Is it possible to specify a vector of adjusted $R^2$ values (or any other measure like AIC, BIC, $C_p$) for the set of all possible models in a data set, and then generate data that is consistent with that pattern of adjusted $R^2$? What restrictions would we have to place on the adjusted $R^2$ vector?
For example, say there are 3 independent variables. That means there are $2^3=8$ possible models (no interactions or transformations of the IV's). Can I specify a vector of adjusted $R^2$ values that correspond to each of the 8 possible models and then generate data that is consistent with that vector?
My intuition is that it should be possible. Adjusted $R^2$ reduces to some function of $RSS$ (and the number of parameters and n, which we can specify). Then, we can create any $X$ matrix we want and solve for $y$ values and the true $\beta$'s.
It will be hard, and the process will likely generate some finite sample correlation between the $X$'s and $\varepsilon$'s, even if they're drawn independently. But in expectation with large N, it seems possible to converge to that pattern of $R^2$.
Is this possible? And what does a sketch of the procedure look like?