3. Mary want to estimates a regression model as below:
$y_i = \alpha + \beta_1 x_{1i} + \beta_2 x_{2i} + \varepsilon_i,$
If she doubts the regression residuals $\hat{\varepsilon}_i$ with heteroskedasticity, she wants to use Breusch-Pagan test to detect whether heteroskedasticity exists in $\hat{\varepsilon}_i^2$. Which independent variables set she will choose to execute Breusch-Pagan test?
(A) \( x_{1i}, x_{2i} \)
(B) \( x_{n}, x_{2n}, x_{1n}^{2}, x_{2n}^{2} \)
(C) \( x_{n}, x_{2n}, x_{1n}^{2}, x_{2n}^{2}, x_{1n} x_{2n} \)
(D) \( x_{n}, x_{2n} \)
(E) \( x_{n}, x_{2n}, x_{1n} x_{2n} \)

4. Consider the simple regression $y_i = \alpha + \beta x_i + \varepsilon_i$And suppose population parameter $\beta = 2$ and least square estimator $\hat{\beta} = 2.3$. Now consider the regression $(x_i - y_i) = \alpha^* + \beta^* x_i + u_i.$What is the value of \( \hat{\beta} \)? (A) 1 (B)-1 (C)-1.3 (D)-0.3 (E) 0.3

5. Consider the linear model with intercept: $y_t = \alpha + \beta x_t + u_t$ If the $A = \sum_{i=1}^{n} (x_i - \bar{x})^2$ is positive definite and $E(u_i \mid x_i) = 0, \forall t = 1, 2, \dots, T$, which statement is true under sample size $T \to \infty$. (A) \( \hat{\beta} \) is consistent, but we cannot judge whether \( \hat{\beta} \) is efficiency (B) \( \hat{\beta} \) is unbiased and efficiency (C) \( \hat{\beta} \) is consistent and efficiency (D) \( \hat{\beta} \) is, but we cannot judge whether \( \hat{\beta} \) is efficiency (E) \( \hat{\beta} \) is unbiased and consistent.

6. Consider the linear population model with intercept:$y_t = \alpha + \beta x_t + \varepsilon_t$The variance structure is known as below: $E(\varepsilon_t^2) = \sigma^2 h_t^2,$Which regression may obtain efficiency $\hat{\beta}$ estimator? (A) Regress \( y_{t} \) on 1, \( x_{t} \) (B) Regress \( y_{t} / h_{t} \) on 1/ht ,xt/ht(C) Regress \( y_{t} / h_{t}^{2} \) on 1/h2t ,xt/h2t(D) Regress \( y_{t} \) on 1/ht ,xt/ht(E) Regress \( y_{t} / h_{t} \) on 1, \( x_{t} \)

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