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I derive a point estimate given value $x_0$ using an estimated linear regression as follows:

$$\hat{y_0} = x_0^T\hat{\beta}.$$

I know that a prediction interval for a given value $x_0$: $$\hat{y}_0\pm t_{n-p}^{(\alpha/2)}\hat{\sigma}\sqrt{x_0^T(X^TX)^{-1}x_0 + 1},$$

where $\hat{\sigma}$ is the sample standard deviation of the residuals and $t_{n-p}^{\alpha/2}$ is a t-statistic with $n-p$ degrees of freedom at the $\alpha/2$ quantile.

I have the following question:

Thus far, I have assumed $\hat{\sigma}$ to be constant - the "standard" case. How do I estimate prediction intervals if my estimate of $\sigma$ changes when $x$ changes, e.g. instead of predicting conditional on $x_0$ $(\hat{\sigma_0})$, I predict conditional on $x_1$ $(\hat{\sigma_1})$. Here, the index may indicate changes across the cross-section or changes in time periods.

shenflow
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  • Related: https://stats.stackexchange.com/questions/352654 – Richard Hardy Nov 25 '22 at 19:45
  • Does my answer here help you? – statmerkur Nov 25 '22 at 19:54
  • @statmerkur I am not sure - I do not see how this relates to what I was asking. But maybe I am just not seeing it. – shenflow Nov 25 '22 at 22:29
  • @Richard Hardy: If I understand your answer correctly, my line of thought ist correct? – shenflow Nov 25 '22 at 22:29
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    You need a model for the variance of the conditional response. Otherwise, you can't even construct prediction intervals for unobserved values of $x.$ – whuber Nov 26 '22 at 17:43
  • @whuber In my question I am assuming that I have some model for the variance of the conditional response, i.e. I got $\hat{\sigma}_1$ and $\hat{\sigma}_2$. My question is then - how do I derive prediction intervals based on them? – shenflow Nov 26 '22 at 21:59
  • I edited the question for clarity. – shenflow Nov 26 '22 at 22:08
  • It should depend in part on the model and how you estimate the variances, because the uncertainty of those variances ought to influence the prediction interval. – whuber Nov 27 '22 at 00:36

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