I am looking to compare regression coefficients between two regression models. Each model has the same four independent variables: two predictors of interest (we'll call them A and B) and two control variables (C and D). The only difference between the two models is that they have different dependent variables: the first model is predicting DV1, while the second model is predicting DV2. All observations are from the same sample, so the regression coefficients are dependent.
I believe that both A and B will more strongly predict DV1 than DV2. In other words, the regression coefficient for A predicting DV1 (controlling for B, C, and D) should be higher in magnitude than the regression coefficient for A predicting DV2 (controlling for B, C, and D). Similarly, the regression coefficient for B predicting DV1 (controlling for A, C, and D) should be higher in magnitude than the regression coefficient for B predicting DV2 (controlling for A, C, and D).
Essentially, I want to test the difference between two dependent regression coefficients from two models that share all of the same IVs, but have different DVs. Is there a formal significance test I can use?