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)?