I cannot figure out why the variance of a prediction of $y$, when given $x_0$ is

$$\sigma^2\bigg(1+\frac 1 n +\frac{(x_0-\bar x)^2}{Sxx}\bigg)$$

Isn't this the variance of a prediction error?

I'm thinking that

$${\rm Var}(\widehat{y_0})\ne{\rm Var}(\widehat{y_0}-y_0)$$

since one involves an error term and one does not. It would be amazing if any of you and show me the proof of this variance! Thanks


marked as duplicate by whuber regression Apr 17 '18 at 13:16

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    $\begingroup$ while you posted your question under the regression tag, you should stil add your actual prediction model, otherwise your question can not be answered accurately. moreover this seems to be a homework problem. assuming ols: you are right, the prediction $\hat{y}_0$ and the error of the prediction, $\hat{y}_0-y_0$ are two different things. However, the variance of both expressions will involve the error term, since your least square estimates are based on $y_i$. For the calculation you just plug in the formulas of your ols-estimates. $\endgroup$ – chRrr Apr 17 '18 at 8:55
  • $\begingroup$ More help about this topic is available through the links at stats.stackexchange.com/…. $\endgroup$ – whuber Apr 17 '18 at 13:16

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