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When centering or mean-zero standardizing independent variables in a multiple linear regression analysis, it works out that $s.e.(\beta_0)= \sqrt{\sigma^2(\beta_0)}$ of the first diagonal element of $\sigma(\mathbf{X}^\top\mathbf{X})^{-1}$ is biased. The mathematics alone would indicate that separate approaches are required for centering vs standardization; however, the regression literature offers little explanation regarding multiple independent variables. Have the additional computational steps for calculating $s.e.(\beta_0)$ for a centered model been published anywhere?

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  • $\begingroup$ To what does "solutions" refer? What is the question? $\endgroup$ – whuber Apr 15 '17 at 14:19
  • $\begingroup$ Straightforward "closed form analytic solutions," i.e., equations. Rephrased question is: What are the additional modifications required to derive $s.e.(\beta_0)$ for a centered model, or when predictors are mean-zero standardized? Using the typical diagonal element of $\sigma(\mathbf{X}^\top\mathbf{X})^{-1}$ will give an erroneous value of $s.e.(\beta_0)$ for a centered(standardized) model; however, all the remaining $s.e.(\beta_j)$ are appropriate. $\endgroup$ – JoleT Apr 15 '17 at 16:09
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    $\begingroup$ This is a question about how to change your units of measurement, which is probably why you can't find much about it: it's routine. With that realization, you can find details in myriad places, beginning with high school science textbooks. $\endgroup$ – whuber Apr 15 '17 at 18:45
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    $\begingroup$ Those textbooks will teach you how, for instance, to convert between degrees Fahrenheit and degrees Centigrade. That's exactly the same thing you need to do here. I will not pretend that this is difficult or poorly documented: I'm trying to direct you (and any other interested readers) to the resources that are most likely available to you if you feel you need any more detail than that. $\endgroup$ – whuber Apr 15 '17 at 19:03
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    $\begingroup$ I am sorry that you seem to think consulting textbooks is beneath you. As an alternative, you could begin with Wikipedia: en.wikipedia.org/wiki/…. $\endgroup$ – whuber Apr 15 '17 at 20:29

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