Global minimum of prediction intervals for multiple linear regression

I'm trying to solve the following problem:

Prove that the width of the prediction interval for $$\hat{y}$$, given the vector $$x_0 = (1, x_1,...,x_{p-1})$$ of explanatory variables, reaches a global minimum when every entry of the vector $$x_0$$ is equal to the mean of the respective explanatory variable in the data set. For this exercise, recall the relevant expression for multiple linear regression $$\hat{y} \, \pm \, t^{1-\frac{\alpha}{2}}_{n-p} \tilde{\sigma} \,\sqrt{1+x_0^{T}(X^TX)^{-1}x_0}$$

Where $$\tilde{\sigma}=\sqrt{\frac{(y-X\hat{\beta})^T(y-X\hat{\beta})}{n-p}}$$. For that purpose, I proposed to minimize the prediction interval and started to do the following algebra:

$$\underset{x_0}{min} \, \sqrt{1+x_0^{T}(X^TX)^{-1}x_0}$$ $$\bar{0}_p = \frac{\partial}{\partial x_0} = \frac{(X^TX)^{-1}x_0}{\sqrt{1+x_0^T(X^TX)^{-1}x_0}} \, \iff \bar{0}_p = (X^TX)^{-1}x_0 \iff \bar{0}_p = x_0$$

But as you can see, I am not reaching the correct solution. Any suggestions?

It is worth to mention that the matrix $$X$$ has dimensions $$n\times p$$, furthermore $$X^TX$$ has dimensions $$p\times p$$.

• Which book did you take this question from? Something doesn't add up here Nov 10 '21 at 10:11
• @Spätzle It was an exercise from one of my lectures, no bibliography was written. But the logic of the problem actually makes sense: in the simple linear regression model it was found that the bands reach their minimum amplitude if the value of X for which a forecast is produced coincides with the mean. In the generalized version happens the same. Nov 10 '21 at 10:57
• I think I know now, but just to make sure - what does your $\tilde{\sigma}$ stands for? Nov 10 '21 at 11:02
• That is the estimator of the variance from the residual sum of squares $\tilde{\sigma} = \sqrt{\frac{(y-X\hat{\beta})^T (y-X\hat{\beta})}{n-p}}$ Nov 10 '21 at 11:22
• "Longitude" must be a mis-translation for "width" or "magnitude." Regardless, one route to a solution is to rewrite the OLS formulas in terms of the centered variables, at which point the result will be immediate.
– whuber
Nov 10 '21 at 11:49