Both significant quadratic (convex) and logaritmic relationship

[Bas Thomas]

Dear community,

I am running a cross-sectional regression on the cultural distance index and firm performance. Since many researchers suggest that the relationship is more than just a simple linear relationship, I am investigating if there is a possible non-linear relationship between the two variables of interest.
After running the regression models, the coefficient on the CD index (X) and its squared term (X^2) appear to be significant at the 5% level and suggest a convex relationship.
However, after running a level-log regression model (so without the X and X^2(!)), the natural logarithm of CD (ln[X]) is also statistically significant. It should be noted that the latter coefficient on lnX is negative and significant at almost the 5% level (alpha = 6.3). Could anyone tell me how it is possible that both relationships appear to be significant and how I should interpret these results?

Thanks in advance.

Best regards,

Bas Thomas

[FernandoRios]

The simplest scenario is that the relationship is in fact nonlinear, and that both log(x) and X and X^2 are capturing part of that nonlinearity. However, it is very likely that a more complex relationship exists across your variables.
For example, open the dataset "webuse motorcycle",
and run models with quadratic cubic or with logs and youwill see that they may appear to be significant
But if you see "deeper" into the the data, (a plot for example), the reason is that the relationship is in fact highly nonlinear.