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st: RE: Not Quite Quadratic Regression


From   "Jacobs, David" <jacobs.184@sociology.osu.edu>
To   "'statalist@hsphsun2.harvard.edu'" <statalist@hsphsun2.harvard.edu>
Subject   st: RE: Not Quite Quadratic Regression
Date   Sat, 4 Aug 2012 17:21:14 +0000

One simple alternative that you might not have tried is to test the joint significance with Stata's test or testparm command.  

Sometimes the coefficients on the two terms of a quadratic specification do not reach significance but their joint effects are significant.  Worth trying if you haven't.

Dave Jacobs

-----Original Message-----
From: owner-statalist@hsphsun2.harvard.edu [mailto:owner-statalist@hsphsun2.harvard.edu] On Behalf Of A. Shaul
Sent: Saturday, August 04, 2012 6:58 AM
To: statalist@hsphsun2.harvard.edu
Subject: st: Not Quite Quadratic Regression

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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