interpreting Stata interaction terms

[Dan Daugaard]

I am hoping to confirm my interpreting and application of the interaction terms Stata provides when we run the var1##var2##var3 regression format.
My regression command is
xtreg ff D1event##D2style##D3rating
where ff = fund flows. My coefficients look like this

VARIABLES
D1event = 1	-9
D2style = 1	-3
1.D1event#1.D2style	-7
D3rating = 1	5
1.D1event#1.D3rating	-1
1.D2 style #1.D3rating	-14
1.D1event#1.D2style#1.D3rating	15
Constant	22

I am hoping to confirm that my interpretation and application of the coefficients are correct. My purpose is to:
1. estimate the unique ff (ie fund flows) associated D2 style fund with a D3rating in a D1 event.
Given the following outputs, am I correct to interpret the 1.D1event#1.D2 style#1.D3rating coefficient as indicating that 15 is estimated to occur in a D1event (compared to a non-D1event) for a D2style fund (compared to a non D2style fund) if the fund has a D3 rating (compared to not having a D3 rating).
2. explain how the interaction components fit together. What is the estimated total ff (ie fund flow) for a D2 style fund with a D3rating in a D1 event?
Is it the sum of all the coefficients? Ie Constant + D1event + D2style + 1.D1event#1.D2style
D3rating + 1.D1event#1.D3rating + 1.D2 style #1.D3rating + 1.D1event#1.D2style#1.D3rating
Thank you for your help, Dan

[Clyde Schechter]

I think interpreting three-way interaction coefficients is hard for anybody to wrap their head around. Use -margins event#style#rating- to get the estimated value of ff in each combination of event style and rating. As for the meaning of the 15 coefficient for 1.D1event#1.D2style#1.D3rating, it is hard to put it into words as it is a difference in differences in differences. I think it's better not to really focus on the coefficients here; work with the -margins- output instead.

[Dan Daugaard]

Thanks Clyde, I'll have a read through https://www.stata.com/manuals13/rmargins.pdf and see if interpreting -margins event#style#rating- helps, thanks, Dan

[Maarten Buis]

An alternative and complementary way of making sense of three way interactions is to use the tricks discussed in this Stata tip: https://www.stata-journal.com/articl...article=st0250 .

Consider the model with a three way interaction below:

. // open example data
. sysuse nlsw88, clear
(NLSW, 1988 extract)

.
. // prepare the data
. gen byte black = race == 2 if !missing(race)

. label variable black "respondent's race"

. label define black 0 "not black" ///
>                    1 "black"

. label value black black

. gen byte highoc = occupation < 3 if !missing(occupation)
(9 missing values generated)

. label variable highoc "high occupation"

. label define highoc 1 "higher" ///
>                     0 "lower"

. label value highoc highoc


. label define south 1 "South" 0 "non-South"

. label value south south

.
. poisson wage ibn.highoc i.highoc#(i.black##i.south), vce(robust) irr nocons
note: you are responsible for interpretation of noncount dep. variable

Iteration 0:   log pseudolikelihood = -13638.358  
Iteration 1:   log pseudolikelihood = -7153.0104  
Iteration 2:   log pseudolikelihood = -7143.3653  
Iteration 3:   log pseudolikelihood = -7143.3612  
Iteration 4:   log pseudolikelihood = -7143.3612  

Poisson regression                              Number of obs     =      2,237
                                                Wald chi2(8)      =   20603.35
Log pseudolikelihood = -7143.3612               Prob > chi2       =     0.0000

-------------------------------------------------------------------------------------
                    |               Robust
               wage |        IRR   Std. Err.      z    P>|z|     [95% Conf. Interval]
--------------------+----------------------------------------------------------------
             highoc |
             lower  |   7.170046   .1896684    74.47   0.000     6.807775    7.551595
            higher  |   11.38247   .3888261    71.20   0.000     10.64534    12.17065
                    |
       highoc#black |
       lower#black  |   1.046585   .0490895     0.97   0.332     .9546616     1.14736
      higher#black  |   1.040409   .1166726     0.35   0.724     .8351213    1.296161
                    |
       highoc#south |
       lower#South  |   .9130233   .0417024    -1.99   0.046     .8348399    .9985285
      higher#South  |   .8572835   .0518683    -2.55   0.011     .7614197    .9652167
                    |
 highoc#black#south |
 lower#black#South  |   .7934442   .0584669    -3.14   0.002     .6867419    .9167254
higher#black#South  |   .9495724   .1476683    -0.33   0.739     .7000957    1.287949
-------------------------------------------------------------------------------------

In this case there is no reference category for the variable highoc, which is possible because we suppressed the constant. We used a poisson with robust standard errors for wage for reasons discussed in https://blog.stata.com/2011/08/22/us...tell-a-friend/ .

The interpretation is that for people in lower occupations who are white and live in the non-south we expect an hourly wage of 7.17 dollars, while for people in higher occupations who are white and live in the non-south the expected wage is 11.38 dollars. So the main effects of highoc are now both constants.

In lower occupations in the non-south blacks can expect to earn 5% ((1.05-1)*100%=5%) more than non-blacks (not significant), while in higher occupations in the non-south blacks can expect to earn 4% more than non-blacks

In lower occupations people from the south can expect to earn 9% less than people from the non-south if they are not black. In higher occupations people from the south can expect to ear 14% less than people from the non-south if they are not black.

In the south for people with a lower occupation the effect of being black is 21% smaller than in the non-south. So the effect of being black for people with a lower occupation in the south is 1.05*.79=.83, or blacks in the south with lower occupation earn 17% less than non-blacks in the south with lower occupations. This change in effect for black is only 5% for higher occupations.

We can expand on this trick even more to show directly the effects of being black for all combinations of occupational status and region:

. poisson wage ibn.highoc#ibn.south (i.highoc#i.south)#i.black, vce(robust) irr nocons
note: you are responsible for interpretation of noncount dep. variable

Iteration 0:   log pseudolikelihood = -13638.358  
Iteration 1:   log pseudolikelihood = -7153.0104  
Iteration 2:   log pseudolikelihood = -7143.3653  
Iteration 3:   log pseudolikelihood = -7143.3612  
Iteration 4:   log pseudolikelihood = -7143.3612  

Poisson regression                              Number of obs     =      2,237
                                                Wald chi2(8)      =   20603.35
Log pseudolikelihood = -7143.3612               Prob > chi2       =     0.0000

-----------------------------------------------------------------------------------------
                        |               Robust
                   wage |        IRR   Std. Err.      z    P>|z|     [95% Conf. Interval]
------------------------+----------------------------------------------------------------
           highoc#south |
       lower#non-South  |   7.170046   .1896684    74.47   0.000     6.807775    7.551595
           lower#South  |   6.546419   .2437565    50.46   0.000     6.085682    7.042038
      higher#non-South  |   11.38247   .3888261    71.20   0.000     10.64534    12.17065
          higher#South  |   9.758006   .4872868    45.62   0.000     8.848191    10.76137
                        |
     highoc#south#black |
 lower#non-South#black  |   1.046585   .0490895     0.97   0.332     .9546616     1.14736
     lower#South#black  |    .830407   .0471932    -3.27   0.001     .7428754    .9282523
higher#non-South#black  |   1.040409   .1166726     0.35   0.724     .8351213    1.296161
    higher#South#black  |   .9879438   .1064409    -0.11   0.910     .7998784    1.220227
-----------------------------------------------------------------------------------------

I leave this as an exercise to ascertain that this model is equivalent to the previous model, i.e. that this model is just a different way of saying the exact same thing as the previous model.