Suppose we have a linear regression model of the following format : $$ y(x) = \beta_0 + \beta_1 x_1+ \beta_2x_2+\beta_3x_3+\epsilon$$

We know that the prediction interval associated with a level $\alpha$, for a new observation $x_0$, is :

$$ IC_{1-\alpha}=\left[\hat{y}_i \pm t_{\alpha/2, n-p-1}\times \sqrt{\hat{\sigma}^2(1+x_0^T(X^TX)^{-1}x_0)}\right]$$

The asked question is :
Without calculation and using approximations, give a prediction interval at $95\%$ for a given $(x_1,x_2,x_3)$.
Available information :

  • Output of the summary of this regression model given by R (so we have the $\beta s$, $\hat{\sigma}^2$ and $std(\beta)$)
  • The means and std errors of our variables.

We can approximate $t_{\alpha/2, n-p-1}$ by 2, how can we approximate $x_0^T(X^TX)^{-1}x_0$ ?

  • $\begingroup$ No approximation for $x_0^T(X^TX)^{-1}x_0$. Need to know$X$ or $(X^TX)^{-1}$. $\endgroup$ – user158565 Dec 17 '18 at 23:50

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