How to conduct a stat test for the difference between two dummy variables in the model

This is a question post.

Imagine that I have a regression model:

Y=b0+b1*black+b2*white+b3*asian+ error

where the ommited category is hispanic.  Because of this omission, each of race variable coefficients corresponds to the difference between each one of the race variables and Hispanic subjects (in other words, Hispanic is being the reference group).

If I want to know if the difference between white and black is statistically significant, I could omit black and see if the coefficient for white subjects is statistically significant (or I can omit white instead).  This, however, requires running of one whole separate model (though it is a mathematically equivalent model as the original model).

Another approach in SAS would be to request stat tests evaluating every possible contrast between the race groups, but this again involves an extra step.

Is there an easy to just to read information I already have from the original model and evaluate the between-group differences statistically (e.g., black vs. white, asian vs. black, white vs. black)?  For example, if I have this table below (w/ hypothetical values), is it possible to know if the black vs. white difference is statistically significant?  Does this table give me sufficient information to know if the white vs black difference (2.1-3) is statistically significant?

coeff. stderr. prob.
Intercept 1.2 (0.2) 0.1
black 2.1 (0.1) 0.6
white 3 (0.2) 0.05
asian 4 (0.01) 0.01

My hunch is that I can pool two standard errors (one from black and the other from black) and use it to evaluate the black and white difference (and somehow I have to figure out what DF should be). However, I don't think pooled standard errors are utilized for stats tests that are being reported already in this table (e.g., black effect is evaluated only based on its own standard error). It would strange that I had to rely on pooled standard errors.

My goal is to create an Excel sheet that does this calculation, so people can conduct the test without necessarily rerunning the models again (and they just rely on result tables).


Example results: variable interpretation of reg results.xlsx

Thank you for your input on this (please email k u e k w a @ GMAIL) or leave a comment below.

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