbf{9}}$represent the player position.

${\mathbf{\beta }}_{\mathbf{0}}$represents the price of the card at a base level (the base level taken here is when the player’s position is offensive lineman)

${\mathbf{\beta }}_{\mathbf{1}}$represents the changes in the price of the card (y) whena player’s position is quarterback.

${\mathbf{\beta }}_{\mathbf{2}}$represents the changes in the price of the card (y) when a player’s position is running back.

localid="1649843613800" ${\mathbf{\beta }}_{\mathbf{3}}$represents the changes in the price of the card (y) when a player’s position is a wide receiver.

${\mathbf{\beta }}_{\mathbf{4}}$represents the changes in the price of the card (y) when a player’s position is a tight end.

localid="1649843745574" ${\mathbf{\beta }}_{\mathbf{5}}$represents the changes in the price of the card (y) when a player position is a defensive lineman.

localid="1649843755924" ${\mathbf{\beta }}_{\mathbf{6}}$represents the changes in the price of the card (y) when a player position is a linebacker.

localid="1649843731417" ${\mathbf{\beta }}_{\mathbf{7}}$represents the changes in the price of the card (y) when a player’s position is a defensive back.