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The variance of the j element of the OLS estimator is given by

$$\operatorname{Var}\left(\hat{\beta}_{j}\right)=\sigma^{2}\left(X_{j}^{T} M_{-j} X_{j}\right)^{-1}$$

where $X_j$ is the column of regressors associated to the $j$ variable, and $M_{-j}$ is the maker of residuals (the projection off) of the space generated by all columns of the matrix $X$ besides the $j$th one.

Show that the variance of $\hat{\beta}$ can also be written as

$$\operatorname{Var}\left(\hat{\beta}_{j}\right)=\frac{\sigma^{2}}{(n-1) \operatorname{Var}\left(X_{j}\right)}\left(\frac{1}{1-R_{X_{j} \mid X_{-j}}^{2}}\right)$$

The variance of the $j$th element of the OLS estimator is given by

$$\operatorname{Var}\left(\hat{\beta}_{j}\right)=\sigma^{2}\left(X_{j}^{T} M_{-j} X_{j}\right)^{-1}$$

where $X_j$ is the column of regressors associated to the $j$ variable, and $M_{-j}$ is the maker of residuals (the projection off) of the space generated by all columns of the matrix $X$ besides the $j$th one.

Show that the variance of $\hat{\beta}$ can also be written as

$$\operatorname{Var}\left(\hat{\beta}_{j}\right)=\frac{\sigma^{2}}{(n-1) \operatorname{Var}\left(X_{j}\right)}\left(\frac{1}{1-R_{X_{j} \mid X_{-j}}^{2}}\right)$$

The variance of the j element of the OLS estimator is given by

$\operatorname{Var}\left(\hat{\beta}_{j}\right)=\sigma^{2}\left(X_{j}^{T} M_{-j} X_{j}\right)^{-1}$

where $X_j$ is the column of regressors associated to the $j$ variable, and $M_{-j}$

  is the maker of residuals  (the projection off) of the space generated by all columns of the matrix $X$ besides the $j$ one.

Show that the variance of $\hat{\beta}$ can also be written as

$\operatorname{Var}\left(\hat{\beta}_{j}\right)=\frac{\sigma^{2}}{(n-1) \operatorname{Var}\left(X_{j}\right)}\left(\frac{1}{1-R_{X_{j} \mid X_{-j}}^{2}}\right)$

How can i deal with these? thanks

The variance of the j element of the OLS estimator is given by

$$\operatorname{Var}\left(\hat{\beta}_{j}\right)=\sigma^{2}\left(X_{j}^{T} M_{-j} X_{j}\right)^{-1}$$

where $X_j$ is the column of regressors associated to the $j$ variable, and $M_{-j}$ is the maker of residuals  (the projection off) of the space generated by all columns of the matrix $X$ besides the $j$th one.

Show that the variance of $\hat{\beta}$ can also be written as

$$\operatorname{Var}\left(\hat{\beta}_{j}\right)=\frac{\sigma^{2}}{(n-1) \operatorname{Var}\left(X_{j}\right)}\left(\frac{1}{1-R_{X_{j} \mid X_{-j}}^{2}}\right)$$