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.