# Variance of the slope when adding predictors

I came across an article with the following question:

Consider a linear regression with one single covariate, y=β0+β1x1+ε and the least-square estimates. The variance of the slope is Var[β1]. Do we decrease this variance if we add one variable, and consider y=β0+β1x1+β2x2+ε ?

They showed, using simulation, that the answer is no as it depends on how correlated x1 and x2 are. Is there a way to prove this without simulation?

• Take a look at the VIF ( variance inflation factor) it may be useful to understand the idea behind this result. – RScrlli Mar 27 '19 at 21:53
• What happens to the variance when $x_2$ is a constant nonzero multiple of $x_1$? – whuber Mar 27 '19 at 22:27

$$Var(\hat{\beta_j}) = \frac{\sigma^2}{SST_j(1-R_j^2)}$$
where $$\sigma^2$$ is the variance of the error term, SST is the total sample variation $$\sum_{i=1}^n(X_{ij}-\bar{X}_j)$$ in $$X_j$$ and $$R_j^2$$ is the R-squared from regressing $$X_j$$ on all other independent variables.
Coming back to your example it's easy to see that the variance of $$\hat{\beta_1}$$ increases when the correlation between $$X_{1}$$ and $$X_{2}$$ increases because a higher correlation implies a higher R-squared from regressing $$X_{1}$$ on $$X_{2}$$ (the denominator decreases).