# Is there a hypothesis test for B1 > B2 in multiple regression? [duplicate]

Is there a way to calculate a p-value for the hypothesis that population coefficient regression coefficient $$B_1$$ is larger than that of $$B_2$$ when doing multiple regression on a sample group?

In the test I'm trying to perform $$x_1$$ and $$x_2$$ are both binary, but I'm interested in the general case as well.

• I thought I had written an answer like this before but I can't locate one so I've written a brief one. [If I do find the supposed original this one would close as duplicate.] Commented Sep 17, 2015 at 3:04
• Your title refers to 'multiple regression' but your body text to 'multivariate regression'. When you say 'multivariate'; are you talking about multiple responses (DVs) or multiple predictors (IVs)? Commented Sep 17, 2015 at 22:04
• @Glen_b my mistake, I am talking about multiple predictors, not multiple response variables. Will edit. Commented Sep 21, 2015 at 3:49
• @Glen_b This one? There are quite a few questions of this form floating around. Commented Sep 21, 2015 at 4:10
• @Affine that looks to be equivalent. Out of curiosity, how did you find that? I had a very hard time trying to find an answer on my own before posting. Commented Sep 21, 2015 at 5:23

Yes; reparameterize it as $\beta_2=\beta_1+\delta$, so that your predictors are no longer $x_1,x_2$ but $x_1^*=x_1+x_2$ (to go with $\beta_1$) and $x_2$ (to go with $\delta$)
[Note that $\delta = \beta_2-\beta_1$, and also $\hat{\delta}=\hat{\beta}_2-\hat{\beta}_1$; further, $\text{Var}(\hat\delta)$ will be correct relative to the original.]
Then test the null of $\delta=0$ against the alternative of $\delta<0$.
[Alternatively, identify the matrix $C$ defining the linear restriction under the null and test the general linear hypothesis $C\beta=0$; for example, see the extensive description via F or t tests here. Since your alternative is one-tailed, you'll want the t-form.]