Interpreting triple interaction term with continuous variable

[Chloe Zapha]

Dear stata community,

I am struggling with the interpretation of Treated x Post x Log(1+Z) with Z being a continuous positive variable.

My variable Treated takes 3 values (0 for control, 1 and 2) and Post takes 2 values (0, 1). Z is a continuous variable like distance (in km).

1- I understand that 1.treated#1.post and 2.treated#1.post are the partial effect of the treatment relative to the control group, 0.treated#1.post, with ln(1+Z)=0, i.e. Z=0
However, Z is stricly positive. Why aren't the coefficients omitted ? What does they mean ?

2- My variable Z increases (exogenously) after the treatment. How do I interpret 1.treated#1.post#c.lnZ and 2.treated#1.post#c.lnZ ? I am not sure what question it answers. Is it :
- "Is the effect of Z different because of the treatment?", or
- "What is the effect of Z being greater after the treatment?"

3- How do I understand the magnitude of 1.treated#1.post#c.lnZ and 2.treated#1.post#c.lnZ ?
One standard deviation of log(1+Z) (.2662474) increases my outcome variable Y by (0.26*30035=) 7809 on my first treated group ?

Any help would be very appreciated.


Here's a (silly) example with the auto data:

use "C:\Program Files (x86)\Stata15\ado\base\a\auto.dta", clear

gen treated=0 if mpg<18
replace treated=1 if mpg>=18 & mpg<24
replace treated=2 if mpg>=24

gen post=0 if rep78<=3
replace post=1 if rep78>3

gen lnZ=log(1+weight)
sum lnZ

    Variable |        Obs        Mean    Std. Dev.       Min        Max
-------------+---------------------------------------------------------
         lnZ |         74    7.979106    .2662474   7.473637   8.484877
r; t=0.00 11:15:39

. 
. reghdfe price ib0.treated##ib0.post##c.lnZ, noabsorb
(MWFE estimator converged in 1 iterations)

HDFE Linear regression                            Number of obs   =         74
Absorbing 1 HDFE group                            F(  11,     62) =       8.02
                                                  Prob > F        =     0.0000
                                                  R-squared       =     0.5873
                                                  Adj R-squared   =     0.5141
                                                  Within R-sq.    =     0.5873
                                                  Root MSE        =  2055.9924

------------------------------------------------------------------------------------
             price |      Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]
-------------------+----------------------------------------------------------------
           treated |
                1  |   121104.6   53046.34     2.28   0.026     15066.45    227142.7
                2  |   156977.3   58501.97     2.68   0.009     40033.52      273921
                   |
            1.post |   286430.3   67555.75     4.24   0.000     151388.3    421472.3
                   |
      treated#post |
              1 1  |  -246480.9   75195.96    -3.28   0.002    -396795.5   -96166.37
              2 1  |  -339319.7    83140.1    -4.08   0.000    -505514.4     -173125
                   |
               lnZ |    20706.8    5809.84     3.56   0.001     9093.091     32320.5
                   |
     treated#c.lnZ |
                1  |  -15026.52   6432.404    -2.34   0.023    -27884.72   -2168.331
                2  |     -19577   7199.596    -2.72   0.008    -33968.79   -5185.217
                   |
        post#c.lnZ |
                1  |  -34890.05   8200.045    -4.25   0.000    -51281.71    -18498.4
                   |
treated#post#c.lnZ |
              1 1  |   30035.15    9177.67     3.27   0.002     11689.25    48381.05
              2 1  |   41973.65   10325.43     4.07   0.000     21333.41    62613.89
                   |
             _cons |  -161752.2   48150.94    -3.36   0.001    -258004.6   -65499.83
------------------------------------------------------------------------------------