Testing for moderation (turning point shift) in U-shaped relationships

[Zeinab Ghamloush]

Dear Stata experts,

I am trying to test the effect of a moderator variable Z on the U-shaped relationship between X and Y. Hence, I introduced the interactions between the moderator, Z and X and its square as well as Z's main term. Normally, there is two types of moderation: 1) Turning point shift and 2) flattening or steepening.
The following specification allows for isolated test of each of the moderation types:

Y = 𝛽0 + 𝛽1X + 𝛽2X2 + 𝛽3XZ + 𝛽4X2Z + 𝛽5Z

To illustrate how the moderator affects the turning point of the U-shaped relationship, we derive the turning point X* by setting the first derivative with respect to X to zero: X* = −𝛽₁ − 𝛽₃Z / 2𝛽₂ + 2𝛽⁴Z equation(1)
To show how the turning point changes as Z changes, we take the derivative of this equation with respect to Z: 𝛿X/𝛿Z = 𝛽₁𝛽₄ − 𝛽₂𝛽₃ / 2 ( 𝛽₂ + 𝛽₄Z )^₂ equation (2)

Thus, the turning point shift depends not only on 𝛽₃, but also on 𝛽₁, 𝛽₂, and 𝛽₄.
I read that, in order to test formally whether a shift in the turning point occurs, one has to assess whether Equation 2 as a whole is significantly different from zero.
My question is how to test whether equation 2 is statistically different from zero using STATA.

Your answer is highly appreciated!
Thank you in advance

[Andrew Musau]

In linear regression, the reported F-statistic is from the test that all slope coefficients are not statistically different from zero. So if you fail to reject the null, you have evidence that all slope coefficients are not statistically different from zero.

[Joseph Coveney]

My question is how to test whether equation 2 is statistically different from zero using STATA.

If your Equation 2 is intended to be Equation 11 of Richard Haans, Constant Pieters & Zi-Lin He, Thinking about U: Theorizing and Testing U- and inverted U-shaped Relationships in Strategy Research. Strategic Management Journal 37(7):1177–95 (July 2016), then according to its Footnote 11, "In [Stata], the nlcom command can be used to test whether Equation 11 is indeed significantly different from zero at specific values of Z."

You can find an example of Stata code for doing exactly this in an earlier post on the list here.

This thread as well as this one might also be of interest.