# Comparing dependent regression coefficients from models with different dependent variables

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?

• If the units are different, you might also consider normalizing your data to be in standard deviation units. Feb 4 '14 at 23:25

It sounds like you are interested in testing hypotheses like $H_0:\beta_2=\alpha_2$. A typical way to test a hypothesis like this by looking to see if a t-statistic is greater than 2 in absolute value: \begin{align} t-stat &= \frac{\hat{\beta}_2-\hat{\alpha}_2}{\sqrt{V(\hat{\beta}_2-\hat{\alpha}_2)}}\\ \strut\\ V(\hat{\beta}_2-\hat{\alpha}_2) &= V(\hat{\beta}_2)+V(\hat{\alpha}_2)-2Cov(\hat{\beta}_2,\hat{\alpha}_2) \end{align}
When you run the two models separately, you can read off estimates of $\sqrt{V(\hat{\beta}_2)}$ and $\sqrt{V(\hat{\alpha}_2)}$ from the regression output---the standard errors of the coefficient estimates. But what do you do to get the covariance? Sometimes it may be reasonable to assume that this covariance is zero, but not often. SUR will calculate this covariance for you, making the calculation of the t-statistic possible.
In R, I think you want systemfit, and in Stata you definitely want sureg.
• (+1) Would there be any advantage to using multivariate regression (like Stata's mvreg) since the RHS variables are all the same? I think it should be a bit faster. Also, sureg carries out a $\chi^2$-based test, while mvreg does an F-test in this case. Feb 4 '14 at 23:41