Confidence interval around difference of regression coefficients I am interested in comparing standardized regression coefficients within the same model. A linear hypothesis test (e.g., car::linearHypothesis in R) can tell me whether the difference between them is significant, and I can calculate the difference to get a point estimate. However, I am not sure how to get a CI around that.
Currently, I believe that I can use $\beta_\delta$ + 1.96 * $\sqrt{SE1^2+SE2^2}$ in the same way that I would deal with the difference between two means, but I am not sure if that is applicable in this case?
 A: Your formula isn't correct, because regression coefficient estimates are typically correlated with each other. It assumes that the variance of the difference of the estimates equals the sum of their individual variances. That's only true for uncorrelated variables. Statistical software can report the variance-covariance matrix for the coefficient estimates. The off-diagonal values of that matrix represent the corresponding covariances between pairs of coefficient estimates. The linear hypothesis tests take the covariances into account.
The Wikipedia page shows the more general formula for the variance of a weighted sum or difference of two correlated variables. In you case for a simple difference, it's the sum of the individual variances minus twice their covariance.
$$\operatorname{Var}(X - Y) =\operatorname{Var}(X) +  \operatorname{Var}(Y) -2 \operatorname{Cov}(X, Y)$$
Taking the square root gives the standard error of the difference. The factor of 1.96 is for 95% confidence limits based on a normal distribution.
