# Multiple correlation in the sampling variance of slope coefficient in multiple regression

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).

• For frequentists, parameters are constants and they have no variance. So $Var(\beta)$ does not exist. – user158565 Jan 5 at 2:55
• @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. – user130451 Jan 5 at 4:52