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I have been given two formulas:

$$ \hat{\sigma}^2 = \frac{1}{n-p} (Y - X \beta)^T(Y - X\beta) $$ $$ = \frac{1}{n-p} \sum ( y_i - \hat{\beta_0} - \hat{\beta_1}x_i)^2 $$

I also have

$$ \hat{\mathrm{MSE}} = \hat{\sigma^2}\left(1 + \frac{1}{n} + \frac{(x^* - \bar{x})^2}{\sum(x_i - \bar{x})}\right) $$

where n is total number of people, p is number of paramters.

Are these formulas right for calculating MSE?

In my lecture notes, my lecturer as calculated $\hat{\sigma}^2 = 0.8$ and said thus MSE = 0.8. So does that mean I don't have to work out MSE if I have $\hat{\sigma}^2$?

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The first is the mean square error of regression, the residual sum of squares divided by its degrees of freedom - which is an estimate of the population variance parameter $\sigma^2$. The second is an estimate of the mean square error of a single predicted response $x^*$. So your lecturer was calculating the MSE for regression (& equating it to $\hat{\sigma}^2$), not the MSE for a predicted response.

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