Consider the following model: $$y_1 = \theta + \gamma + \epsilon_1$$ $$y_2 = \theta + \phi + \epsilon_2$$ $$y_3 = 2\theta + \phi + \gamma + \epsilon_3$$ $$y_4 = \phi − \gamma + \epsilon_4$$ where $\epsilon_i$ are uncorrelated having mean $\theta$ and variance $\sigma^2$. Show that $\gamma − \phi$ is estimable. What is its BLUE?
MY ATTEMPT :
I have shown $\gamma-\phi$ is estimable.
$\rightarrow$ Ok so we have to minimise $\Phi=\mathop{\mathbb{E}}(\gamma-\phi-\alpha-\beta^TY)^2$ Where $\alpha+\beta^TY$ is best linear predictor of $\gamma-\phi$. With $\beta^T=[\beta_1, \beta_2,\beta_3, \beta_4]$
Now $\Phi=\mathop{\mathbb{E}}(\beta^TY)^2+(\gamma-\phi-\alpha)^2$
For $\Phi$ to be minimum, $\hat{\alpha}=\gamma -\phi$
So $\Phi=\mathop{\mathbb{E}}(\beta^TYY^T\beta)$
Now how to find $\beta$ to minimise $Y$? (On a side note, whatever I have done is correct or are there any errors?)