# Why can we mix standard errors with raw covariances when calculating standard error of sum of regression coefficients?

Let's have a look at this post: Standard error for the sum of regression coefficients when the covariance is negative

We have: $$SE_{b_{2+3}} = \sqrt{SE_2^2 + SE_3^2+2Cov(\beta_2,\beta_3)}$$

But why do we mix SE, which is the SD/sqrt(N) with covariance (not divided by the sqrt(N))? I know this formula for variances, but SE is not the sqrt(variance).

Could I kindly ask someone to show me, using algebra, how is this valid?

I mean - why can I take the SE from the model coefficients and use the variance-covariance matrix without any additional steps?

• Re "but SE is not the sqrt(variance)": in this setting, yes it is. You are confusing two formulas applicable in two different situations. No algebra is needed.
– whuber
Jun 10, 2021 at 12:47
• Thank you. So, the variance-covariance matrix contains squared standard errors (diagonal) and covariances (off-diagonal)? Not just variances? Are these covariances, then, the standard covariances, or covariances divided by the sqrt(n)? I'm sorry for dumb questions, but I try to figure it out and learn. Jun 10, 2021 at 14:07
• The division by $\sqrt{n}$ has, in effect, already been performed when the covariance matrix of the estimates is constructed.
– whuber
Jun 10, 2021 at 14:40