In the simple regression model
$\hat{Y_i}=\hat{\beta_1}+\hat{\beta_2}X_{2_i}+\hat{u_i}$
The variance of the $\hat{\beta_2}$ coefficient is apparently given by the formula:
$(\sigma_\hat{\beta_2})^2=\cfrac{(\sigma_u)^2}{\sum{(X_{2_i}}-\bar{X_2})^2}$
But isn't it the case that $(\sigma_u)^2$ is unobservable? If so, it would surely be impossible to calculate the variance of $\hat{\beta_2}$, and consequently, its standard error and test statistic.
What gives?
Best Answer
Yes, and thus $\sigma_u^2$ is usually estimated with $$ \frac{1}{n-2}\sum_{i=1}^n ( \hat y_i - y_i) ^2 $$