Find residual sum of squares Let $Y_1, Y_2,Y_3$ be uncorrelated random variables with common variance $\sigma^2>0$ such that
$$
\begin{aligned}
E[Y_1]&=\beta_1+\beta_2,  \\
E[Y_2]&=2\beta_1  \\
E[Y_3]&=\beta_1-\beta_2
\end{aligned}
$$
where $\beta_1$ and $\beta_2$ are unknown parameters. Find the residual (error) sum of squares under the above linear model. I proceed in this way:  
Note that: 
$$
\begin{aligned}
E[Y_1-\beta_1-\beta_2]&=0,  \\
V[Y_1-\beta_1-\beta_2]&=\sigma^2,  \\
E[Y_2-2\beta_1]&=0,  \\
V[Y_2-2\beta_1]&=\sigma^2,  \\
E[Y_3-\beta_1+\beta_2]&=0,  \\
V[Y_3-\beta_1+\beta_2]&=\sigma^2
\end{aligned}
$$ 
So the residual(error) sum of squares under the above linear model is: 
$$
\sum{e_i^2}=(Y_1-\beta_1-\beta_2)^2+(Y_2-2\beta_1)^2+(Y_3-\beta_1+\beta_2)^2
$$ 
Am I proceeding in the correct way? Can someone please explain what is meant by linear model here?
 A: Your model must be rewritten as an estimating equation of the form 
$Y=X\theta+\epsilon$
where $y_1,y_2,y_3$ are just the same observed variable at different groups (or clusters)
That implies:
$y_i = \theta_1D1_i + \theta2 D2_i + \theta_3 D3_i + \epsilon_i$
Where $Dk_i$ is a dummy variable that =1 if the i-th observation corresponds to the $k-th$ group. So $\theta_1=\beta_1+\beta_2$ $\theta_2=2\beta_1$ $\theta_3=\beta_1-\beta_2$. But estimating the previous equations would yield 3 $\theta$ parameters from which only 2 $\beta$ exist so the system is overidentified. It would more reasonable to write $\theta_3$ as:
$\theta_3 = \theta_2-\theta_1$ thus the estimating equation becomes:
$y_i = \theta_1(D1_i-D3_i) + \theta2 (D2_i+D3_i)  + \epsilon_i\equiv \theta_1x_{1i}+ \theta_2x_{2i} + \epsilon_i$
which is an standard linear regession with 2 explnatory variables. Once the model is estimated ($\hat \theta$), the sum of squared residuals is 
$\sum_i\hat\epsilon_i^2 =\sum_i y_i^2-2 \sum_i y_i \hat y_i+ \sum \hat y_i^2$
The first term is obtained as $\sum (Y1_i + Y2_i +Y3_i)^2$. DEveloping the second term yields :
$-2(\hat\theta_1\sum_i y_i x_{1i}+\hat \theta_2 \sum y_i x_{2i}) $
$\equiv -2(\hat \theta_1 (\sum_i Y1_i-\sum Y3_i)+\hat \theta_2(\sum Y2_i- \sum Y3_i))$
while the third is:
$\hat\theta_1^2\sum_ix_{1i}^2 +\hat\theta_2^2\sum_ix_{2i}^2+2\hat\theta_1\hat\theta_2\sum_i{x_{1i}x_{2i}}$ where the sums are straigforward to calculate following the second term algebra.
But What about $\hat \theta$ values ? It's easy. Lets consider the first group observations, the equation becomes,
 $y_i=Y1_i= \theta_1 + \epsilon_i$, 
the second: $Y2_i=\theta_2 +\epsilon_i$ 
and third: $Y3_i=y_i=\theta_2-\theta_1+\epsilon_i$ 
These 3 equations are mutually exclusive as they are based on different samples so estimating the first and second (by OLS, ML etc) implies that the estimated $\theta$ are just averages of the corresponding $Yk$:
$\hat \theta_1 = \bar{Y1}$,and $\hat \theta_2=\bar{Y2}$ 
and therefore $\hat \theta_3=\bar Y2 -\bar{Y1}$
A: I don't remember encountering this problem before but mathematically you have correctly interpreted the assumptions. Now the next step is to estimate the 2 betas.  How should that be done?  All that is clear from the statement of the problem is that the error terms would be uncorrelated by implication and they would have the same variance.  But it is not clear (1) whether or not they have the same distribution and (2) whether that distribution would be normal.  If the error terms are normal they will be iid with 0 mean and common variance.  You could then apply least squares to your formula you have for the error sum of squares.  If the error terms are very non-normal but have the same distribution a robust ftting procedure like minimum sum of absolute deviations may be used.  Keep in mind that uncorrelated residuals are not necessarily independent in non-normal situations.
This is called a linear model or more correctly a multivariate linear model because the response variables Y$_1$, Y$_2$ and Y$_3$ are all linear functions of the parameters.
