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Let's say we have the following multiple regression model,

$$Y_i = \alpha + \beta_1 x_{i1} +\beta_2x_{i2} + ... + \beta_kx_{ik} + \varepsilon_i$$

with $ \varepsilon_i$ is $iid$ ~ $N(0,\sigma_{\varepsilon}^2) $ and least square estimators are $A, B_1,...,B_k $ for $\alpha, \beta_1,...,\beta_k $.

The sampling variance of a given slope coefficient $B_j$ is given by $$ Var(B_j) = \frac{1}{1-R_j^2}\times \frac{\sigma_\varepsilon ^2}{\sum_{i=1}^{n}(x_{ij}-\bar{x_j})^2} $$ where $R_j^2$ is the squared multiple correlation from the regression of $X_j$ on all the other $X$s. In the matrix notation of the above model, the variance of the parameter estimate vector $(\boldsymbol{B})$ is given by $$ Var({\boldsymbol{B}}) = \sigma_{\varepsilon}^2(X'X)^{-1} $$ My question is, how do we go from the $Var(\boldsymbol{B})$ equation to the $Var(B_j)$ equation where the multiple correlation is incorporated? Note that I took the $Var(B_j)$ equation from John Fox's Applied Regression Analysis & Generalized Linear Models (3rd Ed).

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  • $\begingroup$ For frequentists, parameters are constants and they have no variance. So $Var(\beta)$ does not exist. $\endgroup$ – user158565 Jan 5 at 2:55
  • $\begingroup$ @user158565 - Thanks for pointing out the notation issue. I meant to write the variance of the parameter estimate (not population parameter) vector. It's fixed now. $\endgroup$ – user130451 Jan 5 at 4:52

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