Interpretation of Causal-Moderator Interactions of Month and Treatment Variables when Time Main Effects are Collinear with Individual FEs

[Nicholas Lacourse]

I have some results from a recent paper and I want to make sure my reasoning is correct in its interpretation of the coefficients.

In order to disguise the inflammatory nature of the model, I will relabel variables with synonymous, yet still relevant, labeling.

I have the following model specification that I run in Stata:

reghdfe rating i.treatment##i.t, a(id type_id) cl(id)

The above specification produce the below results.

HDFE Linear regression                            Number of obs   =      1,842
Absorbing 2 HDFE groups                           F(   6,    920) =       2.22
Statistics robust to heteroskedasticity           Prob > F        =     0.0389
R-squared       =     0.8153
Adj R-squared   =     0.6193
Within R-sq.    =     0.0133
Number of clusters (id)      =        921         Root MSE        =     0.6332

(Std. Err. adjusted for 921 clusters in id)

Robust
rating       Coef.   Std. Err.      t    P>|t|     [95% Conf. Interval]

1.treatment    .2392342      .1191     2.01   0.045     .0054949    .4729735

treatment#t
1 2    -.5555135   .2316332    -2.40   0.017    -1.010104   -.1009226
1 3    -.3907255   .1465332    -2.67   0.008    -.6783037   -.1031473
1 4    -.1348838   .1407483    -0.96   0.338    -.4111089    .1413412
1 5    -.3209895   .1362986    -2.36   0.019    -.5884817   -.0534972
1 6     -.229913   .1449132    -1.59   0.113    -.5143118    .0544859

_cons    2.974843   .0184058   161.63   0.000     2.938721    3.010966


Absorbed degrees of freedom:

Absorbed FE  Categories  - Redundant  = Num. Coefs
-
id        921         921           0    *
type_id         22           0          22    

* = FE nested within cluster; treated as redundant for DoF computation

So as can be seen from the above output, 't' main effects do not get produced due to collinearity with 'id' fixed effects and this because in each time-period (month) 't', the subject 'id' is unique to that 't'. So 't' is non-varying within 'id'.

That being said, how do we now interpret the "main" and "interaction" effects in the model's results with the absence of 't' main effects? Are each of the coefficients just simple main effects conditional on different months?

So the table's reported main effect is just the treatment effect conditional on t = 1 (base month), while the interaction effects such as 1.treatment#2.t is the treatment effect conditional on t=2 (treatment effect in t=2 in comparison to non-treated in t=2), and not the treatment effect in t=2 in comparison to the non-treated in t=1, and so on for each month's interaction term?

Is this a correct interpretation?

Any comments would be greatly appreciated.

[Clyde Schechter]

So the table's reported main effect is just the treatment effect conditional on t = 1 (base month),...

Correct.

...while the interaction effects such as 1.treatment#2.t is the treatment effect conditional on t=2 (treatment effect in t=2 in comparison to non-treated in t=2), and not the treatment effect in t=2 in comparison to the non-treated in t=1, and so on for each month's interaction term?

Not correct. The treatment effect conditional on t = 2 is the coefficient of 1.treatment PLUS the coefficient of 1.treatment#2.t.

[Nicholas Lacourse]

Thank you, Clyde, for the feedback. I kept thinking that because month main effects could not be produced, there was not necessarily a way to create a total effect for particular months, so they would instead just be conditioned by month. However, upon reflecting even further, if that was the case, then what would be the differences between the above coefficients with interaction terms and a stratified regression by months (beyond the different model assumptions each approach would take). Anywho, I am now rambling; thanks for clearing up my mistake.

Have a nice Thursday!