I would like to conduct a F-test (Wilks' Lambda test, ideally) to test the equality of the slope coefficients for a single independent variable on two dependent variables. Using the mtcars dataset, the model would be:

data(mtcars)
car.mod<- lm(cbind(mpg, disp) ~ wt, mtcars)

I need to determine whether the slope of wt vs. disp and wt vs. mpg within this model are equivalent. I was hoping there was a way to do this with the linearHypothesis function. The "Mean/Variance" response to this question is very close to what I was hoping for, just with a Chisq test.

The Z-test suggested as a response to this question came close to what I need, but is obviously also the wrong test.

My last option would be fitting the null model, but I'm not sure how to specify a multivariate regression model where the slopes are equivalent, but the intercepts are not.