$Y_1,Y_2,Y_3$ are uncorrelated rvs such that $E(Y_1)=\beta_1+\beta_2$,$E(Y_2)=2\beta_1$ and $E(Y_3)=\beta_1-\beta_2$

Question

Let $Y_1,Y_2,Y_3$ be uncorrelated random variables with common variance $\sigma^2 > 0 $ such that $E(Y_1)=\beta_1+\beta_2$,$E(Y_2)=2\beta_1$ and $E(Y_3)=\beta_1-\beta_2$ where $\beta_1$ and $\beta_2$ are unknown parameters.We need to find the residual (error) sum of squares under the above linear model.

Can anyone give an idea how to approach this? I can't think of an approach.

Answer 1

The residual sum of squares is $(y_1-(b_1+b_2))^2+(y_2-2 b_1)^2+(y_3-(b_1-b_2))^2$ which is minimized to $(y_1-y_2+y_3)^2/3$ by taking $b_1=(y_1+2 y_2+y_3)/6$ and $b_2=(y_1 - y_3)/2$. Because there is just one degree of freedom associated with the sum of squares, $(y_1-y_2+y_3)^2/3$ is also an estimate of $\sigma^2$.