I would like to estimate a confidence interval for a linear combination of regression coefficients that come from several linear regression models that are calculated from different but correlated variables. For instance, suppose $y_1$ and $x_1$ represent cloud cover and temperature anomalies at one level of the atmosphere as measured by a satellite, and $y_2$ and $x_2$ represent cloud cover and temperature anomalies in a lower level level of the atmosphere. I expect $x_1$ to be correlated with $x_2$. I also expect $y_1$ to be negatively correlated with $y_2$ because the satellite only views the highest clouds. I use OLS regression to determine the relationship
$y_i = \beta_i x_i + \epsilon$ for $i \in {1,2}$
The standard error for the regression slopes is $\sigma_1$ and $\sigma_2$. Now I want to calculate a linear combination of the regression slopes: $C=a_1 \beta_1 +a_2 \beta_2$ where $a_1$ and $a_2$ are real-valued constants. I think that the standard error for $C$ can be calculated from the relationship
$\sigma_C^2 = a_1^2 \sigma_1^2 + a_2^2 \sigma_2^2 + 2a_1 a_2 \sigma_{1,2}$
where $\sigma_{1,2}$ represents the covariance between $\beta_1$ and $\beta_2$. My main question is how does one estimate $\sigma_{1,2}$ given that $\beta_1$ and $\beta_2$ are calculated from different variables?