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I usually take a look at the diagonal elements of the matrix $(X'X)^{-1}$ and the $MSE$ in order to determine the standard deviation of the regression coefficients. I encountered a problem from a practice test that has a negative entry in the diagonal for the matrix $(X'X)^{-1}$. Can this be possible?

This is the problem enter image description here

I am testing for $\beta_3=0$, which in order to do, I would need to take $\frac{\hat{\beta_3}}{SE(\beta_3)}$, but the negative number in the diagonal of the inverse matrix throws me off.

Any help is much appreciated.

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  • $\begingroup$ Is this a homework? $\endgroup$
    – patagonicus
    Commented Dec 4, 2021 at 1:43
  • $\begingroup$ No, this is an old practice exam, and my only question is about the standard error for the regression coefficient. I use the diagonal element of the inverse matrix provided, but it has a negative entry, so this would not make sense. $\endgroup$
    – Stiven G
    Commented Dec 4, 2021 at 3:15
  • $\begingroup$ This might help stats.stackexchange.com/questions/96327/… $\endgroup$
    – patagonicus
    Commented Dec 4, 2021 at 4:20

1 Answer 1

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$\boldsymbol{X}^{\prime}\boldsymbol{X}$ is positive semi-definite. Hence, for any vector $\boldsymbol{v}$, $\boldsymbol{v}^{\prime}\boldsymbol{X}^{\prime}\boldsymbol{X}\boldsymbol{v} \ge 0$. If we let $\boldsymbol{v}$ denote a standard basis vector, then this inequality implies the diagonal elements must be non-negative. There is a typo in the problem.

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