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.
Best Answer
To elaborate on (and in fact, make more precise my part of) the discussion in the comments a bit:
Variance-covariance matrices are positive semi-definite, as discussed for example in @DilipSarwate's answer here:
(As the general notation suggests, this issue therefore also has nothing to do with coefficient estimates, but applies for all random variables.)
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 $$ \operatorname{cov}(X_i,X_j)=-\frac{\operatorname{var}\left(X_1\right) +\operatorname{var}\left(X_2\right) }{2} $$