Influence of variance change in variable $x_2$ on the variance of its OLS estimator

Question

So $Var(\hat{β}_1)=σ^2/\sum_{i=1}^n (x_{2i}−\bar{x})^2$. What would happen if the variance of $x_2$ is increased? The $σ^2$ of the error remains the same but the sum must change, but how?

Answer 1

Assuming that you are talking about the sample variance of $X$. The sample variance is calculated by $$ S ^ 2 = \frac{1}{n}\sum_{i=1}^n(x_i - \bar{x}_n) ^2, $$ as such, as the variance of $X$ grows $\sum_{i=1}^n(x_i - \bar{x}_n) ^2$ gets larger, hence the estimated variance of $\hat{\beta}_1$ gets smaller.