Question

Question: Do blondes raise more funds? Refer to the Economic Letters (Vol. 100, 2008) study of whether the color of a female solicitor’s hair impacts the level of capital raised, Exercise 12.75 (p. 756). Recall that 955 households were contacted by a female solicitor to raise funds for hazard mitigation research. In addition to the household’s level of contribution (in dollars) and the hair color of the solicitor (blond Caucasian, brunette Caucasian, or minority female), the researcher also recorded the beauty rating of the solicitor (measured quantitatively, on a 10-point scale).

1. Write a first-order model (with no interaction) for mean contribution level, E(y), as a function of a solicitor’s hair color and her beauty rating.
2. Refer to the model, part a. For each hair color, express the change in contribution level for each 1-point increase in a solicitor’s beauty rating in terms of the model parameters.
3. Write an interaction model for mean contribution level, E(y), as a function of a solicitor’s hair color and her beauty rating.
4. Refer to the model, part c. For each hair color, express the change in contribution level for each 1-point increase in a solicitor’s beauty rating in terms of the model parameters.
5. Refer to the model; part c. Illustrate the interaction with a graph.

Short answer

Answer

1. A first-order model equation in one quantitative variable which is solicitor’s beauty rating here (say x₁) and one qualitative variable which is solicitors’ hair color here with 3 levels (say variables introduced are (k-1) = 2;and x₃ can be written as${\mathbf{\text{E(y)=β}}}_{\mathbf{0}}\mathbf{\text{+}}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{3}}$
2. For minority female the 1-point increase in solicitor’s beauty rating will be represented by${\mathbf{\beta }}_{\mathbf{1}}$_. For blond Caucasian hair color of the solicitor, the 1-point increase in solicitor’s beauty rating will be represented by$\mathbf{(}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{)}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$. For brunette Caucasian hair color of the solicitor, the 1-point increase in solicitor’s beauty rating will be represented byrole="math" localid="1651142480646" $\mathbf{(}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$_.
3. An interaction model for mean contribution level, E(y) as a function of a solicitor’s hair color and beauty rating can be written as.$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{3}}$
4. For minority female the 1-point increase in solicitor’s beauty rating will be represented by_. For blond Caucasian hair color of the solicitor, the 1-point increase in solicitor’s beauty rating will be represented by. For brunette Caucasian hair color of the solicitor, the 1-point increase in solicitor’s beauty rating will be represented by$\mathbf{(}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{+}\mathbf{(}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}\mathbf{)}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$_.
5. Graph

Step-by-step solution

First-order model equation

A first-order model equation in one quantitative variable which is solicitor’s beauty rating here (say x₁) and one qualitative variable which is solicitors’ hair color here with 3 levels (say variables introduced are (k-1) = 2;x₂and x₃) can be written as${\mathbf{\text{E(y)=β}}}_{\mathbf{0}}\mathbf{\text{+}}{\mathit{\beta }}_{\mathbf{1}}{\mathit{x}}_{\mathbf{1}}\mathbf{+}{\mathit{\beta }}_{\mathbf{2}}{\mathit{x}}_{\mathbf{2}}\mathbf{+}{\mathit{\beta }}_{\mathbf{3}}{\mathit{x}}_{\mathbf{3}}$

.

Changes in β

Assume thatx₂= solicitors’ hair color is blond Caucasian and x₃= solicitors’ hair color is brunette Caucasian

The base level for hair is assumed to be minority female.

Therefore, for minority female the 1-point increase in solicitor’s beauty rating will be represented by${\mathit{\beta }}_{\mathbf{1}}$

For blond Caucasian hair color of the solicitor, the 1-point increase in solicitor’s beauty rating will be represented by$\mathbf{(}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{)}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$

For brunette Caucasian hair color of the solicitor, the 1-point increase in solicitor’s beauty rating will be represented by$\mathbf{(}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$

 Step 3: Interaction model

An interaction model for mean contribution level, E(y) as a function of a solicitor’s hair color and beauty rating can be written as$\mathbf{E}\mathbf{\left(}\mathbf{y}\mathbf{\right)}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}{\mathbf{x}}_{\mathbf{3}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{4}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{5}}{\mathbf{x}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{3}}$

Interconnection model and interpretation of β 

Assume that x₂ = solicitors’ hair color is blond Caucasian and x₃= solicitors’ hair color is Brunette Caucasian

The base level for hair is assumed to be minority female.

Therefore, for minority female the 1-point increase in solicitor’s beauty rating will be represented by${\mathbf{\beta }}_{\mathbf{1}}$

For blond Caucasian hair color of the solicitor, the 1-point increase in solicitor’s beauty rating will be represented by$\mathbf{(}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{)}\mathbf{+}\mathbf{(}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_4\mathbf{)}{\mathbf{x}}_{\mathbf{1}}$

For brunette Caucasian hair color of the solicitor, the 1-point increase in solicitor’s beauty rating will be represented byrole="math" localid="1651143179432" $\mathbf{(}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{3}}\mathbf{)}\mathbf{+}\mathbf{(}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_5\mathbf{)}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}$

Graph