What is the most efficient way to calculate $R^2$? Hello I am working on a question from an old exam paper and wondered what is the best way to tackle parts ii and iii. Given the data it is easy to find $\hat{\beta_0}=-1.071$ and $\hat{\beta_1}=2.741$.
Now for part ii) I have the formula $R^2=1-SSE/SST$ where $SST=\sum(y_i-\bar{y})^2$ (easy to work out) and $SSE=\sum e_i^2=\sum (y_i-\hat{y_i})$.
Likewise I have for part iii) An unbiased estimate of $\sigma^2$ is $\sum e_i^2/(n-2)$.

Question: I wondered if there is a nice and more efficient way to work
  out $\sum e_i^2$ or do I have to calculate each predicted value based
  on the model take it away from the actual value square that value and
  then sum all the values up?



Yes, you are correct that computing SSE using the definition is a tedious job. If the regression coefficients are estimated without rounding errors, then the following procedure may be used for computing SSE.  
By definition,
$SSE = \sum(y_i-\hat{y_i})^2$, and $\hat{y}_i=\hat{\beta}_0 + \hat{\beta}_1 x$. 
Plug-in $\hat{y_i}$ into the definition of SSE and simplify. It results in the following: 
$$SSE = \sum y_{i}^{2}-\hat{\beta_0}\sum y_i - \hat{\beta_1}\sum x_i y_i.$$
