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Question: Company donations to charity. The amount a company donates to a charitable organization is often restricted by financial inflexibility at the firm. One measure of financial inflexibility is the ratio of restricted assets to total firm assets. A study published in the Journal of Management Accounting Research (Vol. 27, 2015) investigated the link between donation amount and this ratio. Data were collected on donations to 115,333 charities over a recent 10-year period, resulting in a sample of 419,225 firm-years. The researchers fit the quadratic model,$\mathit{E}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{=}{\mathit{β}}_{\mathbf{0}}\mathbf{+}{\mathit{β}}_{\mathbf{1}}\mathit{x}\mathbf{+}{\mathit{β}}_{\mathbf{2}}{\mathit{x}}^{\mathbf{2}}$, where y = natural logarithm of total donations to charity by a firm in a year and x = ratio of restricted assets to the firm’s total assets in the previous year. [Note: This model is a simplified version of the actual model fit by the researchers.]

1. The researchers’ theory is that as a firm’s restricted assets increase, donations will initially increase. However, there is a point at which donations will not only diminish, but also decline as restricted assets increase. How should the researchers use the model to test this theory?
2. The results of the multiple regression are shown in the table below. Use this information to test the researchers’ theory at. What do you conclude?

Step by step solution

01

Testing the curvilinear relationship

The researchers’ theory is that as a firm’s restricted assets increase, donations will initially increase and after a point donation will decline as restricted assets increase. This relationship can be tested by drawing a scatterplot for the data. If the researchers’ theory is right, then the curve will slope upwards initially and then after a point, it will start moving downwards and the slope will become negative.

02

Significance of β2

To test the curvilinear relation amongst the variables, the significance of${\mathit{β}}_{\mathbf{2}}$is tested.

Here,${\mathit{H}}_{\mathbf{0}}\mathbf{}\mathbf{:}{\mathit{β}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$,whilelocalid="1649839803799" ${\mathit{H}}_a\mathbf{}\mathbf{:}{\mathit{β}}_{\mathbf{2}}\mathbf{≠}\mathbf{0}$

Here, t-test statistic

$\mathbf{=}\frac{\widehat{{\mathbf{β}}_{\mathbf{2}}}}{\mathbf{s}{\mathbf{β}}_{\mathbf{2}}}\mathbf{=}\frac{\mathbf{-}\mathbf{0}\mathbf{.}\mathbf{279}}{\mathbf{0}\mathbf{.}\mathbf{039}}\mathbf{=}\mathbf{-}\mathbf{7}\mathbf{.}\mathbf{1538}$

Value of ${\mathit{t}}_{\mathbf{0}\mathbf{.}\mathbf{025}\mathbf{,}\mathbf{419224}}$is 1.96

${\mathit{H}}_{\mathbf{0}}$is rejected if t statistic >${\mathit{t}}_{\mathbf{0}\mathbf{.}\mathbf{035}\mathbf{,}\mathbf{385}}$. For $\mathit{Î\pm }\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{05}$, since t<${\mathit{t}}_{\mathbf{0}\mathbf{.}\mathbf{05}\mathbf{,}\mathbf{199}}$

Not sufficient evidence to reject at 95% confidence interval.

Therefore,${\mathit{β}}_{\mathbf{2}}\mathbf{=}\mathbf{0}$

This means that the parabola doesn’t have a curvature and it essentially is a straight line.

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