Hi all,

I am doing an empirical analysis with difference-in-differences (DD) together with difference-in-difference-in-differences (DDD). All the variables of interest are indicator variables with two values, 0 and 1. Thus, in my estimation models, 7 variables of interests are included along with control variables. However, as can be seen from the outputs, for some reason, there are some variables are omitted in DD and DDD estimation models. Since I am using FE model with cluster-robust standard errors with annual panel data, I do understand a time-invariant variable (regul_subject) of interest are omitted in DD models and DDD models. However, I do not understand why two variables (i.e., regul_subjectxAfter or regul_subjectxNA) in DDD models are omitted. I personally guess that since DDD variable is coincidentally the same as regul_subjectxAfter (actually they are exactly the same; i.e., collinearity) in the first DDD model, regul_subjectxAfter is omitted. I think this is correct. As for the second DDD model, even though regul_subjectxNA are not exactly the same as other variables in the respective DDD models, they are omitted from the second DDD regression models (Why?). According to the STATA, it says due to collinearity they are omitted, but I still do not understand why. Are they critical to interpret the results estimated from DDD estimation? Do you have any idea as to how to understand these omissions? I mean why they are omitted from each regression model. I have attached outputs as follows. Thank you.

** DD models **
(Model 1) . xtreg lnEMP regul_subject*After regul_subject After controls i.t, fe vce(cluster ID)
note: regul_subject omitted because of collinearity.
note: 2016.t omitted because of collinearity.

Fixed-effects (within) regression Number of obs = 168
Group variable: ID Number of groups = 27

R-squared: Obs per group:
Within = 0.3927 min = 1
Between = 0.3436 avg = 6.2
Overall = 0.3901 max = 10

F(17,26) = 4.76
corr(u_i, Xb) = -0.2896 Prob > F = 0.0002

(Std. err. adjusted for 27 clusters in ID)
-------------------------------------------------------------------------------------
| Robust
lnEMP | Coefficient std. err. t P>|t| [95% conf. interval]
--------------------+----------------------------------------------------------------
regul_subjectxAfter | .4961581 .1473858 3.37 0.002 .1932022 .7991139
regul_subject | 0 (omitted)
After | -.0781401 .2126817 -0.37 0.716 -.5153136 .3590335


(Model 2) . xtreg lnENV regul_subject*After regul_subject After controls i.t, fe vce(cluster ID)
note: regul_subject omitted because of collinearity.
note: 2016.t omitted because of collinearity.

Fixed-effects (within) regression Number of obs = 184
Group variable: ID Number of groups = 28

R-squared: Obs per group:
Within = 0.5178 min = 1
Between = 0.2465 avg = 6.6
Overall = 0.2101 max = 10

F(17,27) = 136.00
corr(u_i, Xb) = -0.3198 Prob > F = 0.0000

(Std. err. adjusted for 28 clusters in ID)
-------------------------------------------------------------------------------------
| Robust
lnENV | Coefficient std. err. t P>|t| [95% conf. interval]
--------------------+----------------------------------------------------------------
regul_subjectxAfter | -.1360702 .0764059 -1.78 0.086 -.2928422 .0207018
regul_subject | 0 (omitted)
After | .1172538 .0874287 1.34 0.191 -.0621352 .2966427


(Model 3) . xtreg NetResult regul_subject*After regul_subject After controls i.t, fe vce(cluster ID)
note: regul_subject omitted because of collinearity.
note: 2016.t omitted because of collinearity.

Fixed-effects (within) regression Number of obs = 139
Group variable: ID Number of groups = 23

R-squared: Obs per group:
Within = 0.3165 min = 1
Between = 0.2484 avg = 6.0
Overall = 0.0834 max = 10

F(17,22) = 4.19
corr(u_i, Xb) = -0.9952 Prob > F = 0.0010

(Std. err. adjusted for 23 clusters in ID)
-------------------------------------------------------------------------------------
| Robust
NetResult | Coefficient std. err. t P>|t| [95% conf. interval]
--------------------+----------------------------------------------------------------
regul_subjectxAfter | 2022815 1074758 1.88 0.073 -206097.2 4251727
regul_subject | 0 (omitted)
After | -1693723 805878.9 -2.10 0.047 -3365014 -22432.4


*** first DDD models ***
. gen DDD = regul_subject*kreport*After
. gen regul_subjectxkreport = regul_subject*kreport
. gen Afterxkreport = After*kreport

(Model 1 related). xtreg lnEMP DDD regul_subject After kreport regul_subjectxAfter regul_subjectxkreport Afterxkreport controls i.t, fe vce(cluster ID)
note: regul_subject omitted because of collinearity.
note: regul_subjectxAfter omitted because of collinearity.
note: 2016.t omitted because of collinearity.

Fixed-effects (within) regression Number of obs = 168
Group variable: ID Number of groups = 27

R-squared: Obs per group:
Within = 0.4204 min = 1
Between = 0.3294 avg = 6.2
Overall = 0.3804 max = 10

F(19,26) = .
corr(u_i, Xb) = -0.3235 Prob > F = .

(Std. err. adjusted for 27 clusters in ID)
---------------------------------------------------------------------------------------
| Robust
lnEMP | Coefficient std. err. t P>|t| [95% conf. interval]
----------------------+----------------------------------------------------------------
DDD | .5027745 .1780385 2.82 0.009 .1368111 .8687378
regul_subject | 0 (omitted)
After | .2525456 .229819 1.10 0.282 -.2198542 .7249453
kreport | -.0545024 .1676641 -0.33 0.748 -.399141 .2901361
regul_subjectxAfter | 0 (omitted)
regul_subjectxkreport | .0199282 .2441491 0.08 0.936 -.4819275 .5217839
Afterxkreport | -.3408959 .1809803 -1.88 0.071 -.7129062 .0311144


(Model 2 related). xtreg lnENV DDD regul_subject After kreport regul_subjectxAfter regul_subjectxkreport Afterxkreport controls i.t, fe vce(cluster ID)
note: regul_subject omitted because of collinearity.
note: regul_subjectxAfter omitted because of collinearity.
note: 2016.t omitted because of collinearity.

Fixed-effects (within) regression Number of obs = 184
Group variable: ID Number of groups = 28

R-squared: Obs per group:
Within = 0.5426 min = 1
Between = 0.2604 avg = 6.6
Overall = 0.2308 max = 10

F(19,27) = .
corr(u_i, Xb) = -0.3241 Prob > F = .

(Std. err. adjusted for 28 clusters in ID)
---------------------------------------------------------------------------------------
| Robust
lnENV | Coefficient std. err. t P>|t| [95% conf. interval]
----------------------+----------------------------------------------------------------
DDD | -.1477384 .0816066 -1.81 0.081 -.3151814 .0197045
regul_subject | 0 (omitted)
After | .2428912 .1135572 2.14 0.042 .009891 .4758914
kreport | -.0044123 .0716012 -0.06 0.951 -.1513258 .1425012
regul_subjectxAfter | 0 (omitted)
regul_subjectxkreport | .2005685 .1282398 1.56 0.129 -.0625578 .4636948
Afterxkreport | -.1320063 .0807913 -1.63 0.114 -.2977763 .0337637


(Model 3 related). xtreg NetResult DDD regul_subject After kreport regul_subjectxAfter regul_subjectxkreport Afterxkreport controls i.t, fe vce(cluster ID
> )
note: regul_subject omitted because of collinearity.
note: regul_subjectxAfter omitted because of collinearity.
note: 2016.t omitted because of collinearity.

Fixed-effects (within) regression Number of obs = 139
Group variable: ID Number of groups = 23

R-squared: Obs per group:
Within = 0.4135 min = 1
Between = 0.2556 avg = 6.0
Overall = 0.0903 max = 10

F(19,22) = .
corr(u_i, Xb) = -0.9944 Prob > F = .

(Std. err. adjusted for 23 clusters in ID)
---------------------------------------------------------------------------------------
| Robust
NetResult | Coefficient std. err. t P>|t| [95% conf. interval]
----------------------+----------------------------------------------------------------
DDD | 1633557 936311.4 1.74 0.095 -308234.1 3575348
regul_subject | 0 (omitted)
After | -739589.3 777525.4 -0.95 0.352 -2352078 872899.6
kreport | 180216.2 604930.7 0.30 0.769 -1074333 1434766
regul_subjectxAfter | 0 (omitted)
regul_subjectxkreport | 4637721 927462.8 5.00 0.000 2714281 6561161
Afterxkreport | -973113.2 547589.5 -1.78 0.089 -2108744 162518.1


*** second DDD models ***
. gen DDDNA = regul_subjectxAfter*NA
. gen regul_subjectxNA = regul_subject*NA
. gen AfterxNA = After*NA


(Model 1 related). xtreg lnEMP DDDNA regul_subject After NA regul_subjectxAfter regul_subjectxNA AfterxNA controls i.t, fe vce(cluster ID)
note: regul_subject omitted because of collinearity.
note: regul_subjectxNA omitted because of collinearity.
note: 2016.t omitted because of collinearity.

Fixed-effects (within) regression Number of obs = 168
Group variable: ID Number of groups = 27

R-squared: Obs per group:
Within = 0.4049 min = 1
Between = 0.3303 avg = 6.2
Overall = 0.3761 max = 10

F(19,26) = .
corr(u_i, Xb) = -0.3462 Prob > F = .

(Std. err. adjusted for 27 clusters in ID)
-------------------------------------------------------------------------------------
| Robust
lnEMP | Coefficient std. err. t P>|t| [95% conf. interval]
--------------------+----------------------------------------------------------------
DDDNA | .4843287 .2345998 2.06 0.049 .002102 .9665554
regul_subject | 0 (omitted)
After | .1288884 .3121713 0.41 0.683 -.5127887 .7705656
NA | .180748 .3733715 0.48 0.632 -.5867281 .948224
regul_subjectxAfter | .1234151 .2150666 0.57 0.571 -.3186606 .5654907
regul_subjectxNA | 0 (omitted)
AfterxNA | -.2782617 .189522 -1.47 0.154 -.6678298 .1113063


(Model 2 related). xtreg lnENV DDDNA regul_subject After NA regul_subjectxAfter regul_subjectxNA AfterxNA controls i.t, fe vce(cluster ID)
note: regul_subject omitted because of collinearity.
note: regul_subjectxNA omitted because of collinearity.
note: 2016.t omitted because of collinearity.

Fixed-effects (within) regression Number of obs = 184
Group variable: ID Number of groups = 28

R-squared: Obs per group:
Within = 0.5323 min = 1
Between = 0.2265 avg = 6.6
Overall = 0.1956 max = 10

F(20,27) = 1908.41
corr(u_i, Xb) = -0.3737 Prob > F = 0.0000

(Std. err. adjusted for 28 clusters in ID)
-------------------------------------------------------------------------------------
| Robust
lnENV | Coefficient std. err. t P>|t| [95% conf. interval]
--------------------+----------------------------------------------------------------
DDDNA | -.175323 .0916629 -1.91 0.066 -.3633996 .0127537
regul_subject | 0 (omitted)
After | .0996518 .1050469 0.95 0.351 -.1158867 .3151903
NA | .0885052 .0851095 1.04 0.308 -.0861251 .2631355
regul_subjectxAfter | -.0408048 .0868052 -0.47 0.642 -.2189143 .1373047
regul_subjectxNA | 0 (omitted)
AfterxNA | .023657 .088674 0.27 0.792 -.1582871 .2056011


(Model 3 related). xtreg NetResult DDDNA regul_subject After NA regul_subjectxAfter regul_subjectxNA AfterxNA controls i.t, fe vce(cluster ID)
note: regul_subject omitted because of collinearity.
note: regul_subjectxNA omitted because of collinearity.
note: 2016.t omitted because of collinearity.

Fixed-effects (within) regression Number of obs = 139
Group variable: ID Number of groups = 23

R-squared: Obs per group:
Within = 0.3412 min = 1
Between = 0.2562 avg = 6.0
Overall = 0.0899 max = 10

F(20,22) = 8.36
corr(u_i, Xb) = -0.9941 Prob > F = 0.0000

(Std. err. adjusted for 23 clusters in ID)
-------------------------------------------------------------------------------------
| Robust
NetResult | Coefficient std. err. t P>|t| [95% conf. interval]
--------------------+----------------------------------------------------------------
DDDNA | -1194225 1587386 -0.75 0.460 -4486262 2097813
regul_subject | 0 (omitted)
After | -1104117 858674.6 -1.29 0.212 -2884899 676665.1
NA | 548503.5 869980.3 0.63 0.535 -1255725 2352732
regul_subjectxAfter | 2371697 1687083 1.41 0.174 -1127098 5870493
regul_subjectxNA | 0 (omitted)
AfterxNA | -674618.9 577149.5 -1.17 0.255 -1871554 522315.8



Thank you!