Consider the Model Where $x _ { 1 }$ Is a Quantitative Variable And

question 88

Multiple Choice

Consider the model
$$y = \beta _ { 0 } + \beta _ { 1 } x _ { 1 } + \beta _ { 2 } x _ { 1 } ^ { 2 } + \beta _ { 3 } x _ { 2 } + \beta _ { 4 } x _ { 3 } + \beta _ { 5 } x _ { 1 } x _ { 2 } + \beta _ { 6 } x _ { 1 } x _ { 3 } + \beta _ { 7 } x _ { 1 } 2 _ { x _ { 2 } } + \beta _ { 8 } x _ { 1 } x _ { 3 } + \varepsilon$$
where $x _ { 1 }$ is a quantitative variable and $x _ { 2 }$ and $x _ { 3 }$ are dummy variables describing a qualitative variable at three levels using the coding scheme
$$x _ { 2 } = \left\{ \begin{array} { l l } 1 & \text { if level } 2 \\0 & \text { otherwise }\end{array} \quad x _ { 3 } = \left\{ \begin{array} { l l } 1 & \text { if level } 3 \\0 & \text { otherwise }\end{array} \right. \right.$$
The resulting least squares prediction equation is
$$\hat { y } = 8.8 - 1.1 x _ { 1 } + 3.2 x _ { 1 } ^ { 2 } + 1.6 x _ { 2 } - 4.4 x _ { 3 } + .02 x _ { 1 } x _ { 2 } + 1.3 x _ { 1 } x _ { 3 } + .01 x _ { 1 } 2 ^ { 2 } - .06 x _ { 1 } x _ { 3 }$$
What is the equation of the response curve for $E ( y )$ when $x _ { 2 } = 0$ and $x _ { 3 } = 0$ ?

• A) \(\hat { y } = 8.8 - 1.1 x _ { 1 } + 3.2 x _ { 1 } { } ^ { 2 }\)
• B) \(\hat { y } = 8.8 - 1.3 x _ { 1 } + 3.2 x _ { 1 } ^ { 2 }\)
• C) \(\hat { y } = 8.8 - 1.6 x _ { 2 } - 4.4 x _ { 3 }\)
• D) \(\hat { y } = 8.8 - .22 x _ { 1 } + 3.15 x _ { 1 } 2\)

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Definitions:

Sulfur Dioxide

A colorless gas with a sharp smell, produced by burning fossil fuels and volcanic eruptions, often associated with air pollution and acid rain.

Marginal Social Cost

The total cost to society of producing one additional unit of a good or service, including both the private costs and any external costs.

Pigouvian Taxes

Taxes designed to reduce external costs.

External Costs

costs inflicted upon a third party when goods and services are produced and consumed.