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I was reading in "A Guide to Econometrics" that given $Y = X \beta + \epsilon$, the variance covariance matrix of $\beta^\text{OLS}$ is given by $\sigma^2 (X' X)^{-1}$ where $\sigma^2$ is the variance of the error term...

it then says that in the case of a single regressor $y = \beta_1 + \beta_2 x$, that this simplifies to $\sigma^2 / \sum(x-\bar{x})^2$. I don't quite see this..isn't the matrix $X$ in this case still a two by two matrix? Namely:

$$ X = \left( \begin{array}{cc} 1 & x_1 \\\ 1 & x_2 \end{array} \right) $$

How are we getting something like $\sum(x-\bar{x})^2$ ?

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This question would be a better fit at math.stackexchange.com, as it has a broader remit than MO; this site is (primarily) for research-level questions. –  David Roberts Sep 15 '11 at 5:28
    
Or even stats.stackexchange.com, which is more statistics-focussed. –  David Roberts Sep 15 '11 at 7:16
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