The sandwich estimator for OLS regressions where heteroskedasticity is suspected is $$ var(\hat\beta) = (X'X)^{-1}X'ee'X(X'X)^{-1} $$

If I want confidence intervals on predictions, I can just take many draws from $N(\hat\beta, var(\hat\beta))$, multiply by $X$, and take the quantiles.

But what if I want prediction intervals? My model is $$ y = X\beta + e $$ $$ e = N(0, \sigma^2 f(X)) $$ and I need to be able to take draws from the error to make a prediction interval. Is there some trick for getting a prediction interval in such situations without going back and estimating $f$, which would be maybe some sort of two-step conditional heteroskedasticity model?

Perhaps a trick that involves taking on some simplifying assumptions?



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