109年 - [無官方正解]109 國立臺北大學_碩士班一般入學考試試題_經濟學系：統計學#146713

1. Jack use OLS method to estimate the parameters of linear model with intercept:
$y_t = \alpha + \beta x_t + \varepsilon_t.$
If the heteroskedasticity ($Var(\varepsilon_t | x_t) = \sigma^2_x x_t$) exits in the errors but other Gauss-Markov conditions satisfy, which statement is true?      (1)      (A). The estimated parameter, \( \hat{\beta} \), is biased, and variance of \( \hat{\beta}_1 \) obtained by OLS method is incorrect. (B). The estimated parameter, \( \hat{\beta} \), is biased, and variance of \( \hat{\beta}_1 \) obtained by OLS method is accurate. (C). The estimated parameter, \( \hat{\beta} \), is unbiased, and variance of \( \hat{\beta}_1 \) obtained by White heteroskedasticity-robust method is accurate. (D). The estimated parameter, \( \hat{\beta}_1 \), is unbiased and efficiency. (E). Jack cannot use OLS to estimate parameters because of inefficiency.

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). \( E(uv) = 0 \)

1-1 (5 分) Define
$\tilde{\beta} = \frac{y_B - y_S}{x_B - x_S}$
Whether $\tilde{\beta}$ is an unbiased estimator?      (6)     

1-2 (10 分) Please calculate $\text{var}(\tilde{\beta})$. Whether $\tilde{\beta}$ is efficiency?     (7)     

2.1 (10 分) What is the true value of $R^2$ in Equation (1)? What is the value of $s^2$ (estimated error variance.)?      (8)     

2.2 (5 分) What is the Durbin-Watson statistic in Equation (1)?      (9)     

1. (8分)假設隨機抽樣所來自的母體可以區分成兩類人群，A群與B群。A群人所構成的子母體佔總母體比例為 \( 60\% \) 且其子母體的變異數 \( \sigma_{A} \) 為7；B群人所構成的子母體佔總母體比例為 \( 40\% \) 且其子母體的變異數 \( \sigma_{B} \) 為5。考慮隨機從總母體抽出N個人(不分A、B群)，並計算其樣本平均數 \( \overline{Y} \) 。若A、B兩群子母體有相同的母體平均，即 \( \mu_{A} = \mu_{B} = \mu \) ，則 \( \sqrt{N} (\overline{Y} -\mu) \) 的漸近分配為何。(10)

2. 阿正手上展示一個有正反兩面不同花樣的鋼板(正面以H表示，反面以T表示)，令p代表隨機投擲一次會出現正面的機率。小飛與小貝欲估計p的值，並要求阿正給點提示。阿正於是擲投此鋼板三次得到三個正面，以HHH表示。
2.1 小飛是個頻率學派統計學家(frequentist statistician)，且打算使用最大概似估計法(Maximum Likelihood Estimation)來估計 p。 (7 分)
請寫下小飛的樣本概似函數，並求解其最大概似估計值(簡稱 MLE 值)。     (11)     (過程要清楚寫下一階條件，如有角解也需清楚呈現)

(8分)在看到阿正的樣本後，小貝對 \( p \) 認知的事後機率分配為何？      (12)     

(7分)若小貝打算用E(p|HHH)來推估p，則她的估計值為多少？      (13)     

(6分)比較小飛與小貝的估計，誰的值比較合理？理由為何？      (14)      (此題得分以你的統計邏輯依狀況給分，沒有一定對錯，請盡量陳述發揮。)

(7 分)請問一個月(以 30 天計)故障不超過 3 次的機率為何？     (15)    
(註：exp(-0.05)=0.9512, exp(-0.1)=0.9048, exp(-0.2)=0.8187, exp(-0.5)=0.6065)

(7 分)二次故障間隔超過 30 天的機率為何？     (16)