Question

Can money spent on gifts buy love? Refer to the Journal of Experimental Social Psychology (Vol. 45, 2009) study of whether buying gifts truly buys love, Exercise 9.9 (p. 529). Recall those study participants were randomly assigned to play the role of gift-giver or gift-receiver. Gift-receivers were asked to provide the level of appreciation (measured on a 7-point scale where 1 = “not at all” and 7 = “to a great extent”) they had for the last birthday gift they received from a loved one. Gift-givers were asked to recall the last birthday gift they gave to a loved one and to provide the level of appreciation the loved one had for the gift.

1. Write a dummy variable regression model that will allow the researchers to compare the average level of appreciation for birthday gift-giverswith the average for birthday gift-receivers.
2. Express each of the model’s β parameters in terms ofand.
3. The researchers hypothesize that the average level of appreciation is higher for birthday gift-givers than for birthday gift-receivers. Explain how to test this hypothesis using the regression model.

Short answer

1. The dummy variable regression model for appreciation for birthday gift-givers and birthday gift-receivers can be written as
2. anddenotes the difference between the mean levels for different dummy variables. Here,denotes the base level for mean when both the variablesandare 0. This means thatwhile
3. $$\begin{gathered} {\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0} \\ {\mathbf{H}}_{\mathbf{a}}\mathbf{:}\mathbf{}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{>}{\mathbf{\beta }}_{\mathbf{2}} \end{gathered}$$

Here, the null hypothesis becomes that the means for the two groups are equal meaningwhile the alternate hypothesis implies that the means ( and ) differ, specifically mean appreciation for birthday gift-givers is higher than for birthday gift-receivers.

Step-by-step solution

Dummy variable regression model

The dummy variable regression model for appreciation for birthday gift-givers $\left({\mu }_G\right)$and birthday gift-receivers $\left({\mu }_R\right)$can be written as $\mathbf{y}\mathbf{=}{\mathbf{\beta }}_{\mathbf{0}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{\beta }}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{\epsilon }$Where, ${\mathbf{x}}_{\mathbf{1}}$denotes appreciation for birthday gift-givers $\left({\mu }_G\right)$and ${\mathbf{x}}_{\mathbf{2}}$denotes appreciation by gift-receivers. The value both ${\mathbf{x}}_{\mathbf{1}}$and ${\mathbf{x}}_{\mathbf{2}}$can take are 0 or 1; 1 for the presence of appreciation they had felt and 0 otherwise.

Interpretation of β's

${\mathbf{\beta }}_{\mathbf{1}}$and ${\mathbf{\beta }}_{\mathbf{2}}$denotes the difference between the mean levels for different dummy variables.

Here, ${\mathbf{\beta }}_{\mathbf{0}}$denotes${\mathbf{x}}_{\mathbf{1}}$ the base level for mean when both the variables${\mathbf{x}}_1$ and ${\mathbf{x}}_{\mathbf{2}}$are 0.

This means that${\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\mu }}_{\mathbf{G}}\mathbf{-}{\mathbf{\mu }}_{\mathbf{A}}$ while${\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}{\mathbf{\mu }}_{\mathbf{R}}\mathbf{-}{\mathbf{\mu }}_{\mathbf{A}}$

Hypothesis testing

$$\begin{gathered} {\mathbf{H}}_{\mathbf{0}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{=}{\mathbf{\beta }}_{\mathbf{2}}\mathbf{=}\mathbf{0} \\ {\mathbf{H}}_{\mathbf{a}}\mathbf{:}{\mathbf{\beta }}_{\mathbf{1}}\mathbf{>}{\mathbf{\beta }}_{\mathbf{2}} \end{gathered}$$

Here, the null hypothesis becomes that the means for the two groups are equal meaning ${\mathbf{\mu }}_{\mathbf{1}}\mathbf{=}{\mathbf{\mu }}_{\mathbf{2}}$while the alternate hypothesis implies that the means (${\mathbf{\mu }}_{\mathbf{1}}$and ${\mathbf{\mu }}_{\mathbf{2}}$) differ, specifically mean appreciation for birthday gift-givers is higher than for birthday gift-receivers.