Continuous by Continuous Interaction in Poisson Model

[Maxence Morlet]

Hi all,

I have a regression of the type:

y = exp(b1*x1 + b2*x2 + b3*x1*x2)*error

y is a count variable: number of employees in a given firm. x1 and x2 are both continuous. However, they are on completely different scales. They are in level, i.e. not logged or exponentiated. x1 is the GDP growth rate, x2 is a continuous variable with range [-0.65 ; 0.756] I am interested in the interaction effect between x1 and x2.

I understand that in light of McCabe et al (2022), and the seminal paper by Ai and Norton (2003), the interaction effect in nonlinear models is completely different from b3.

I applied the methodology of Leitgoeb (2014): https://www.stata.com/meeting/german...4_leitgoeb.pdf

I obtained the total interaction effect. My question is the following:

x1 is the regressor of interest, x2 acts as the moderator. How do I interpret the interaction effect? In particular, how do I interpret its magnitude?

Furthermore, given that x1 and x2 are on completely different scales, can I say the following: "A unit rise in x2 changes the semi-elasticity of y to x1 by the magnitude of the interaction effect".

Many thanks in advance for your help!

[Maarten Buis]

the interaction effect in nonlinear models is completely different from b3.

That statement isn't completely wrong, but it is not completely right either. The somewhat trivial point Ai and Norton made was that if you interpret coefficient wrongly, then that is wrong. I have no quarrels with that. They also made the point that interaction effects in non-linear model (in particular logistic regression) are often misinterpreted. Again, no quarrel. However, I do have a quarrel with their final point, that it is humanly impossible to interpret such an interaction effect. If they said that economists are incapable of interpreting such an effect, basically saying economist can add or subtract two numbers, but cannot multiply or divide two numbers, then that would fit nicely with the stereotypes sociologists have about economists (there is a lot of bling, but the basics are sorely lacking). However, I don't think that that was the point they were trying to make.

As always with interactions it really helps to center your continuous variables. In this example I centered grade at 12 (highschool) and hours at 40 (full-time). So we see that we expect someone with only high school and who works 40 hours per week earns 7.02 dollars an hour (the constant). The hourly wage increases by (1.09-1)*100%=9% when someone attains an extra year of education if that person works 40 hours per week, i.e. if the other variable in the interaction is 0. This is why centering hours at 40 is so important: otherwise we would look at the effect of grade on wage when someone works 0 hours per week... Similarly, working an extra hour per week is associated with a wage increase of 1% if someone has only finished high school. This effect of working hours decreases -0.125% for every year of education you attain. So rather than adding a main effect and an interaction effect, as you would in a linear model, we now need to multiply the main effect and the interaction effect in a model with a log link-function, like logit or poisson, and that is it. I think even economists can just about manage that.

, i

. sysuse nlsw88, clear
(NLSW, 1988 extract)

. gen gradec = grade - 12
(2 missing values generated)

. gen hoursc = hours - 40
(4 missing values generated)

. poisson wage c.gradec##c.hoursc, irr vce(robust)
note: noncount dependent variable encountered; results correspond to an exponential-mean model rather than a poisson
      model.

Iteration 0:  Log pseudolikelihood = -7074.3485  
Iteration 1:  Log pseudolikelihood = -7074.3466  
Iteration 2:  Log pseudolikelihood = -7074.3466  

Poisson regression                                      Number of obs =  2,240
                                                        Wald chi2(3)  = 421.14
                                                        Prob > chi2   = 0.0000
Log pseudolikelihood = -7074.3466                       Pseudo R2     = 0.0769

-----------------------------------------------------------------------------------
                  |               Robust
             wage |        IRR   std. err.      z    P>|z|     [95% conf. interval]
------------------+----------------------------------------------------------------
           gradec |   1.093891   .0058122    16.89   0.000     1.082558    1.105342
           hoursc |   1.011665   .0017104     6.86   0.000     1.008318    1.015023
                  |
c.gradec#c.hoursc |     .99875   .0005389    -2.32   0.020     .9976944    .9998067
                  |
            _cons |   7.024667   .1274847   107.42   0.000     6.779193     7.27903
-----------------------------------------------------------------------------------
Note: _cons estimates baseline incidence rate.

[Maxence Morlet]

Thank you so much for the response!