# How to prove variance of OLS estimator in matrix form?

I am reading Wooldridge's Introductory Econometrics (2000), don't judge me, old version = cheap second hand book, and in the page P94 Theorem 3.2 of Multiple Regression Analysis, it says that:

$$Var(\hat{\beta_{j}}) = \dfrac{\sigma^2}{SST_j(1-R^2_j)}\tag{1}$$

where $$SST_j$$ is the total sample variation in $$x_j$$, $$R^2_j$$ is the $$R$$-squared from regressing $$x_j$$ on all other independent variables. I do understanding the meaning of this formula, and it has a algebraic proof in appendix. So this equation, along with the expected value of $$\beta_j$$ shows several properties such as: variance of betas in misspeficied models, and also why adding irrelevant independent variables in regression model to boost $$R$$-squared is at the cost of high variance of OLS estimators.

But if we take a look at the same formula in matrix form, for instance, in Hayashi (2000), the same variance of $$\hat{\beta}$$ is:

$$Var(\hat{\beta}) = \sigma^2(X'X)^{-1}\tag{2}$$

Now, the question is that how to show the above question in the following form:

$$Var(\hat{\beta_{j}}) = \dfrac{\sigma^2}{nS^2_j(1-R^2_j)}\tag{3}$$

where $$R_j$$ is the $$R$$-squared from regressing $$x_j$$ on all other $$x$$’s, $$n$$ is the sample size and $$S^2_j$$ is the sample variance of the regressor $$X$$.

• I'm fairly certain the $R_j$ you have listed in (3) is a centered $R^2$? – StatsStudent Jan 5 at 16:55
• Got it...yes it is a centered r squared...but I was wondering since the text just mentioned that by doing some operation, we can get (3) from (2)...I was wondering how.. – commentallez-vous Jan 5 at 17:12
• Well, you have to understand that (2) is a covariance matrix of the vector $\beta$ whereas (3) is just an individual variance of a specific estimated coefficient $\beta_j$. You'd essentially be working with the $j$-th diagonal element of the matrix in (2) (variances). – StatsStudent Jan 5 at 17:24
• But (2) is the inverse of the var-cov matrix X'X, so I have no idea about the properties or what the inverse of X'X is doing here. – commentallez-vous Jan 5 at 17:28
• Do you have much knowledge in matrix algebra (I just don't have a good sense of your level here). If not, I'd highly recommend you check out chapter 5 of Applied Linear Statistical Models Fifth Edition by Kutner et al. for a good crash course. I think this will also many of your questions. If I have time later, I'll try to work this out for you and write up a complete answer. – StatsStudent Jan 5 at 17:38