10. Let \( \{X_n\} \) be a sequence of random variables indexed by the sample size n. Suppose that \( X_n \) takes two values, 1 and n, with probabilities \( 1 - \frac{1}{n} \) and \( \frac{1}{n} \), respectively. Which of the following is correct?
(A) \( X_n \) converges in probability.
(B) \( X_n \) converges in mean square.
(C) \( X_n \) converges in both probability and mean square.
(D) \( X_n \) converges in neither probability nor mean square.

1. Consider the linear model $yi = \beta0 + \beta1 x{1i} + \beta2 x{2i} + ui$, where $E(ui \mid x{1i}, x{2i}) = 0$ and $Var(ui \mid x{1i}, x{2i}) = \sigma^2$ (homoskedasticity). Let $R1^2$ be the $R^2$ from the auxiliary regression of $x{1i}$ on $(1, x{2i})$. Suppose $\hat{\sigma}^2 = 4.00$, $\sum{i=1}^{n} (x{1i} - \bar{x}1)^2 = 200$, and $R1^2 = 0.75$. Using the standard homoskedastic OLS variance formula, what is the implied standard error of $\hat{\beta}_1$?(A) 0.071(B) 0.100(C) 0.283(D) 0.400

2. Consider $yi = \beta0 + \beta1 Di + \beta2 xi + \beta3 (Di xi) + ui$, where $Di \in \{0, 1\}$ and $xi \in \mathbb{R}$, with $E(ui \mid Di, xi) = 0$. Define the conditional mean $\mu(D, x) = E(yi \mid Di = D, xi = x)$. Which statement is correct?(A) \( \beta_{1} = \mu (1,x) - \mu (0,x) \) for all \( x \)(B) The marginal effect of \( x \) when \( D = 1 \) is \( \beta_{2} \)(C) \( \beta{3} = \frac{\partial\mu(1,x)}{\partial x}\Big|{x = 0} - \frac{\partial\mu(0,x)}{\partial x}\Big|_{x = 0} \)(D) \( \beta{2} + \beta{3} \) equals the intercept difference between \( D = 1 \) and \( D = 0 \)

3. The true model is $yi = \beta0 + \beta1 xi + \beta2 zi + ui$, $E(ui \mid xi, zi) = 0$, but you estimate the short regression $yi = \alpha0 + \alpha1 xi + ei$ (omitting $zi$). Assume the standard OVB formula:$$plim(\hat{\alpha}1) = \beta1 + \beta2 \frac{Cov(xi, zi)}{Var(xi)}.$$Suppose $\beta1 = 1.0$, $\beta2 = -2.0$, $Var(xi) = 4.0$, and $Cov(xi, zi) = 1.0$. Which is closest to $plim(\hat{\alpha}1)$?(A) 0.0(B) 0.5(C) 1.0(D)-0.5

4. Consider the simple regression $yi = \beta0 + \beta1 xi + u_i$. Which condition, if violated alone, implies OLS is generally not BLUE but can remain unbiased?(A) \( E(u{i}\mid x{i}) = 0 \)(B) \( \operatorname{Var}(ui \mid xi) = \sigma^2 \) (constant variance)(C) \( \operatorname{Var}(x_i) > 0 \)(D) Linear-in-parameters conditional mean: \( E(y{i} \mid x{i}) = \beta{0} + \beta{1}x_{i} \)

5. In the multiple regression $yi = \beta0 + \beta1 x{1i} + \beta2 x{2i} + ui$, you want to test $H0: \beta1 - \beta2 = 0$ (a single linear restriction).Suppose you have $\hat{\beta}1 = 0.80$, $\hat{\beta}2 = 0.50$, $\widehat{Var}(\hat{\beta}1) = 0.040$, $\widehat{Var}(\hat{\beta}2) = 0.010$, $\widehat{Cov}(\hat{\beta}1, \hat{\beta}2) = 0.005$.Compute the $t$-statistic $t = \frac{(\hat{\beta}1 - \hat{\beta}2) - 0}{\sqrt{\widehat{Var}(\hat{\beta}1 - \hat{\beta}2)}}$, $\widehat{Var}(\hat{\beta}1 - \hat{\beta}2) = \widehat{Var}(\hat{\beta}1) + \widehat{Var}(\hat{\beta}2) - 2\widehat{Cov}(\hat{\beta}1, \hat{\beta}2)$.Which is closest?(A) 1.50(B) 2.12(C) 3.00(D) 4.24

6. Consider the structural model $yi = \beta0 + \beta1 xi + ui$, where $Cov(xi, ui) \neq 0$. You have one instrument $zi$ satisfying $Cov(zi, xi) \neq 0$ (relevance), $Cov(zi, ui) = 0$ (exogeneity).Assume all variables are mean-zero for simplicity. The just-identified IV estimand is$$\beta1^{IV} = \frac{Cov(zi, yi)}{Cov(zi, x_i)}$$Suppose $Cov(zi, yi) = -0.18$ and $Cov(zi, xi) = 0.12$. What is $\beta_1^{IV}$?(A)-0.67(B)-0.5(C) 0.5(D)-1.50

7. In a just-identified IV setup with one endogenous regressor $xi$ and one instrument $zi$, the first stage (with an intercept) is $xi = \pi0 + \pi1 zi + vi$. You estimate $\hat{\pi}1$ with $t$-statistic $t{\pi1} = 1.9$ (two-sided). A common rule-of-thumb uses the first-stage $F$ statistic for $H0: \pi1 = 0$, which in this single-instrument case satisfies $F = t{\pi1}^2$. Which statement is most accurate?(A) \( F \approx 1.9 \), so the instrument is strong(B) \( F \approx 3.6 \), suggesting potential weak-instrument concerns and distorted 2SLS inference(C) \( F \approx 19 \), so 2SLS is more efficient than OLS(D) \( F \approx 0.5 \), so 2SLS is unbiased in finite samples

8. Let a categorical variable $Gi \in \{1,2,3,4\}$ and define dummies $D{ki} = \mathbf{1}\{Gi = k\}$, $k = 1,2,3,4$. Consider the regression with an intercept: $yi = \beta0 + \beta1 D{1i} + \beta2 D{2i} + \beta3 D{3i} + \beta4 D{4i} + ui$. Which statement is correct?(A) The model is identified because the dummies are mutually exclusive(B) The model is not identified because \( \sum{k=1}^{4} D{ki} = 1 \) for all \( i \); drop one dummy or drop the intercept(C) The model is identified if \( n \) is large enough(D) Identification fails only if one group has zero observations

9. Suppose you estimate a log-level model $\ln(yi) = \beta0 + \beta1 xi + ui, \quad yi > 0,$with $E(ui \mid xi) = 0$. Using the standard approximation for small changes, which interpretation of $\beta_1$ is correct?(A) A one-unit increase in \( x \) changes \( y \) by exactly \( \beta_{1} \) units(B) A one-percent increase in \( x \) changes \( y \) by approximately \( \beta_{1} \) percent(C) A one-unit increase in \( x \) changes \( E(y \mid x) \) by approximately \( 100\beta_{1} \) percent(D) \( \beta_{1} \) is the price elasticity of demand

10. Consider the wage equation $\ln(wagei) = \beta0 + \beta1 educi + ui$, where ui contains unobserved ability abilityi and local labor-market quality Li. Assume Cov(educi, abilityi) > 0and Li directly affects wages. A proposed instrument is distance to nearest college, zi. Suppose zi is strongly correlated with Li (e.g. urban areas have smaller distance and higher wages even holding education fixed), and $L_i$ is omitted from the regression. Which statement is most correct?(A) The exclusion restriction may fail because \( z{i} \) can affect \( \ln (wage{i}) \) through \( L{i} \), so \( Cov(z{i},u_{i}) \) may be nonzero.(B) If is always consistent regardless of \( L_i \)(C) Adding an intercept is sufficient to control for \( L_{i} \)(D) If OLS is biased, IV must be unbiased in finite samples

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