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As discussed in 33433, the prediction interval for single linear regression is most precise at the mean of the $x$ values. Does this also hold true for multiple linear regression, that is: the prediction interval is smaller near the means of the $x$ values?

The calculation for prediction interval in MLR is given in 9131 as

$$\hat{y} \pm t \hat{\sigma} \sqrt{1 + \mathbf{X}^* (\mathbf{X}'\mathbf{X})^{-1} (\mathbf{X}^*)'}.$$

Looking at this equation, it appears that the minimum prediction interval occurs when $\mathbf{X}^*$ is all zeros. Is that correct, or am I misunderstanding something?

On a related note, the formula for the prediction interval for single linear regression is given in 33433 as

$$s_\text{predictions(new)}=\sqrt{s^2_\text{error}\left(1+\frac{1}{N}+\frac{(x_\text{new}-\bar x)^2}{\sum(x_i-\bar x)^2}\right)}$$

Is this the equivalent to the MLR formula for $k=1$?

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  • $\begingroup$ After some more thought, I'm guessing that the prediction interval assumes that $X$ has been normalized, so that $X* = 0$ is the mean. $\endgroup$ – Dr. Pain Nov 10 '15 at 19:04

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