Polynomial regression help

#1
Hello,
I am working in retailing here in Europe, and we are trying to support evidence that sales are positively correlated with space allocated, but that there is a "saturation point" beyond which marginal sales generated by additional retail space are getting close to zero.
We are working with a panel of c. 190 stores across 19 product categories, and are trying to "force" the identification of the saturation point by using a polynomial regression on sales vs. m² with degree 3 :
Sales = aX^3 + bX^2 + cX + d
This means that we obtain virtually every time a nice "S-curve" that would be expected...
Judging from the F-stats our models are always very significant (p<0,0001), but t-stats are less satisfactory (even if they always relate to the same "X").
The models also seem to make sense vs. intuition (good news),
Has anyone ever done something similar ?
Are t-stats additive (whether it is for a, b or c, there is always at least one that is relevant at 0,001 level) ?
Is the approach robust in terms of modeling ?
Thanks for your help.
 
#2
Hi Grace, I'm not sure if a polynomial based regression is the way to show the saturation point you're looking for. Of course the value of the coefficient will be smaller with ^3 compared to ^2 but that's only logical when there's a positive correlation. The coefficient is merely correcting for the larger X value in relation the Y value.

What I would do in this scenario is split your sample into different clusters according to the size of the store. Assuming there are no additional variables to include (e.g.store location, or other sales affecting variables) and attach a dummy variable. This way you could compare the value of the coefficients with each different store-size.

I'd bet the coefficient remains significant in all cases but the coefficient value decreases when testing larger store-sizes.
GL!