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 \( y_{t} / h_{t}, x_{t} / h_{t} \) (C) Regress \( y_{t} / h_{t}^{2} \) on \( y_{t} / h_{t}^{2}, x_{t} / h_{t}^{2} \) (D) Regress \( y_{t} \) on \( y_{t} / h_{t}, x_{t} / h_{t} \) (E) Regress \( y_{t} / h_{t} \) on 1, \( x_{t} \)

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