ML Estimation of Linear Regression Model with $A R(1)$ Errors and Two Observations. This is based on Magee (1993). Consider the regression model $y_i=x_i \beta+u_i$, with only two observations $i=1,2$, and the nonstochastic $\left|x_1\right| \neq\left|x_2\right|$ are scalars. Assume that $u_i \sim N\left(0, \sigma^2\right)$ and $u_2=\rho u_1+\epsilon$ with $|\rho|<1$. Also, $\epsilon \sim N\left[0,\left(1-\rho^2\right) \sigma^2\right]$ where $\epsilon$ and $u_1$ are independent.
(a) Show that the OLS estimator of $\beta$ is $\left(x_1 y_1+x_2 y_2\right) /\left(x_1^2+x_2^2\right)$.
(b) Show that the ML estimator of $\beta$ is $\left(x_1 y_1-x_2 y_2\right) /\left(x_1^2-x_2^2\right)$.
(c) Show that the ML estimator of $\rho$ is $2 x_1 x_2 /\left(x_1^2+x_2^2\right)$ and thus is nonstochastic.
(d) How do the ML estimates of $\beta$ and $\rho$ behave as $x_1 \rightarrow x_2$ and $x_1 \rightarrow-x_2$ ? Assume $x_2 \neq 0$. Hint: See the solution by Baltagi and Li (1995).