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Suppose I have a multiple regression model $$Y(X) = \beta_0 + \sum_{i=1}^n\beta_iX_i + \epsilon$$

I train the regression, then predict $\hat{Y_i} $ for a new data point $X_i$, and get its standard error $se(\hat{Y_i})$. Hence, the 95% prediction interval of $\hat{Y_i}$ is roughly $$(\hat{Y_i}-2\times se(\hat{Y_i}), \hat{Y_i}+2\times se(\hat{Y_i}))$$

I care a lot about $se(\hat{Y_i})$ and I want to be confident of its value as I have another model to train on both $se(\hat{Y_i})$ and $\hat{Y_i}$

So my question is, what is the distribution of $se(\hat{Y_i})$? What is a 95% confidence interval for it such that I am certain with probability 95% that the true standard error is in this range?

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    $\begingroup$ xkcd.com/2110 $\endgroup$
    – Tim
    Commented Feb 14, 2020 at 8:18
  • $\begingroup$ @Tim The second model that is fitted on $\hat{Y_i}$ and $se(\hat{Y_i})$ is highly non trivial without a closed form. It uses the entire distribution of $\hat{Y_i}$, and so it's essential to be certain of the entire normal distribution, not just the expected value. $\endgroup$
    – AspiringMat
    Commented Feb 14, 2020 at 8:23
  • $\begingroup$ Why would $se(\hat{Y}_i)$ would have a normal distribution with support on all of $\mathbb{R}$? $\endgroup$
    – Dave
    Commented Feb 14, 2020 at 11:39
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    $\begingroup$ The answer depends on what formula you use to estimate the standard error. Even when the $\epsilon_i$ are assumed iid Normal, there are at least two formulas in common use. Could you tell us specifically (a) what assumptions you make about the errors and (b) what formula you use for the SE estimate? $\endgroup$
    – whuber ♦
    Commented Feb 14, 2020 at 17:34

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