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Excel

A relatively common transformation of your independent variables (x's) is adding the term x? to your model. This transformation takes the following form:

y_{i}=\hat{\beta}_{0}+\hat{\beta}_{1} x_{i}+\hat{\beta}_{2} x_{i}^{2}+\hat{\epsilon}_{i}

а. [3 Points] Why can you include both x; and x? in the model? That is, why does this not cause the model to have perfect multicollinearity?

b. Draw a graph with x on the x-axis and y =f(x) on the y-axis. Graph the relationship between

y=f(x)=\hat{\beta}_{0}+\hat{\beta}_{1} x+\hat{\beta}_{2} x^{2} \text { and } y \text { under the following scerarios: }

\text { I. }[3 \text { Points }] \hat{\boldsymbol{\beta}}_{1}, \hat{\boldsymbol{\beta}}_{2}>\mathbf{0}

\text { II. } \quad[3 \text { Points }] \hat{\boldsymbol{\beta}}_{1}>0, \hat{\beta}_{2}<0

\text { III. } \quad[3 \text { Points }] \hat{\boldsymbol{\beta}}_{1}<\mathbf{0}, \hat{\boldsymbol{\beta}}_{2}>\mathbf{0}

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