# How to construct the standard error for the sum of three parameter estimates corresponding to two dummy variables and their interaction term?

I have a simple multiple linear regression model with an interaction term, and I would like to construct a standard error to support interpretation. My goal is to construct a standard error for an estimate of the sum of coefficients β1, β2, and β3. Suppose the dependent variable Y is income. Suppose X1 refers to gender such that X1=0 represents an individual who is "not female" and X1=1 represents one who is female. Suppose X2 refers to education, where X2=0 means the individual has no college degree and X2=1 means they have a college degree. A third term is included where X1 and X2 are multiplied, and the parameter for this interaction term is β3.

Y = β0 + β1*X1 + β2*X2 + β3X1*X2 + ε


I want to compare the income of a college-educated female with the income of a non-female with no college degree. My understanding is this would require β1, β2, and β3 to be summed. How would I construct the standard error for the sum of these three coefficients? Below is the covariance matrix. I have consulted this thread, and a few textbooks including Wooldridge, but have not been successful. Can anyone provide some clarity here along with a citation from a published source? It would also be helpful to know if I am approaching this incorrectly. Thanks.

X1 X2 X1*X2
X1 2.105816e-03 -1.477463e-05 -2.021945e-03
X2 -1.477463e-05 1.889486e-04 -2.801293e-05
X1*X2 -2.021945e-03 -2.801293e-05 2.290629e-03
• Apply the bilinearity property as explained at stats.stackexchange.com/questions/38721. That is also explained in the thread you reference, so could you please explain how it was "not ... successful"?
– whuber
Commented Sep 6, 2023 at 21:21
• Thank you for the reply. The approach I take at the moment uses Var(X1) + Var(X2) + Var(X1*X3) + 2*Cov(X1, X2) + 2*Cov(X1,X1*X2) + 2*Cov(X2, X1*X2), which ultimately produces a SE of ~0.02135248. Is this the correct application of the bilinearity property? In textbooks I have looked for specific examples, but not found a citation I prefer. E.g., Example 7.9 in Wooldridge's Introductory Econometrics textbook has a specific example with this model setup and comparison of interest, but the book only explains coefficient interpretation, not SEs. Commented Sep 7, 2023 at 2:54
• The form of your formula looks correct but the notation is confusing: the variances and covariances concern the parameter estimates $\hat\beta_i$ rather than the variables themselves.
– whuber
Commented Sep 7, 2023 at 12:04
• Thank you very much! Commented Sep 7, 2023 at 13:42