# Standard error for the sum of regression coefficients when the covariance is negative

I have a question about appropriately calculation the standard error for the sum of two coefficients in a linear regression model. My question is similar to this and this, but I can't seem to solve the problem from the answers presented there.

I have a model of the following form:

$Y = \alpha + \beta_1X1 + \beta_2X2 + \beta_3X1X2$

I would like to be able to calculate the marginal effect of $X2$ on $Y$ for $X1 = 0$ and $X1=1$. I think I am right in saying that the point estimates for these marginal effects is simply $\beta_2$ for the first case and $\beta_2 + \beta_3$ for the second case.

My question regards the appropriate calculation of the standard error. I think that this is the correct formula for the standard error of the $\beta_2 + \beta_3$ point estimate.

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

However, the problem arises from the fact that the model that I am estimating produces a covariance matrix that looks like this:

              Constant        beta_1        beta_2        beta_3
Constant  3.938580e-06 -6.259416e-06 -1.397691e-06  2.242824e-06
beta_1   -6.259416e-06  1.187334e-04  2.222736e-06 -4.738965e-05
beta_2   -1.397691e-06  2.222736e-06  5.457572e-07 -8.701802e-07
beta_3    2.242824e-06 -4.738965e-05 -8.701802e-07  2.004982e-05


Plugging the relevant values into the formula above results in a negative number below the square root sign, which clearly can't be right!

1.409763e-08 + 4.019951e-10 + 2*(-8.701802e-07) = -1.725861e-06


In more general terms, to me this seems like a more general problem where the covariance of two estimated parameters is negative and larger (in absolute terms) than the sum of the variances of those parameters. On the other hand, I may just be making a mistake somewhere. If anyone has any suggestions as to where I might be going wrong, it would be most appreciated.

• Can you post code that created the numbers? Are you sure you need to square the diagonal entries? Maybe these are variances already? – Christoph Hanck Jan 5 '16 at 17:56
• Thank you. You have pointed out my stupid mistake! Indeed, these are the variances, where I had been treating them as the standard errors. Problem solved. In the spirit of generalisability: is it ever the case that a negative covariance two parameters can be (in absolute terms) greater than the sum of the variances of two parameters? – user2728808 Jan 5 '16 at 17:59
• No, this cannot happen because covariance matrices are positive definite! – Christoph Hanck Jan 5 '16 at 18:06
• The diagonal elements in the covariance matrix are already the squared standard errors, but you are squaring them again. That's why you are getting erroneous results. It also looks like you used the wrong diagonal element in your first term. – whuber Jan 5 '16 at 22:57
• @user2728808, I elaborated upon my last comment a bit in an answer. – Christoph Hanck Jan 6 '16 at 8:56

The variance of a weighted sum $\sum_i a_i X_i$ of random variables must be nonnegative for all choices of real numbers $a_i$. Since the variance can be expressed as $$\operatorname{var}\left(\sum_i a_i X_i\right) = \sum_i \sum_j a_ia_j \operatorname{cov}(X_i,X_j) = \sum_i \sum_j a_ia_j \Sigma_{i,j},$$ we have that the covariance matrix $\Sigma = [\Sigma_{i,j}]$ must be positive semidefinite (which is sometimes called nonnegative definite).
If you specialize this to your problem $a_1=a_2=1$, you obtain the well-known result that $$\operatorname{var}\left(X_1+X_2\right) = \operatorname{var}\left(X_1\right) +\operatorname{var}\left(X_2\right) +2\operatorname{cov}(X_i,X_j)\geq0$$ Thus, the smallest possible covariance is $$\operatorname{cov}(X_i,X_j)=-\frac{\operatorname{var}\left(X_1\right) +\operatorname{var}\left(X_2\right) }{2}$$