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試題隨卷繳交接背面4國立臺北大學 115 學年度碩士班一般入學考試試題系(所)組別：經濟學系科 目：統計學第4頁 共4頁☑可 ☐不可使用計算機

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

41. 有一個線圈有 80 匝，線圈通以 0.4 安培的電流，產生 0.2 韋伯的磁通量，則此線圈的電感量為何？ 台糖 110 (A)7.4 亨利 (B)40 亨利 (C)160 亨利 (D)320 亨利

42. 如下圖所示，請求 a、b 兩端的總電感 Lab 為何？ 台糖 110 (A)4H (B)5H (C)8H (D)12H

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