the margin effect of interaction term in nonlinear model (e.g. xtgee, family(binomial) link(probit)

[David Wong]

Dear Stata Statisticians,

I have one unbalanced panel data with large N and Small T.
The model is fractional response model for panel data (Papke and Wooldridge 2008).

 xtgee y  c_x1 c_x2  y1998 y2000 y2002  c.c_x1#c.c_x2 , fam(bin) link(probit) corr(uns) vce(robust)

Where y is one fractional response variable, arranging from 0 to 1;
c_x1 is one centered continuous variable;
c_x2 is one centered continuous variable;

Since the model I use is basically one nonlinear model if I am not wrong, I cannot directly get the average marginal effect of

c.c_x1#c.c_x2

by

margins,dydx(*)

. I read Williams (2015) in which I get the information that

People often ask what the ME of an interaction term is. Stata’s margins command replies: there is not one. You just have the MEs of the component terms. The value of the interaction term cannot change independently of the values of the component terms, so you cannot estimate a separate effect for the interaction. The older mfx command will report MEs for interaction terms, but the numbers it gives are wrong because mfx is not aware of the interdependencies between the interaction term itself and the variables used to compute the interaction term.

I am not an expert in Econometrics and I sincerely look forward the answer that

Whether now can we have the marginal effect of interaction effect in nonlinear model?

Many thanks and Best regards,
David

Reference:
Papke, L.E. and J.M. Wooldridge (2008), “Panel Data Methods for Fractional Response Variables with an Application to Test Pass Rates," Journal of Econometrics 145, 121-133.

Williams, R. (2012). Using the margins command to estimate and interpret adjusted predictions and marginal effects. Stata Journal, 12(2), 308.

http://www.statalist.org/forums/foru...ctional-models

[Clyde Schechter]

No. I don't think I can say it more clearly than Richard Williams did. But I can say it more emphatically. It is not that Stata refuses to provide you with the marginal effect of an interaction term (in any model, linear or otherwise). It is that there is no such thing. It does not exist. The interaction term is one of the "ingredients" in calculating the marginal effects of its constituent terms, but it does not, itself, have a marginal effect.

[David Wong]

No. I don't think I can say it more clearly than Richard Williams did. But I can say it more emphatically. It is not that Stata refuses to provide you with the marginal effect of an interaction term (in any model, linear or otherwise). It is that there is no such thing. It does not exist. The interaction term is one of the "ingredients" in calculating the marginal effects of its constituent terms, but it does not, itself, have a marginal effect.

Dear Clyde,

Thank you for your answer, which is very helpful. Since there is no marginal effect of interaction in nonlinear model, what is the usual way that we report in our empirical results?

I have been aware that I can use

marginsplot

to visualize the interaction effect between two variables on the dependent variable. However, I am not sure whether I report the coefficient of this interaction in the nonlinear model in the table simultaneously is correct or not. I really appreciate your further guidance.

Best regards,
David

[Clyde Schechter]

Since you are talking about the interaction between two continuous variables, I think that the most useful way is to report things is to select "interesting" values of variables x1 and x2. (If no particular values are especially interesting, then just sets of values that more or less span the range of observed values). Let's say they are 1, 2, 3, 4, 5 for x1, and 20, 40, 60, 80 for x2. Then the predicted values of the outcome at all combinations of those are of interest. And so would be the marginal effects of those at all combinations of them:

margins, at(x1 = (1 2 3 4 5)) at(x = (20 40 60 80))
marginsplot
margins, dydx(x1 x2) at(x1 = (1 2 3 4 5)) at (x2 = (20 40 60 80))
marginsplot

Whether to present those results as tables or as graphs is a matter of a) tastes, b) preferences of your intended audience, c) relevance to the focus of your research interests.

Presenting the interaction coefficient itself, with continuous variables, is usually not particularly informative here. (With dichotomous variables, it is easier to understand what it means, and it is often germane.)

[David Wong]

Since you are talking about the interaction between two continuous variables, I think that the most useful way is to report things is to select "interesting" values of variables x1 and x2. (If no particular values are especially interesting, then just sets of values that more or less span the range of observed values). Let's say they are 1, 2, 3, 4, 5 for x1, and 20, 40, 60, 80 for x2. Then the predicted values of the outcome at all combinations of those are of interest. And so would be the marginal effects of those at all combinations of them:

margins, at(x1 = (1 2 3 4 5)) at(x = (20 40 60 80))
marginsplot
margins, dydx(x1 x2) at(x1 = (1 2 3 4 5)) at (x2 = (20 40 60 80))
marginsplot

Whether to present those results as tables or as graphs is a matter of a) tastes, b) preferences of your intended audience, c) relevance to the focus of your research interests.

Presenting the interaction coefficient itself, with continuous variables, is usually not particularly informative here. (With dichotomous variables, it is easier to understand what it means, and it is often germane.)

Dear Clyde,

Your answer is very informative and helpful. I will report both the tables and graphs in the draft of paper. Thank you so much for your kind help.

Best regards,
David

[Phil Bromiley]

Let me try to add a little to Clyde's comments. Think about y = b1 x1 + b2 x2 + b3 x1*x2. There is no margin on an interaction because what you really want to talk about is the influence of x1 or x2. The interaction makes the influence of x1 vary with x2 so the way to think about it is the influence (margin) of x1 at different values of x2 (or vice versa).

There is a substantial literature on interactions in non-linear models. The short answer is the simple things that work with linear models cannot be depended on in non-linear models. You might find William Greene, Testing hypotheses about interaction terms in nonlinear models, Economics letters, 2010 helpful.

[David Wong]

Let me try to add a little to Clyde's comments. Think about y = b1 x1 + b2 x2 + b3 x1*x2. There is no margin on an interaction because what you really want to talk about is the influence of x1 or x2. The interaction makes the influence of x1 vary with x2 so the way to think about it is the influence (margin) of x1 at different values of x2 (or vice versa).

There is a substantial literature on interactions in non-linear models. The short answer is the simple things that work with linear models cannot be depended on in non-linear models. You might find William Greene, Testing hypotheses about interaction terms in nonlinear models, Economics letters, 2010 helpful.

Dear Phil,

Thank you so much for providing this paper to me. I am reading it now.

Best regards,
David