# Perform a t-test in multiple regression matrix

I have the following matrices: $$X'X$$ $$X'y$$$$y'y$$

I know that the B matrix can be computed as follows : $$B = (X'X)^{-1}X'y$$

If I want to perform a t test for a specific B, say $$B_{1}$$, I know that I must follow the formula : $$t = \frac{B_{1} - B_{0}}{se(B_{1})}$$

However, if I only have the 3 aforementioned matrices, how can I compute the standard error of a given B?

## 1 Answer

If $\epsilon$ designate your residuals, estimator variance matrix is given by $Var(\hat{\beta}) = \sigma^2(X'X)^{-1}, \sigma^2 = Var(\epsilon)$ in my class notes but I can't demonstrate it so can't be a 100% sure.

EDIT : this thread gives (and proves) $Var(\beta_1) = \frac{\sigma^2}{\sum\limits_i (x_i - \bar{x})^2}$ so I guess my notes are still good enough :)

• Even I am trying to get my head around this. From what I understand, to obtain 'e' you need Y and predicted Y matrices (which are not there). I guess there is some way you can get those from y'y. – Anurag Priyadarshi Apr 19 '16 at 8:26
• How can you not have $\hat{Y}$ if you have $X$ and $\beta$ ? Because I would say that you can not derive estimators variance without some kind of information on the residuals. – Riff Apr 19 '16 at 9:05
• according to the question only 3 matrices are available: X'X, X'y and y'y. Matrix X is not there. – Anurag Priyadarshi Apr 19 '16 at 9:34
• Is that a self-study case ? – Riff Apr 19 '16 at 9:42
• homework case rather ;) – Anurag Priyadarshi Apr 19 '16 at 9:53