I am interested in comparing two design matrices in which the parameter values of interest are perfectly negatively correlated. Of course, via traditional GLM, these design matrices yield identical residuals, as the sign of the beta corresponding to the parameter of interest flips between models (but the absolute value remains the same). The direction of these parameter values are informative, and it seems to me that it is, therefore, inappropriate for these to be assigned a negative beta.
I have considered using a non-negative least squares algorithm to solve this problem. However, in this case, beta weights of all predictors become non-negative (specifically, they take on the value of "0"). I wonder if it is possible to constrain only the value of the beta weight corresponding to this particular parameter.
Any advice would be appreciated.