Quadratic term and significance level

[Neyla Tad]

Dear all,
I am running a differences-in-differences model. Theory indicates that there is a non-linear relation between the outcome y and the x variable (control variable). When I run the regression (1) below without the quadratic term, I have a beta (treat*postevent multiplier) significant and negative, as expected. Treat is a dummy for the treatment group (in a panel with fixed effects, it will be dropped). Postevent is a dummy for the period post-shock. X is a continuous variable used as an important control. Treat*postevent is my differences-in-differences estimator.

(1) xtreg y treat postevent treat*postevent x, fe rob

However, when I add the quadratic term x squared (see equation 2 below), I have a beta (treat*postevent multiplier) not significant.

(2) xtreg y treat postevent treat*postevent x x^2, fe rob

What intrigues me is the fact that x squared is not significant at all. Then, which result is valid? The difference-in-differences estimator in the regression without the squared x, or the differences-in-differences estimator with the x squared (although x squared is not significant)?
I would appreciate any help.
Thank you all.

[Maarten Buis]

\(x\) and \(x^2\) are strongly correlated unless you centered \(x\) at the mean before squaring. So the statistical power will be low and the statement that the parameter of \(x^2\) is not significant is pretty much meaningless. So first step would be to center \(x\) at the mean and add that and its square to the model. Even if that square term is still not significant then that still does not mean that the effect of \(x\) is linear; a parabola is only one very specific way in which an effect can be non-linear and there are many others. You could try the fp prefix.

[Neyla Tad]

Thank you, Mr. Buis, for the excelent explanation.
I centered x at the mean, and then squared it. The results remain the same: I have a diff-in-diff estimator and a x squared beta with no significance.
However, I am interested in the diff-in-diff estimator and NOT on the x effect. Which diff-in-diff I should interpret? The diff-in-diff in the regression without the x squared or the diff-in-diff in the regression with the (insignificant x squared)?

Also, I cannon install the fp command in STATA 12 (my latest version). I found a fracpoly command, but still having problems with the sintax, since I have a lot of controls in my regression, and fracpoly seems to allow only one.

[Maarten Buis]

If you want to control for x and you have good reasons to suspect that the relationship between x and y is non-linear, then you have to take that seriously. Adding a square term is one simple way of adding non-linearity, which is nice if it works. However, if your square term is not significant that does not tell you the relationship is linear; you only compared a linear with a very specific non-linear effect. I suggested in #2 you look into the fp prefix, to more fully explore the nonlinearity of the relationship between x and y, and I do so again.

[Neyla Tad]

Thank you again!
However, as I mentioned before, I cannot find a way to use the fp command in STATA 12. Is it available for this STATA version? If not, can I use BIC and AIC criteria to choose the best specification?

[Maarten Buis]

Ok, I missed the version. In Stata 12 it was called fracpoly.

Before you go and look BIC and AIC you first need to decide which models you want to compare...