st: Not Quite Quadratic Regression

["A. Shaul" <3c5171@gmail.com>]

Hello Statalisters,

Theory predict an u-shaped relation between two variables, y and x.
When I perform a quadratic linear regression with a model like

     y = ax + bx^2 + constant + error,

the coefficients a and b are not significant. However, if I change the
exponent to something less than 2, e.g. 1.5, I obtain significance. In
other words a model like

     y = ax + bx^1.5 + constant + error,

yields significant estimates of a and b. The curvature is still quite
marked using the exponent of 1.5. I can even use an exponent of 1.1
and obtain significance and a nice shape. But I don't think I can
simply choose the exponent based on whatever yields significance. Or
can I? This is my question.

I have tried to run a non-linear regression where the exponent was a
free parameter. Although it tend to yield an exponent around 1 to 2,
everything turns out highly insignificant. If I plug the estimated
exponent into an OLS model, like the ones above, I get significance. I
have also tried to use splines as well as a piecewise constant
formulation. Again the results are less than ideal (although I get the
same overall picture).

The non-linearity is rather apparant in a scatterplot (although
extremely noisy), and the problem shows up when controlling for other
covariates where a simple graphical/nonparametric approach is
unfeasible.

Needless to say, I have been searching high and low for an answer
before posting here. This is my first message to Statalist (although I
am an avid reader of the archives). I hope my question is fine.

Thank you in advance
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