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I have a number of explanatory variables $x_1,...,x_n$ and an outcome variable $y = f(x_1,...,x_n)$. Here $f$ is assumed to be known (estimated). I heard that for a confidence interval for $y$ one can use a covariance matrix $C$ such that $c_{ij} = Cov(x_i,x_j)$. Could you please refer me to these methods?

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Usually the diagonal elements of the variance-covariance matrix are used for CIs; any good book on linear regression should have the requisite formulae. – J. M. Apr 12 '11 at 2:36
Diagonal elements are used for CIs only for single factors - like a rule of $3\sigma$ - that method I've already read. Here the CI for $y$ is needed, not for $x_i$'s. – Ilya Apr 12 '11 at 7:49

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