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The maximum likelihood parameter estimates for the linear model where $\Pr(Y|X\beta) \sim \mathcal{N}(0,\sigma^2)$ are:

$$\hat{\beta} = (X'X)^{-1}X'Y$$

How do you compute the statistical power of the parameter estimates? It seems that there should be a closed-form, but I've searched and haven't found a proof anywhere else online.

Michael K
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  • Can you show that $\hat \beta$ is normal? Are you looking for a test on a single parameter (say a t-test), or some overall test of the regression (F-test)? Is this for some subject? – Glen_b Jun 23 '14 at 04:58
  • I'm looking to test all of the individual parameters $\beta_i$, so a t-test will work, but I'm not sure how to compute the standard error for each of the $\beta_i$. – Michael K Jun 23 '14 at 22:57
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    The $\beta_i$ are constants; I presume you mean their estimates. This is standard calculation; e.g. [Wikipedia](http://en.wikipedia.org/wiki/Ordinary_least_squares#Finite_sample_properties), [CrossValidated](http://stats.stackexchange.com/a/44841/805). (I simply searched in obvious ways in both places to locate those links.) – Glen_b Jun 23 '14 at 23:35

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