# How to decide whether a variable belongs to a linear model?

I have a set of inputs $x$ and noisy outputs $y$. I think that either $$y = a_0 + a_1 x$$ or $$y = a_0 + a_1 x + a_2 x^2.$$ How can I determine which model was more likely to have generated the data?

Could this be done by cross-validating ordinary least squares and lasso regression on these data, and then doing a paired hypothesis test to see which has the better median coefficient of determination?

• If you mean "Given the correct model is one of these two, which has a greater probability of having generated the data?", that's a question I'd approach from a Bayesian viewpoint. If you mean something other than that, I think you'd need to be more precise about what you actually mean by 'more likely' (I presume it's not intended to be formally a reference to likelihood). – Glen_b -Reinstate Monica Mar 19 '15 at 8:00
• Yes, that is what I mean. How can I approach it this way? – rhombidodecahedron Mar 19 '15 at 21:54

• Isn't the less complicated model a subset of the other one, not vice versa? Or am I misunderstanding? The way I'm seeing it is MC = {1, x, x^2}; LC = {1, x}, therefore LC $\subset$ MC. – rhombidodecahedron Mar 19 '15 at 21:48