Inverted U-shaped/Quadratic moderation in regression model

[Yannick Beunderman]

Hi,

I want to test for an inverted U-shaped moderating effect in my regression model. I was thinking of:

xtpoisson divestment_rate c.tmt_deg_rep##c.tmt_age_diversity##c.tmt_age_dive rsity c.tmt_deg_rep##c.roa_av_2y log_emp tmt_average_tenure tmt_average_age tmt_members, re

in which

c.tmt_deg_rep##c.tmt_age_diversity##c.tmt_age_dive rsity

must present the quadratic moderating effect.

Is this the right way to perform a quadratic moderating effect?

Thank you in advance!

[Clyde Schechter]

Almost. You have to get rid of the blank space in the middle of the second occurrence of c.tmt_age_diversity. Otherwise it looks right.

[Yannick Beunderman]

Clyde Schechter thank you for your response and help!

Now that I have executed the regression, I was wondering which margins command is the right one to show a quadratic moderation effect, since

margins, at(c.tmt_deg_rep=(0(.2)1) c.tmt_age_diversity=(0.0199753 0.2202453) )

would provide a graph showing just a normal interaction effect instead of a quadratic moderation effect of tmt_age_diversity, isn't it?

Could you maybe help me with this issue as well, please? Thank you in advance!

[Clyde Schechter]

margins, at(c.tmt_deg_rep=(0(.2)1) c.tmt_age_diversity=(0.0199753 0.2202453) )

would provide a graph showing just a normal interaction effect instead of a quadratic moderation effect of tmt_age_diversity, isn't it?

No, that -margins- command is correct for your purposes. It does account for the quadratic effect moderation. That's the beauty of factor-variable notation: -margins- knows that tmt_age_diversity appears as both a linear and quadratic term in the regression model, and it calculates the predictive margins accordingly.

[Nitin Jain]

No, that -margins- command is correct for your purposes. It does account for the quadratic effect moderation. That's the beauty of factor-variable notation: -margins- knows that tmt_age_diversity appears as both a linear and quadratic term in the regression model, and it calculates the predictive margins accordingly.

Hi, Clyde, Can you suggest any paper that talks about when to moderate only the linear term and when to moderate both the linear and quadratic terms of an inverted relationship. Are there any tests for these. Thanks.

[Clyde Schechter]

Can you suggest any paper that talks about when to moderate only the linear term and when to moderate both the linear and quadratic terms of an inverted relationship.

Although it does not mean they do not exist, I do not know of any papers dealing with this question. My advice is based on understanding the underlying mathematics. In a quadratic polynomial, the linear coefficient is not the slope of any relationship. It is not a marginal effect. If y = c0 + c1*x + c2*x^2, then the marginal effect of x at any value of x is dy/dx = 2*c2*x + c1. So while c1 is a piece of the marginal effect, it is not the entire thing. In fact, the primary impact of the linear coefficient on a quadratic is its effect on the location of the turning point (vertex of the parabola), which occurs where x = -c1/(2*c2). In any event you can see that if we have a moderator and apply it only to c1, and not c2, it will only partially capture the modification of the marginal effect of x, and it will also have a (probably unintended) impact of shifting the location of the turning point. Perhaps somebody can imagine a highly unusual situation where this is what you are trying to accomplish with a moderator, but I have never seen a situation where that would be the case and my imagination does not reach that far.

So my advice is quite simple: never apply a moderator to only the linear term of a quadratic polynomial.

[Nitin Jain]

Although it does not mean they do not exist, I do not know of any papers dealing with this question. My advice is based on understanding the underlying mathematics. In a quadratic polynomial, the linear coefficient is not the slope of any relationship. It is not a marginal effect. If y = c0 + c1*x + c2*x^2, then the marginal effect of x at any value of x is dy/dx = 2*c2*x + c1. So while c1 is a piece of the marginal effect, it is not the entire thing. In fact, the primary impact of the linear coefficient on a quadratic is its effect on the location of the turning point (vertex of the parabola), which occurs where x = -c1/(2*c2). In any event you can see that if we have a moderator and apply it only to c1, and not c2, it will only partially capture the modification of the marginal effect of x, and it will also have a (probably unintended) impact of shifting the location of the turning point. Perhaps somebody can imagine a highly unusual situation where this is what you are trying to accomplish with a moderator, but I have never seen a situation where that would be the case and my imagination does not reach that far.

So my advice is quite simple: never apply a moderator to only the linear term of a quadratic polynomial.

Ok, thanks, Clyde.