E)
① $H_0: \gamma_2 = \gamma_3 = 0$
$H_1: \text{At least one } \gamma_j \neq 0, j = 2,3$

② Significance level
$\alpha = 0.05$

③ Estimators / Statistics
$\hat{\gamma}_2, \hat{\gamma}_3, R_{\mathrm{UR}}^2, R_{\mathrm{R}}^2$

④ Assumptions

$\operatorname{Cov}(\hat{\gamma}_1, u_1) = 0$ or $\hat{\gamma}_3$ is fixed in repeated sampling.
$u_i \sim \mathrm{NIID}(0, \hat{\gamma}_2)$ or $\hat{\gamma}_1$ is large
$E(u_i) = 0$
$V(u_i) = \hat{\gamma}_2$
$\operatorname{Cov}(u_i, u_j) = 0 \quad i \neq j$
No exact multicollinearity

(Cont.)

⑤ Test statistic

F_{obs} = \frac{(R_{re}^2 - R_{R}^2)/m}{(1 - R_{re}^2)/(n-k)} \sim F_m, n-k,

$R_{re}^2$ = coefficient of determination for the unrestricted model
$R_{R}^2$ = coefficient of determination for the restricted model
$m$ = no. of linear restrictions
$n$ = no. of observations
$k$ = no. of regression coefficients in the regression line of the unrestricted model, including the intercept.

⑥ Figure under the null and decision rule

\text{Reject } H_0 \text{ if } F_{obs} \> F_2, 60-3 \approx F_{2,40} = 3.23

→ F-value not in table, so we take a conservative approximation.

⑦ $C_{R^2} = 0.5628$ $m = 2$ $n = 60$ $R^2 = 0$ $k = 3$

→ No explanatory variables

F_{obs} = \frac{(0.5628 - 0)/2}{(1 - 0.5628)/(60 - 3)} \approx 36.6876 F_{obs} = 36.6876 \> 3.23 = \text{Critical value} \rightarrow \text{Reject } H_0

⑧ At 5% significance we reject the null hypothesis that both $\beta_2$ and $\beta_3$ are equal to zero, in favor of the alternative hypothesis that at least one of them is not zero.