Question

Question: Write a regression model relating E(y) to a qualitative independent variable that can assume three levels. Interpret all the terms in the model.

Short answer

Answer:

The regression model for one qualitative independent variable with two-level is$\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{3}}$ .Here${\mathit{\beta }}_{\mathbf{0}}$denotes$\mathit{\mu }\mathit{x}\mathit{i}$the mean for base level and role="math" localid="1649848834273" ${\mathit{\beta }}_{\mathbf{1}}\mathbf{,}{\mathit{\beta }}_{\mathbf{2}}$_, and ${\mathit{\beta }}_{\mathbf{3}}$denotes the difference between the mean levels for different dummy variables. While,${\mathit{x}}_{\mathbf{1}}\mathbf{,}{\mathit{x}}_{\mathbf{2}}$ , and${\mathit{x}}_3$ denotes the dummy variables used in the model which can take the value of either 0 or 1.

Step-by-step solution

Regression model for one qualitative independent variable

A regression model relating the mean value of y to a qualitative independent variable that can assume two levels can be written as

$\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{\beta }}_{\mathbf{0}}\mathbf{+}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{3}}$

Where,${\mathit{x}}_{\mathbf{1}}$ is the dummy variable for$\mathbf{\text{i+1level}}$

Meaning${\mathit{x}}_{\mathbf{1}}\mathbf{=}\left(\begin{array}{c}1ifyisobservedatleveli+1\\ 00therwise\end{array}\right)$

Interpretation of the terms in the model

Here ${\mathit{\beta }}_{\mathbf{0}}$denotes $\mathit{\mu }\mathit{x}\mathit{i}$ the mean for base level and _{${\mathit{\beta }}_{\mathbf{1}}\mathbf{,}\mathbf{}{\mathit{\beta }}_{\mathbf{2}}$} and ${\mathit{\beta }}_{\mathbf{3}}$denotes the difference between the mean levels for different dummy variables. While,${\mathit{x}}_{\mathbf{1}}\mathbf{,}\mathbf{}{\mathit{x}}_{\mathbf{2}}$ , and${\mathit{x}}_{\mathbf{3}}$denotes the dummy variables used in the model which can take the value of either 0 or 1.