I'm estimating a simple OLS regression model of the type: $y = \beta X + u$
After estimating the model, I need to generate a weighted combination of coefficients (e.g. $w_1 \beta_1 + w_2 \beta_2$) and estimate standard errors for the combined statistic. What's the right way to calculate the standard errors of the sum of coefficients?
I've got this far: I have plenty of cases, so it's safe to say that the asymptotic normality assumption is satisfied. Let's call $s_1$ and $s_2$ the standard errors for $\beta_1$ and $\beta_2$, respectively. If the $\beta$'s were independent estimates, we could use the basic sum-of-normals function to say that the variance of $\beta_1+\beta_2$ is $w_1^2s_1^2 + w_2^2s_2^2$. But unless I'm deeply mistaken, the $\beta_1$ and $\beta_2$ aren't independent. Is there a simple way to fold the variance-covariance matrix of $X$ in to solve this problem?