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As the question says, what is the distribution of the estimate of residual variance in a standard gaussian linear regression?

I'm confused because I know in theory the observed $y$ subtract the predicted $y$ should be normal, so the sum of these squared should be chi squared. In practice though, say we want to calculate t-statistics, then we estimate the residual variance using our estimated predicted points. Thus I'm not sure exactly how the estimated residuals are distributed, and hence the estimated residual variance.

Would greatly appreciate any help explaining this.

mrepic1123
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  • I asked a related question here and got no response unfortunately. – Noah Sep 29 '23 at 20:33
  • Just to be clear @Zhanxiong , is the answer that it is distributed as $\frac{\sigma^2 \chi^2_{N-p-1}}{N-p-1}$ if I'm understanding correctly? – mrepic1123 Sep 29 '23 at 21:57
  • @mrepicfoulgermrepic1123 I would prefer to say $(N - p - 1)\sigma^{-2}\hat{\sigma}^2 \sim \chi_{N - p - 1}^2$ instead, because $\sigma^2\chi^2_{N - p - 1}/(N - p - 1)$ is not a standard distribution name (of course, as long as you interpret it as what I wrote, then it's fine). – Zhanxiong Sep 29 '23 at 22:52

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