2. Consider the simple linear regression model:

$$y_t = \beta y_{t-1} + \varepsilon_t,$$

with the error structure as below:

$$\varepsilon_t = \rho \varepsilon_{t-1} + u_t,$$

where $u_t \sim i.i.d(0, \sigma^2)$. What is the probability limit ($\rho$ lim) of OLS estimator $\hat{\beta}$? (2)
(A). \(\frac{\beta}{1 + \beta\rho}\)
(B). \(\beta\)
(C). \(\frac{\beta + \rho}{1 + \beta\rho}\)
(D). \(\frac{\rho}{1 + \beta\rho}\)
(E). \(\frac{1}{1 + \beta\rho}\)

3. Consider the model: $$y_t = \alpha + \rho x_t + \varepsilon_t,$$ where $t = 1, \dots, T$. If we want to obtain consistent estimator $\hat{\rho}$ for the large sample, which necessary condition it must be have? (3) (A). \(\operatorname{cov}(\varepsilon_t x_t) = 0\) (B). \( Var(\varepsilon_t \mid 1, x_t) = \sigma^2 \) (C). \( E(\varepsilon_t \mid 1, x_t) = 0 \) (D). \( E(\varepsilon_t \mid x_t) = 0 \) (E). \( E(\varepsilon_t\varepsilon_s|x_t,x_s) = 0 \)

4. An economist uses data to estimate the linear model with intercept $$y_t = \alpha + \beta_1 x_{1t} + \varepsilon_t,$$ but the true population model is as below: $$y_t = \alpha + \beta_1 x_{1t} + \beta_2 x_{2t} + u_t.$$ Under which condition satisfied, the OLS estimator $\hat{\beta}_1$ is still consistent? (4) (A). \(E(x_{2t}u_t) = 0\) (B). \(\alpha = 0\) (C). \(E(x_{2t}\varepsilon_t) = 0\) (D). \(E(x_{2t}|x_{1t}) = 0\) (E). \(E(x_{1t}u_t) = 0\)

5. Suppose that there is an exact relationship between two scalar random variable $y$ and $x$: $$y = \beta x + v$$ However, $y$ is not known and unobserved. A variable $x^*$ is observed, such that $$x^* = x + u$$ Under which condition satisfied, the OLS estimator $\hat{\beta}_1$ is still consistent? (5) (A). \(E(y\alpha^{*}) = 0\) (B). \(E(x^{*}u) = 0\) (C). \(E(xu) = 0\) (D). \(E(xv) = 0\) (E). \(E(uv) = 0\)

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