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?
Best Answer
You need to add a third term: $2 \cdot w_{1} \cdot w_{2} \cdot Cov(\beta_{1},\beta_{2})$. You can find the estimated covariance in the off-diagonal part of the variance-covariance matrix. If you have an intercept and 2 regressors, that would (typically) be either V[2,3] or V[3,2] since $Cov(\beta_{1},\beta_{2})=Cov(\beta_{2},\beta_{1})$.