Question

Production technologies, terroir, and quality of Bordeaux wine. In addition to state-of-the-art technologies, the production of quality wine is strongly influenced by the natural endowments of the grape-growing region—called the “terroir.” The Economic Journal (May 2008) published an empirical study of the factors that yield a quality Bordeaux wine. A quantitative measure of wine quality (y) was modeled as a function of several qualitative independent variables, including grape-picking method (manual or automated), soil type (clay, gravel, or sand), and slope orientation (east, south, west, southeast, or southwest).

1. Create the appropriate dummy variables for each of the qualitative independent variables.
2. Write a model for wine quality (y) as a function of grape-picking method. Interpret the$\mathbf{\beta }$’s in the model.
3. Write a model for wine quality (y) as a function of soil type. Interpret the$\mathbf{\beta }$’s in the model.
4. Write a model for wine quality (y) as a function of slope orientation. Interpret the$\mathbf{\beta }$’s in the model.

Short answer

1. To represent the 3 qualitative independent, 7 dummy variables will be created.
2. A model for wine quality (y) as a function of the grape-picking method can be written as ${\mathbf{x}}_{\mathbf{6}}\mathbf{=}\mathbf{0}$where ${\mathbf{x}}_{\mathbf{1}}$represents the grape-picking method.
3. A model for wine quality (y) as a function of soil type can be written as $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_6\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{3}}$where localid="1649839735830" ${\mathbf{x}}_{\mathbf{2}}$and localid="1649839743019" ${\mathbf{x}}_{\mathbf{3}}$both represent soil type.
4. A model for wine quality (y) as a function of the grape-picking method can be written as $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{4}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{5}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{6}}$where ${\mathbf{x}}_{\mathbf{4}}\mathbf{}\mathbf{,}\mathbf{}{\mathbf{x}}_{\mathbf{5}}$and localid="1662363239978" ${\mathbf{x}}_{\mathbf{6}}$ represents slope orientation.

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 a grape-picking method, where ${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{1}$when it is manual and ${\mathbf{x}}_{\mathbf{1}}\mathbf{=}\mathbf{0}$when it is automated.

Since the soil type is categorized into three types, (k-1) = 2 no of dummy variables will be used

${\mathbf{x}}_{\mathbf{2}}$= soil type where value of ${\mathbf{x}}_{\mathbf{2}}$= 1 if soil type is clay; role="math" localid="1649840223066" ${\mathbf{x}}_{\mathbf{2}}$= 0 if soil type is gravel

role="math" localid="1649840195626" ${\mathbf{x}}_{\mathbf{3}}$= soil where value of role="math" localid="1649840208029" ${\mathbf{x}}_{\mathbf{3}}$= 1 if soil type is sand; role="math" localid="1649840215484" ${\mathbf{x}}_{\mathbf{3}}$= 0 if soil type is gravel

Similarly, slope orientation has 4 types hence (k-1) = 3dummy variables will be introduced in the model

${\mathbf{x}}_{\mathbf{4}}$= slope orientation where ${\mathbf{x}}_{\mathbf{4}}$= 1 if slope orientation is east; 0 otherwise

${\mathbf{x}}_{\mathbf{5}}$= slope orientation where ${\mathbf{x}}_{\mathbf{5}}$= 1 if slope orientation is west; 0 otherwise

${\mathbf{x}}_{\mathbf{6}}$= slope orientation where ${\mathbf{x}}_{\mathbf{6}}$= 1 if slope orientation is southeast; 0 otherwise

Therefore, to represent the 3 qualitative independent, 7 dummy variables will be created.

Dummy variable model

A model for wine quality (y) as a function of the grape-picking method can be written as $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$where ${\mathbf{x}}_{\mathbf{1}}$represents the grape-picking method.

${\mathbf{\beta }}_{\mathbf{0}}$represents the wine quality (y) at a base level (here base level means the level when ${\mathbf{x}}_{\mathbf{1}}$= 0, meaning the wine quality when the grapes are picked automatically)

${\mathbf{\beta }}_{\mathbf{1}}$ represents the changes in wine quality (y) when the grape-picking is manual.

Dolt variable imitation

A model for wine quality (y) as a function of soil type can be written as $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{3}}$where ${\mathbf{x}}_{\mathbf{2}}$and ${\mathbf{x}}_{\mathbf{3}}$both represent soil type.

${\mathbf{\beta }}_{\mathbf{0}}$represents the wine quality (y) at a base level (here base level means the level when ${\mathbf{x}}_{\mathbf{2}}$= 0 and ${\mathbf{x}}_{\mathbf{3}}$= 0, meaning the wine quality when the soil type is gravel)

${\mathbf{\beta }}_{\mathbf{1}}$represents the changes in wine quality (y) when the soil type is clay.

${\mathbf{\beta }}_{\mathbf{2}}$represents the changes in wine quality (y) when the soil type is sand.

Dunce variable representation

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

${\mathbf{\beta }}_{\mathbf{0}}$represents the wine quality (y) at a base level (here base level means the level when ${\mathbf{x}}_{\mathbf{4}}$= 0, ${\mathbf{x}}_{\mathbf{5}}$= 0, and ${\mathbf{x}}_{\mathbf{6}}$= 0 meaning the wine quality when the slope orientation is southwest)

${\mathbf{\beta }}_{\mathbf{1}}$represents the changes in wine quality (y) when the slope orientation is east.

${\mathbf{\beta }}_{\mathbf{2}}$represents the changes in wine quality (y) when the slope orientation is west.

${\mathbf{\beta }}_{\mathbf{3}}$represents the changes in wine quality (y) when the slope orientation is southeast.