When using \(N=50\) observations to estimate the model \(y_{i}=\beta_{1}+\beta_{2} x_{i}+\beta_{3} z_{i}+e_{i}\), you obtain \(S S E=2132.65\) and \(s_{y}=9.8355\).

a. Find \(R^{2}\).

b. Find the value of the \(F\)-statistic for testing \(H_{0}: \beta_{2}=0, \beta_{3}=0\). Do you reject or fail to reject \(H_{0}\) at a \(1 \%\) level of significance?

c. After augmenting this model with the squares and cubes of predictions \(\hat{y}_{i}^{2}\) and \(\hat{y}_{i}^{3}\), you obtain \(S S E=1072.88\). Use RESET to test for misspecification at a \(1 \%\) level of significance.

d. After estimating the model \(y_{i}=\beta_{1}+\beta_{2} x_{i}+\beta_{3} z_{i}+\beta_{4} z_{i}^{2}+e_{i}\), you obtain \(S S E=401.179\). What is the \(R^{2}\) from estimating this model?

e. After augmenting the model in (d) with the squares and cubes of predictions \(\hat{y}_{i}^{2}\) and \(\hat{y}_{i}^{3}\), you obtain \(S S E=388.684\). Use RESET to test for misspecification at a \(5 \%\) level of significance.