Hello,
I am trying to estimate the ATT using individual-level data with 1 pre-treatment and 2 post-treatment periods (T= 0, 1, 2). Based on the regression formulation similar toa two-period DiD approach, I use the following specification for the two post-treatment periods:
Y_it = a_i + a*Treat_i + t1*Post1_t + t2*Post2_t + d1*(Post1_t * Treat_i) + d2*(Post2_t + Treat_i) + u_it
Post1 = 1 if in post-treatment period 1
Post2 = 1 if in post-treatment period 2
Running
yields equivalent coefficients in all three cases and equivalent s.e. for FE and RE (see below):
Running the same regressions clustering at individual-level (cluster(id)), the coeff. and s.e. in all three cases are equivalent:
(By adding additional time-varying covariates, the results between the three cases do differ.) However, I am not sure of the correct interpretation.
1) As I understand, in this specification I am essentially computing first-differences for each post-treatment period. This means that for OLS vs. FE, the coeff. are equivalent because assuming a common intercept (OLS) or individual FE (FE) does not change the coeff.: In both cases the time-invariant variables drop out taking first differences.
2) Given that the RE is essentially a weighted average between the BE (Between Estimator) and the FE, I do not understand why FE and RE yield the same results? When computing the BE, I get results for d1 and _cons only, due to multicollinearity in the specification (since by exploiting the variation between individuals only, Post1_t, Post2_t and the interactions with Treat_i are multicollinear).
3) To take into account serial correlation within cluster (for each i over t), I computed cluster-robust s.e. However, I do not understand why the results in all three cases are equivalent.
Any help is much appreciated.
I am trying to estimate the ATT using individual-level data with 1 pre-treatment and 2 post-treatment periods (T= 0, 1, 2). Based on the regression formulation similar toa two-period DiD approach, I use the following specification for the two post-treatment periods:
Y_it = a_i + a*Treat_i + t1*Post1_t + t2*Post2_t + d1*(Post1_t * Treat_i) + d2*(Post2_t + Treat_i) + u_it
Post1 = 1 if in post-treatment period 1
Post2 = 1 if in post-treatment period 2
Running
Code:
reg y d1 d1 Post1 Post2 Treat xtreg y d1 d1 Post1 Post2, fe xtreg y d1 d1 Post1 Post2 Treat, re
Code:
---------------------------------------------------------------------------------------------------
(1) (2) (3)
OLS FE RE
---------------------------------------------------------------------------------------------------
d1 -0.0191 (0.0418) -0.0191 (0.0355) -0.0191 (0.0355)
d2 -0.0574 (0.0418) -0.0574 (0.0355) -0.0574 (0.0355)
a 0.0600* (0.0296) 0.0600* (0.0296)
t1 -0.0233 (0.0280) -0.0233 (0.0238) -0.0233 (0.0238)
t2 -0.0331 (0.0280) -0.0331 (0.0238) -0.0331 (0.0238)
_cons 0.0987*** (0.0198) 0.126*** (0.0125) 0.0987*** (0.0198)
---------------------------------------------------------------------------------------------------
N 1000 1000 1000
---------------------------------------------------------------------------------------------------
Code:
---------------------------------------------------------------------------------------------------
(1) (2) (3)
OLS FE RE
---------------------------------------------------------------------------------------------------
d1 -0.0191 (0.0349) -0.0191 (0.0349) -0.0191 (0.0349)
d2 -0.0574 (0.0329) -0.0574 (0.0329) -0.0574 (0.0329)
a 0.0600* (0.0300) 0.0600* (0.0300)
t1 -0.0233 (0.0181) -0.0233 (0.0181) -0.0233 (0.0181)
t2 -0.0331 (0.0179) -0.0331 (0.0178) -0.0331 (0.0179)
_cons 0.0987*** (0.0144) 0.126*** (0.0104) 0.0987*** (0.0144)
---------------------------------------------------------------------------------------------------
N 1000 1000 1000
---------------------------------------------------------------------------------------------------
1) As I understand, in this specification I am essentially computing first-differences for each post-treatment period. This means that for OLS vs. FE, the coeff. are equivalent because assuming a common intercept (OLS) or individual FE (FE) does not change the coeff.: In both cases the time-invariant variables drop out taking first differences.
2) Given that the RE is essentially a weighted average between the BE (Between Estimator) and the FE, I do not understand why FE and RE yield the same results? When computing the BE, I get results for d1 and _cons only, due to multicollinearity in the specification (since by exploiting the variation between individuals only, Post1_t, Post2_t and the interactions with Treat_i are multicollinear).
3) To take into account serial correlation within cluster (for each i over t), I computed cluster-robust s.e. However, I do not understand why the results in all three cases are equivalent.
Any help is much appreciated.
