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The observations $y_1, y_2, y_3$ were taken on the random variables $Y_1, Y_2, Y_3$ where

$Y_1=\theta+e_1$

$Y_2=2\theta - \phi+e_2$

$Y_3=\theta +2 \phi+e_3$

and

$E(e_i)=0, var(e_i)=\sigma^2 (i=1,2,3), cov(e_i,e_j)=0 (i\neq j)$.

I need to formulate this model in matrix notation and find the vector of least squares estimates of the regression parameter vector $\left(\begin{array}{c} \theta\\ \phi\end{array}\right)$.

Attempt: So from what I'm given I know that $E(Y)=X\beta$ gives

$\left(\begin{array}{c} \theta\\ 2\theta - \phi\\ \theta +2 \phi\end{array}\right)=X \left(\begin{array}{c} \theta\\ \phi\end{array}\right)$. But how can I find the least squares estimates without having the value of X?

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  • $\begingroup$ But you do have it, don't you see? Find a $3\times 2$ matrix that satisfies the last matrix equation you wrote. $\endgroup$ Nov 10 '13 at 18:24
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Writing what you have in matrices:

$$ \begin{bmatrix}Y_1\\Y_2\\Y_3\end{bmatrix}=\begin{bmatrix}1 & 0 \\ 2 & -1 \\ 1 & 2\end{bmatrix}\begin{bmatrix}\theta \\ \phi\end{bmatrix}+\begin{bmatrix}e_1\\e_2\\e_3\end{bmatrix} $$ The first matrix on the right hand side is your $\mathbf{X}$, where the model function is formulated as: $$ \mathbf{Y}=\mathbf{X}\boldsymbol{\beta}+\mathbf{e}. $$

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