(9)

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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}) = $ (9) and $ p \lim (\hat{\mu}) = $ (10) .

III. 計算與申論題 (20%。每題 10 分，共 2 題) 請在答案卷標示大題「III」及其小題號 (1) 至 (2)，並陳述對應答案。陳述過程中若有明確數值者，請寫下該數值並四捨五入進位至小數第四位。 (1) (10%) Consider the following regression model with a lagged dependent variable: $$y_t = \beta_1 + \beta_2 x_t + \beta_3 y_{t-1} + \varepsilon_t,$$ in which the regression residual is suspected to be an AR(1) process. That is, $\varepsilon_t = \rho \varepsilon_{t-1} + v_t$, where $0 < |\rho| \le 1$ and $v_t$ is white noise. The OLS regression output for this model gives you the Durbin-Watson statistic equaling 0.39 from a sample size $T = 50$, Can you test the existence of an AR(1) process in $\varepsilon_t$? If you can, state your test steps and conclusion. If you cannot, state your reason(s)?

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