Marginal effect of triple interaction terms

[Man Yang]

Hello folks, I am running a regression model to obtain diff-in-diff effect and I have pair-wise and triple interaction terms in the model. However, instead of using the interaction operator ##, I have the multiplicative terms in the model to represent interactions. My question is how to get marginal effect afterwards? Below are my model:

gen close_arv = close * arvbefore10
gen close_asian = close * ASIAN
gen asian_arv = ASIAN * arvbefore10
gen asian_arv_close = ASIAN * arvbefore10 * close

svy: reg TEACH i.arvbefore10 close_arv i.waob i.ASIAN close_asian asian_arv asian_arv_close female if agep>=35 & !missing(agep) & schlcat==5

below is my output:

. svy: reg TEACH i.arvbefore10 close_arv i.waob i.ASIAN close_asian asian_arv asian_arv_close
>  female if agep>=35 & !missing(agep) & schlcat==5
(running regress on estimation sample)

BRR replications (80)
----+--- 1 ---+--- 2 ---+--- 3 ---+--- 4 ---+--- 5 
..................................................    50
..............................

Survey: Linear regression                       Number of obs     =     78,579
                                                Population size   =  8,124,606
                                                Replications      =         80
                                                Design df         =         79
                                                F(  13,     67)   =     131.93
                                                Prob > F          =     0.0000
                                                R-squared         =     0.0338

-----------------------------------------------------------------------------------
                  |                BRR *
            TEACH |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
------------------+----------------------------------------------------------------
      arvbefore10 |
arrive before 10  |   .0374327   .0048218     7.76   0.000     .0278352    .0470302
        close_arv |  -.0154932   .0070831    -2.19   0.032    -.0295917   -.0013947
                  |
             waob |
   latin america  |   -.025105   .0100193    -2.51   0.014    -.0450479   -.0051621
           asian  |  -.0550817   .0102091    -5.40   0.000    -.0754024    -.034761
          europe  |  -.0431038   .0101057    -4.27   0.000    -.0632187   -.0229889
          africa  |  -.0513495   .0110327    -4.65   0.000    -.0733095   -.0293894
northern america  |  -.0409606   .0117507    -3.49   0.001    -.0643498   -.0175714
         oceania  |  -.0552619   .0144687    -3.82   0.000    -.0840612   -.0264627
                  |
            ASIAN |
           asian  |  -.0170005   .0031304    -5.43   0.000    -.0232314   -.0107695
      close_asian |  -.0023289   .0021116    -1.10   0.273     -.006532    .0018742
        asian_arv |  -.0292225    .006891    -4.24   0.000    -.0429386   -.0155063
  asian_arv_close |   .0119716   .0110631     1.08   0.282    -.0100491    .0339922
           female |    .064774   .0016792    38.57   0.000     .0614317    .0681164
            _cons |   .0710566   .0100407     7.08   0.000     .0510711    .0910421
-----------------------------------------------------------------------------------

For collinearity concern, I can only include arv, the interaction between arv and close in the model while dropping close because it collinear with another covariate I have in the model. I have the other two pairwise interactions: close_asian, asian_arv, all of which are generated based on multiplication. and lastly I have a triple interaction between asian, arv and close, which is generated as the multiplication of the three variables. My question is how could I obtain marginal effects of close and arv when asian == 1 versus asian == 0?

Thanks!

[Clyde Schechter]

Well, by using product variables instead of factorvariables with the ## operator, you have made it unnecessarily difficult to calculate any marginal effects at all. So step #1 is to go back and re-do it with factor-variable notation, ##, and then using -margins, dydx()- for each of the variables.

You will not, of course, be able to get marginal effects of the interaction terms themselves because there is no such thing. It is possible to do some calculations with what you have done that would give you the "marginal effects" of the product terms. But those are not the marginal effects of the interactions (because no such things exist). In fact, they are not the marginal effects of anything.

[Man Yang]

Thanks Clyde. Is it definitely impossible to calculate it? I may misinterpret something here, but basically what I want to calculate is whether, say, the effect of variables arv and close operate differently in Asians versus non-Asians and I feel this is something that the triple interaction can tell me about. Correct me if I were wrong.

[Clyde Schechter]

You can ask -margins- for the marginal effects of arv and close in Asians and in non-Asians (and, either on average or at specific value of each other.) That will tell you about differences of the effects of arv and close in Asians and in non-Asians.

The use of the three-way interaction in your model is appropriate to set the stage for that. But there is no such thing as the marginal effect of an interaction term. It's not just impossible to calculate: it does not exist.