5. If \( X_{1}, X_{2}, \ldots, X_{n} \) are normal random variables with variance \( \sigma^{2} \) and covariance between any pair of these random variables is \( \rho \sigma^{2} \), then the variance of \( \frac{1}{n}(X_{1} + X_{2} + \dots + X_{n}) \) equals (5).

6. Suppose \( X \) and \( Y \) are random variables with \( E(Y|X = x) = -X + 3 \) and \( E(X|Y = y) = -\frac{1}{4} Y + 5 \). Then, the correlation coefficient between \( X \) and \( Y \), \( \operatorname{Corr}(X,Y) \), equals (6).

7. Let \( Y = X_{1} + X_{2} + \dots + X_{15} \) be the sum of a random sample of size 15 from the distribution whose density function is \( f(x) = 1.5x^{2}, -1 < x < 1 \). What is the approximate value of \( P(-0.3 \leq Y \leq 1.5) \) when one uses the central limit theorem? (7).

8. Let \( X_{1} \) and \( X_{2} \) be a random sample of size 2 from the standard normal distribution, then the \( 75^{\text{th}} \) percentile of the statistic \( W = \frac{X_{1}}{\sqrt{X_{2}^{2}}} \) equals (8).

9. Two random samples are drawn independently from a normal distribution \( N(\mu, \sigma^2) \). The first sample has \( n_1 \) observations, and its sample mean is \( \overline{X}_1 = \frac{1}{n_1} \sum_{i=1}^{n_1} X_{1i} \); the second sample has \( n_2 \) observations, and its sample mean is \( \overline{X}_2 = \frac{1}{n_2} \sum_{i=1}^{n_2} X_{2i} \). One estimator of \( \mu \) is proposed: \( \hat{\mu} = \frac{(n_1 \overline{X}_1 + n_2 \overline{X}_2)}{(n_1 + n_2)} \). Calculate \( Var(\hat{\mu}) = \underline{\hspace{2cm}}(9) \) and \( p \lim (\hat{\mu}) = \underline{\hspace{2cm}}(10) \). (9)

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II. 填空題 (30%。每格 5 分，共 6 格) 請在答案卷標示大題「II」及其小題號 (1)、(2)、... 至 (6)，並填入對應答案，不需另外標示 Question 1、2 等題號。答案可得明確數值者，請寫下該數值並四捨五入進位至小數第四位。 # Question 1 The Okun's Law describes the negative relationship between output gap ($y_t$) and the cyclical rate of unemployment ($u_t^c$) as $y_t = \beta_1 + \beta_2 u_t^c + \varepsilon_t$, where $\varepsilon_t$ is white noise. Using data from Taiwan over the period 1978-2017 (sample size $T = 40$), someone estimated this equation by OLS and got the estimated Okun's coefficient $\hat{\beta}_2 = -2.9516$ and $R^2 = 0.5762$. If the standard deviation of output gap is 0.0195, then the regression sum of squared residuals (SSR) = (1), and the correlation coefficient between $y_t$ and $u_t^c$ = (2). (1)

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Question 2 A gender-wage regression output shows: ln $wage_i = 1.6573male_i + 1.1936female_i + 0.0364exper_i - 0.0321male_i \times exper_i$, where the logarithmic wage is significantly explained by two gender dummy variables, male and female, year of working experience (exper), and an interaction term: $male \times exper$. Based on the estimation result, we can conclude that, first, given zero year of working experience, male workers will be paid by (3) % higher than female workers. Second, the marginal return of an additional year of working experience of male workers is (4) % lower than that of female workers. (3)

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Question 3 A multiple regression model $y_i = \beta_1 + \beta_2 x_{2,i} + \beta_3 x_{3,i} + \varepsilon_i$ ($\varepsilon_i$ is white noise and $i = 1,2,\dots,20$) is estimated by least squares, and the R-squared is reported by $R^2 = 0.8$. An F-statistic equaling (5) can be calculated and used to test the overall significance of $\beta_2 = \beta_3 = 0$. Based on 5% level of significance, the critical value of this test is (6), so we (6) (reject OR do not reject) the null hypothesis of $\beta_2 = \beta_3 = 0$. (註：第 (6) 小題需 2 格全對才可得 5 分)。 (5)

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