Let $y_1=\alpha_1+e_1,y_2=2\alpha_1-\alpha_2+e_2$ and $y_3=\alpha_1+2\alpha_2+e_3$ where $e_i \sim N(0,\sigma^2)$ iid for $i=1,2,3$. Find the least square estimate of $\alpha_1$ and $\alpha_2$. Derive the test statistic for $H: \alpha_1=\alpha_2$.
So, I have found the least square estimates by minimizing the squares of the errors equating the partial derivatives to $0$ and got the estimate of $\alpha_1$ and $\alpha_2$. But I am struggling to find how to form a test statistic for testing the given hypothesis. Any help would be appreciated. Thank you in advance.
