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I have 2 questions about multiple linear regression model.

Does the residual vector in the multiple linear regression model have the form $y-(X(X^T X)^{-1} X^T)^2 y$?

And, is it true that the sample distribution for a linear combination of model parameters in the multiple linear regression model is used in deriving the prediction interval for a future observation $x_0$?

Thanks in advance

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Yes, the residual vector is correct but the expression can be simplified because: $$(X(X^TX)^{-1}X^T)^2y=X(X^TX)^{-1}(X^TX)(X^TX)^{-1}Xy=X(X^TX)^{-1}X^Ty=X\hat{\beta}=\hat{y}$$ and residuals are $r=y-\hat{y}$. You don't have to square the term in the parentheses.

Since $\hat y_0=\hat{\beta}x_0$, prediction interval will use sampling distribution of parameter estimates.

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  • $\begingroup$ oh thank you so much, that was really helpful...can i do one more question? Is it true or false that the sample distribution for a linear combination of model parameters in the multiple linear regression model is used in deriving the prediction interval for a future observation x0? $\endgroup$ – pianca Strict Nov 28 '19 at 15:21
  • $\begingroup$ yes, i tried to address that question in my last sentence. $\endgroup$ – gunes Nov 28 '19 at 16:09
  • $\begingroup$ it's a different question..idk whether its true or false $\endgroup$ – pianca Strict Nov 28 '19 at 16:21
  • $\begingroup$ I still don't realize the difference, but ok it's true. $\endgroup$ – gunes Nov 28 '19 at 16:23
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    $\begingroup$ Thanks for the upvote; do you know how to accept an answer? You'll click the "tick" sign under the arrows upper left. $\endgroup$ – gunes Nov 28 '19 at 18:04

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