TWFE
Two-way fixed effect model has been widely used, especially in difference-in-difference situation, when we have a panel data with pre and post data. The basic setup is:
\[ y_{it} = X_{it} \beta + c_i + f_t + u_{it} \]
with \(X\) has \(k\) variables, \(t\) has \(T\) periods, and \(i\) has \(N\) units.
For a linear model, we know we can put in \(k\) unit dummies and \(T\) time dummies to estimate the model. When we have a lot of dummies, this becomes inefficient, sometimes infeasible. When we have nonlinear models, such as Poisson, or logit, then this becomes even more difficult.
TWM
The recent Wooldrige paper (https://www.researchgate.net/publication/353938385_Two-Way_Fixed_Effects_the_Two-Way_Mundlak_Regression_and_Difference-in-Differences_Estimators) showed that two-way Mundlak regression (TWM) can be useful, not only in the common event time situation, but also in staggered TWFE situation. But in this post I’ll only talk about common event time sitation (which is the case that all treated units get treated at the same period, and stay treated afterwards). The staggered situation (treated units get treated at different periods) will be a different post.
The idea of Mundlak (1978) is the following.
In the one-way fixed effect case,
\[ y_{it} = x_{it} \beta + c_i + u_{it} \]
Mundlak models \(c_i\) as a function of the mean of \(x_{it}\) for each unit:
\[ c_i = \bar x_i \theta + \eta_i \]
Then
\[ y_{it} = x_{it} \beta + \bar x_i \theta + \eta_i + u_{it} \]
This \(\eta_i + u_{it}\) becomes the composite error term. This becomes a model including \(\bar x_i\)’s, which are additional \(k\) variables, instead of including a set of unit dummies. This model can be estimated by either Pooled OLS (POLS), or random effect. The estimated coefficients \(\beta\) are identical. The standard errors are the same too, after clustering on units. In other words, if we include means of \(x\)’s as controls, then POLS and random effect models are identical.
The by-product of this equation is that if \(\theta = 0\), then \(c_i\) is uncorrelated with \(x_{it}\); therefore a random effect is good. Otherwise we should do fixed effect. This is similar to a Hausman test (equivalent?).
Here is an example:
clear
webuse nlswork
xtset idcode year
drop if union==.
drop if age==.
bysort idcode: egen mean_age=mean(age)
bysort idcode: egen mean_union=mean(union)
xtreg ln_wage age union, fe cluster(idcode)
reg ln_wage age union mean_age mean_union, cluster(idcode)
xtreg ln_wage age union mean_age mean_union, re cluster(idcode)(National Longitudinal Survey of Young Women, 14-24 years old in 1968)
Panel variable: idcode (unbalanced)
Time variable: year, 68 to 88, but with gaps
Delta: 1 unit
(9,296 observations deleted)
(9 observations deleted)
Fixed-effects (within) regression Number of obs = 19,229
Group variable: idcode Number of groups = 4,150
R-squared: Obs per group:
Within = 0.0963 min = 1
Between = 0.0433 avg = 4.6
Overall = 0.0562 max = 12
F(2,4149) = 331.52
corr(u_i, Xb) = 0.0127 Prob > F = 0.0000
(Std. err. adjusted for 4,150 clusters in idcode)
------------------------------------------------------------------------------
| Robust
ln_wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
age | .0153507 .0006912 22.21 0.000 .0139956 .0167058
union | .1055274 .0098582 10.70 0.000 .0862001 .1248546
_cons | 1.248435 .0215661 57.89 0.000 1.206154 1.290716
-------------+----------------------------------------------------------------
sigma_u | .42353003
sigma_e | .26213464
rho | .72302816 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Linear regression Number of obs = 19,229
F(4, 4149) = 234.77
Prob > F = 0.0000
R-squared = 0.0769
Root MSE = .44953
(Std. err. adjusted for 4,150 clusters in idcode)
------------------------------------------------------------------------------
| Robust
ln_wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
age | .0153507 .0006912 22.21 0.000 .0139956 .0167059
union | .1055274 .0098587 10.70 0.000 .0861991 .1248556
mean_age | -.0064208 .001548 -4.15 0.000 -.0094556 -.0033859
mean_union | .1900785 .0223172 8.52 0.000 .1463249 .2338321
_cons | 1.405264 .042734 32.88 0.000 1.321482 1.489046
------------------------------------------------------------------------------
Random-effects GLS regression Number of obs = 19,229
Group variable: idcode Number of groups = 4,150
R-squared: Obs per group:
Within = 0.0963 min = 1
Between = 0.0750 avg = 4.6
Overall = 0.0768 max = 12
Wald chi2(4) = 1023.99
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
(Std. err. adjusted for 4,150 clusters in idcode)
------------------------------------------------------------------------------
| Robust
ln_wage | Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
age | .0153507 .0006912 22.21 0.000 .0139959 .0167055
union | .1055274 .0098587 10.70 0.000 .0862047 .12485
mean_age | -.0061464 .0014254 -4.31 0.000 -.0089401 -.0033527
mean_union | .212183 .0210098 10.10 0.000 .1710046 .2533614
_cons | 1.362101 .0383784 35.49 0.000 1.286881 1.437322
-------------+----------------------------------------------------------------
sigma_u | .38505558
sigma_e | .26213464
rho | .68331727 (fraction of variance due to u_i)
------------------------------------------------------------------------------Clearly we see in the these three models, the coefficients and standard errors on age and union are the same.
What Wooldridge (2021) shows is that this (Mundlak) works for two-way fixed effect too.
That is, for the TWFE, \[ y_{it} = X_{it} \beta + c_i + f_t + u_{it} \] it’s equivalent to estimate
\[ y_{it} = X_{it} \beta + X_{i \cdot} \lambda + X_{\cdot t} \theta + u_{it} \]
where \(X_{i \cdot}\) is the unit specific averages over time, and \(X_{\cdot t}\) is the time specific averages over units. This is basically some kind of control function approach. That is, from FWL theorem, these two models are equivalent. See Wooldridge 2021 for proof.
Wooldrige calls this POLS of regressing \(y_{it}\) on regressors \(X_{it}\) and two sets of addtional controls \(X_{i \cdot}\) and \(X_{\cdot t}\) Two-way Mundlak (TWM) model.
One advantage of TWM is that it has potentially huge cut on the number of regressors, if you use the dummy variable approach for the TWFE.
The other advantage is that you can actually include time-invariant or unit-invariant variables in the regression and the \(\beta\)’s remain unchanged. We all know that in TWFE model you cannot include those time-invariant or unit-invariant variables in the model.
We can see in the following example of POLS (which is TWM), we add in “race” which is unit-invariant. It can be estimated in TWM, but would not be estimatible in TWFE. Notice that coefficients on age and union remain unchanged. This is basically to say that race is orthogonal to the surface of all the \(X\)’s and the means of \(X\)’s. We did not add in time invariant variables, but that would be the same story.
reg ln_wage age union mean_age mean_union i.race, cluster(idcode)
Linear regression Number of obs = 19,229
F(6, 4149) = 189.66
Prob > F = 0.0000
R-squared = 0.1069
Root MSE = .4422
(Std. err. adjusted for 4,150 clusters in idcode)
------------------------------------------------------------------------------
| Robust
ln_wage | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
age | .0153507 .0006913 22.21 0.000 .0139955 .0167059
union | .1055274 .0098592 10.70 0.000 .0861981 .1248566
mean_age | -.0070345 .0015342 -4.59 0.000 -.0100424 -.0040266
mean_union | .2227511 .0220026 10.12 0.000 .1796142 .265888
|
race |
Black | -.1790995 .0143624 -12.47 0.000 -.2072575 -.1509415
Other | .0882461 .0715315 1.23 0.217 -.0519939 .2284861
|
_cons | 1.466452 .0426087 34.42 0.000 1.382916 1.549988
------------------------------------------------------------------------------Applications of TWM to DID in common treatment timing
There is an extension of Mundlak device that if \(x_{it}\) can be seen as an interaction between a time-invariant variable and an unit-invariant variable,
\[ x_{itj} = z_{ij} * m_{tj} \]
Then
\[ \bar x_{i \cdot j} = z_{ij} \bar m_j \] \[ \bar x_{\cdot t j} = \bar z_{j} m_{tj} \]
In other words, the two averages of \(x\) are constant times \(z_{ij}\) and constant times \(m_{tj}\) respectively. By Mundlak, we can just include \(z_{ij}\) and \(m_{tj}\) as control in addtional to \(x_{it}\) to get the same estimates as TWFE. This makes things much easier, as we’ll see later.
Wooldridge 2021 goes on showing that TWM works in common treatment time, and also in staggered timing setting, as long as we allow flexible interactions of treatment cohorts and treatment periods. It’s truely remarkable that he showed that TWM can be so powerful! In this post I am going to just use simulation to show TWM in common treatment timing; staggered timing setting will be another post. All the codes are based on Wooldridge’s do files he shared in his dropbox.
In a common timing setting, treatment comes in at one time, all the treated units are treated at the same time. There is the parallel trend assumption that assumes that the treatment group and control group have the same trend before treatment. Also there is no anticipation assumption.
\[ y_{it} = w_{it} \beta + c_i + f_t + u_{it} \]
where \(w_{it}\) is the treatment indicator, with \(T\) periods and treatment happens at \(t=q\).
\[ w_{it} = d_i \cdot p_t \]
where \(d_i\) is the treatment group dummy, and \(p_t\) is the post treatment dummy.
\[ \bar w_{i \cdot} = d_i \cdot \bar p \] \[ \bar w_{ \cdot t} = \bar d \cdot p_t \]
Therefore, we just need to include \(d_i\) and \(p_t\) as controls in the regression to be the same estimator as TWFE, by the spirit of Mundlak.
That is, instead of doing TWFE, we can do TWM by regressing \(y\) on \(w\), \(d_i\) and \(p_t\).
Moreover, in a TWM, we can include any time-invariant or unit-invariant controls, or both and have no effect on \(\beta\).
We can also allow the effect to change across time post treatment by doing
\[y_{it} = \alpha + \beta_q (w_{it} \cdot f_q) + \dots + \beta_T (w_{it} \cdot f_T) + \eta d_i + \theta_q f_q + \dots + \theta_T f_T + e_{it} \]
In the TWM, we allow effects change from \(t=q\) to \(t=T\), by interacting \(w_{it}\) and \(f_t\), which is an dummy for time \(t\). Very flexible, yet easy to do, given everything is linear.
Even more, if we have another variable, \(x_i\), say gender, which is time-invariant, we can allow the effect to be different across gender, across time!
Now I use Wooldridge’s code to run on a simulated data set.
clear
set obs 100
set seed 1234567
gen id = _n
gen w1 = rnormal()
gen w2 = rnormal()
expand 4
bysort id: gen year=2011+_n
gen post=(year>2013)
gen d=(id>50)
gen w=d*post
gen x1 = rnormal() + w1 + 2*w2
gen u = rnormal()
gen f2014=(year==2014)
gen f2015=(year==2015)
gen y1 = 1 +2*x1 + 3*w1 + u
# let's allow each year's treatment effect being different. ATT for post period would be roughly 3, because of balanced panel.
gen logy = y1 + 2.5*d*f2014 + 3.5*d*f2015
xtset id year
tab year wNumber of observations (_N) was 0, now 100.
(300 observations created)
Unknown #command
Panel variable: id (strongly balanced)
Time variable: year, 2012 to 2015
Delta: 1 unit
| w
year | 0 1 | Total
-----------+----------------------+----------
2012 | 100 0 | 100
2013 | 100 0 | 100
2014 | 50 50 | 100
2015 | 50 50 | 100
-----------+----------------------+----------
Total | 300 100 | 400 In this data set, we have 100 units, four years (2012 to 2015). Half units are treated, treatment happens on 2014. Treatment dummy \(d\), \(w\) is \(d * post\). Treatment effect is 2.5 in year 2014, and 3.5 in year 2015. If we estimate ATT for the entire post treatment period, we should expect to see ATT about 3, given the balanced panel.
All the following code are from Wooldridge. I just used it on the simulated data.
The difficulty is that we have to be careful and mindful about what we want to estimate, then come up with the correct code for all those interaction terms.
* Basic DID regression. Only need the post-treatment dummy but including
* all time dummies is harmless (and necessary for heterogeneous trends).
* Standard errors change a little because of different degrees-of-freedom.
reg logy c.d#c.post d post, vce(cluster id)
reg logy w d post, vce(cluster id)
reg logy w d f2014 f2015, vce(cluster id)
reg logy w d i.year, vce(cluster id)
* Using fixed effects is equivalent:
xtreg logy w i.year, fe vce(cluster id)
xtreg logy w post, fe vce(cluster id)
* So is RE with d included:
xtreg logy w d post, re vce(cluster id)
* Allow a separate effect in each of the treated time periods:
* It is the ATT for each treated period:
xtreg logy c.w#c.f2014 c.w#c.f2015 i.year, fe vce(cluster id)
xtreg logy c.d#c.f2014 c.d#c.f2015 i.year, fe vce(cluster id)
* POLS version:
reg logy c.w#c.f2014 c.w#c.f2015 d f2014 f2015, vce(cluster id)
lincom (c.w#c.f2014 + c.w#c.f2015)/2
* RE still gives same estimates:
xtreg logy c.w#c.f2014 c.w#c.f2015 d f2014 f2015, re vce(cluster id)
* Putting in time-constant controls in the levels and even interacted with d
* does not change the estimates. It does boost the R-squared and
* slightly changes the standard errors:
reg logy c.w#c.f2014 c.w#c.f2015 d f2014 f2015 x1 c.d#c.x1, vce(cluster id)
reg logy c.w#c.f2014 c.w#c.f2015 d i.year x1 c.d#c.x1, vce(cluster id)
* Now add covariate interacted with everything:
sum x1 if d
gen x1_dm_1 = x1 - r(mean)
xtreg logy c.w#c.f2014 c.w#c.f2015 c.w#c.f2014#c.x1_dm_1 c.w#c.f2015#c.x1_dm_1 i.year c.f2014#c.x1 c.f2015#c.x1, fe vce(cluster id)
reg logy c.w#c.f2014 c.w#c.f2015 c.w#c.f2014#c.x1_dm_1 c.w#c.f2015#c.x1_dm_1 i.year c.f2014#c.x1 c.f2015#c.x1 d x1 c.d#c.x1, vce(cluster id)
reg logy c.w#c.f2014 c.w#c.f2015 c.w#c.f2014#c.x1_dm_1 c.w#c.f2015#c.x1_dm_1 i.year i.year#c.x1 d x1 c.d#c.x1, vce(cluster id)
* Now use margins to account for the sampling error in the mean of x1. It is
* important to have w defined as the time-varying treatment variable.
* In practice, it seems to have little effect:
reg logy c.w#c.f2014 c.w#c.f2015 c.w#c.f2014#c.x1 c.w#c.f2015#c.x1 i.year c.f2014#c.x1 c.f2015#c.x1 d x1 c.d#c.x1, vce(cluster id)
margins, dydx(w) at(f2014 = 1 f2015 = 0) subpop(if d == 1) vce(uncond)
margins, dydx(w) at(f2014 = 0 f2015 = 1) subpop(if d == 1) vce(uncond)
* Apply Callaway & Sant'Anna (2021):
gen first_treat = 0
replace first_treat = 2014 if d
csdid logy x1, ivar(id) time(year) gvar(first_treat)
* Show imputation is equivalent to POLS/ETWFE:
reg logy i.year c.f2014#c.x1 c.f2015#c.x1 d x1 c.d#c.x1 if w == 0
predict tetilda, resid
sum tetilda if (d & f2014)
sum tetilda if (d & f2015)
* Regression produces same ATT estimates, but use the full pooled regression for
* valid standard errors:
reg tetilda c.w#c.f2014 c.w#c.f2015 c.w#c.f2014#c.x1_dm_1 c.w#c.f2015#c.x1_dm_1 if w == 1, nocons
* Now test parallel trends. First, unconditionally.
xtdidreg (logy) (w), group(id) time(year)
* estat trendplots
estat ptrends
* With two control periods, same as the following:
sort id year
reg D.logy d if year <= 2013, vce(robust)
* Test/correct for common trends in a general equation:
gen t = year - 2011
xtreg logy c.w#c.f2014 c.w#c.f2015 c.w#c.f2014#c.x1_dm_1 c.w#c.f2015#c.x1_dm_1 i.year i.year#c.x1 c.d#c.t, fe vce(cluster id)
reg logy c.w#c.f2014 c.w#c.f2015 c.w#c.f2014#c.x1_dm_1 c.w#c.f2015#c.x1_dm_1 i.year i.year#c.x1 d x1 c.d#c.x1 c.d#c.t, vce(cluster id)
* Allow the trend to also vary with x1:
reg logy c.w#c.f2014 c.w#c.f2015 c.w#c.f2014#c.x1_dm_1 c.w#c.f2015#c.x1_dm_1 i.year i.year#c.x1 d x1 c.d#c.x1 c.d#c.t c.d#c.t#c.x1, vce(cluster id)
test c.d#c.t c.d#c.t#c.x1
* Imputation. Note that the estimates on c.d#c.t and c.d#c.t#c.x1
* are the same as full regression with all data, as is the joint
* F statistic:
drop tetilda
reg logy i.year i.year#c.x1 d x1 c.d#c.x1 c.d#c.t c.d#c.t#c.x1 if w == 0, vce(cluster id)
test c.d#c.t c.d#c.t#c.x1
predict tetilda, resid
sum tetilda if (d & f2014)
sum tetilda if (d & f2015)
* Again, can use regression to obtain the estimates but not the standard errors:
reg tetilda c.w#c.f2014 c.w#c.f2015 c.w#c.f2014#c.x1_dm_1 c.w#c.f2015#c.x1_dm_1 if w == 1, nocons Unknown #command
Linear regression Number of obs = 400
F(3, 99) = 34.04
Prob > F = 0.0000
R-squared = 0.0348
Root MSE = 7.0626
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.d#c.post | 3.206857 .4314949 7.43 0.000 2.350677 4.063036
|
d | -.0636461 1.356731 -0.05 0.963 -2.755695 2.628403
post | -.1255431 .3021942 -0.42 0.679 -.725162 .4740758
_cons | .11305 .9272947 0.12 0.903 -1.726904 1.953004
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(3, 99) = 34.04
Prob > F = 0.0000
R-squared = 0.0348
Root MSE = 7.0626
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
w | 3.206857 .4314949 7.43 0.000 2.350677 4.063036
d | -.0636461 1.356731 -0.05 0.963 -2.755695 2.628403
post | -.1255431 .3021942 -0.42 0.679 -.725162 .4740758
_cons | .11305 .9272947 0.12 0.903 -1.726904 1.953004
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(4, 99) = 27.19
Prob > F = 0.0000
R-squared = 0.0381
Root MSE = 7.0597
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
w | 3.206857 .4320407 7.42 0.000 2.349594 4.064119
d | -.0636461 1.358447 -0.05 0.963 -2.7591 2.631808
f2014 | -.7011363 .3419429 -2.05 0.043 -1.379625 -.0226474
f2015 | .4500501 .3583234 1.26 0.212 -.2609413 1.161041
_cons | .11305 .9284678 0.12 0.903 -1.729232 1.955331
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(5, 99) = 21.95
Prob > F = 0.0000
R-squared = 0.0383
Root MSE = 7.0677
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
w | 3.206857 .4325887 7.41 0.000 2.348507 4.065207
d | -.0636461 1.36017 -0.05 0.963 -2.762519 2.635227
|
year |
2013 | .3260046 .3325494 0.98 0.329 -.3338456 .9858548
2014 | -.538134 .3934578 -1.37 0.175 -1.31884 .2425716
2015 | .6130524 .3830314 1.60 0.113 -.146965 1.37307
|
_cons | -.0499523 .9214237 -0.05 0.957 -1.878257 1.778352
------------------------------------------------------------------------------
Fixed-effects (within) regression Number of obs = 400
Group variable: id Number of groups = 100
R-squared: Obs per group:
Within = 0.2539 min = 4
Between = 0.0129 avg = 4.0
Overall = 0.0383 max = 4
F(4,99) = 27.29
corr(u_i, Xb) = -0.0027 Prob > F = 0.0000
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
w | 3.206857 .4320407 7.42 0.000 2.349594 4.064119
|
year |
2013 | .3260046 .3321282 0.98 0.329 -.3330098 .985019
2014 | -.538134 .3929594 -1.37 0.174 -1.317851 .2415828
2015 | .6130524 .3825463 1.60 0.112 -.1460024 1.372107
|
_cons | -.0817754 .2003881 -0.41 0.684 -.4793889 .3158381
-------------+----------------------------------------------------------------
sigma_u | 6.7557992
sigma_e | 2.3305153
rho | .89365429 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Fixed-effects (within) regression Number of obs = 400
Group variable: id Number of groups = 100
R-squared: Obs per group:
Within = 0.2207 min = 4
Between = 0.0129 avg = 4.0
Overall = 0.0348 max = 4
F(2,99) = 50.25
corr(u_i, Xb) = -0.0028 Prob > F = 0.0000
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
w | 3.206857 .4309511 7.44 0.000 2.351756 4.061957
post | -.1255431 .3018134 -0.42 0.678 -.7244064 .4733201
_cons | .0812269 .1077378 0.75 0.453 -.1325482 .295002
-------------+----------------------------------------------------------------
sigma_u | 6.7557992
sigma_e | 2.3738231
rho | .89010353 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Random-effects GLS regression Number of obs = 400
Group variable: id Number of groups = 100
R-squared: Obs per group:
Within = 0.2207 min = 4
Between = 0.0129 avg = 4.0
Overall = 0.0348 max = 4
Wald chi2(3) = 102.12
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
w | 3.206857 .4314949 7.43 0.000 2.361142 4.052571
d | -.0636461 1.356731 -0.05 0.963 -2.72279 2.595498
post | -.1255431 .3021942 -0.42 0.678 -.7178329 .4667467
_cons | .11305 .9272947 0.12 0.903 -1.704414 1.930514
-------------+----------------------------------------------------------------
sigma_u | 6.685563
sigma_e | 2.3738231
rho | .88804221 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Fixed-effects (within) regression Number of obs = 400
Group variable: id Number of groups = 100
R-squared: Obs per group:
Within = 0.2575 min = 4
Between = 0.0129 avg = 4.0
Overall = 0.0387 max = 4
F(5,99) = 21.98
corr(u_i, Xb) = -0.0027 Prob > F = 0.0000
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.813593 .5368268 5.24 0.000 1.748412 3.878774
|
c.w#c.f2015 | 3.600121 .5765523 6.24 0.000 2.456116 4.744126
|
year |
2013 | .3260046 .3325494 0.98 0.329 -.3338456 .9858548
2014 | -.341502 .4488412 -0.76 0.449 -1.2321 .5490964
2015 | .4164204 .4300378 0.97 0.335 -.4368679 1.269709
|
_cons | -.0817754 .2006423 -0.41 0.684 -.4798932 .3163424
-------------+----------------------------------------------------------------
sigma_u | 6.7557992
sigma_e | 2.3288408
rho | .89379082 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Fixed-effects (within) regression Number of obs = 400
Group variable: id Number of groups = 100
R-squared: Obs per group:
Within = 0.2575 min = 4
Between = 0.0129 avg = 4.0
Overall = 0.0387 max = 4
F(5,99) = 21.98
corr(u_i, Xb) = -0.0027 Prob > F = 0.0000
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.d#c.f2014 | 2.813593 .5368268 5.24 0.000 1.748412 3.878774
|
c.d#c.f2015 | 3.600121 .5765523 6.24 0.000 2.456116 4.744126
|
year |
2013 | .3260046 .3325494 0.98 0.329 -.3338456 .9858548
2014 | -.341502 .4488412 -0.76 0.449 -1.2321 .5490964
2015 | .4164204 .4300378 0.97 0.335 -.4368679 1.269709
|
_cons | -.0817754 .2006423 -0.41 0.684 -.4798932 .3163424
-------------+----------------------------------------------------------------
sigma_u | 6.7557992
sigma_e | 2.3288408
rho | .89379082 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(5, 99) = 21.97
Prob > F = 0.0000
R-squared = 0.0384
Root MSE = 7.0672
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.813593 .5368268 5.24 0.000 1.748412 3.878774
|
c.w#c.f2015 | 3.600121 .5765523 6.24 0.000 2.456116 4.744126
|
d | -.0636461 1.36017 -0.05 0.963 -2.762519 2.635227
f2014 | -.5045043 .3923088 -1.29 0.201 -1.28293 .2739216
f2015 | .2534181 .4206018 0.60 0.548 -.5811472 1.087983
_cons | .11305 .9296453 0.12 0.903 -1.731568 1.957668
------------------------------------------------------------------------------
( 1) .5*c.w#c.f2014 + .5*c.w#c.f2015 = 0
------------------------------------------------------------------------------
logy | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
(1) | 3.206857 .4325887 7.41 0.000 2.348507 4.065207
------------------------------------------------------------------------------
Random-effects GLS regression Number of obs = 400
Group variable: id Number of groups = 100
R-squared: Obs per group:
Within = 0.2550 min = 4
Between = 0.0129 avg = 4.0
Overall = 0.0384 max = 4
Wald chi2(5) = 109.84
corr(u_i, X) = 0 (assumed) Prob > chi2 = 0.0000
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.813593 .5368268 5.24 0.000 1.761432 3.865754
|
c.w#c.f2015 | 3.600121 .5765523 6.24 0.000 2.470099 4.730143
|
d | -.0636461 1.36017 -0.05 0.963 -2.729531 2.602239
f2014 | -.5045043 .3923088 -1.29 0.198 -1.273415 .2644069
f2015 | .2534181 .4206018 0.60 0.547 -.5709463 1.077782
_cons | .11305 .9296453 0.12 0.903 -1.709021 1.935121
-------------+----------------------------------------------------------------
sigma_u | 6.6895238
sigma_e | 2.3287614
rho | .89191109 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(7, 99) = 182.67
Prob > F = 0.0000
R-squared = 0.8438
Root MSE = 2.8557
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.503103 .3066694 8.16 0.000 1.894605 3.111602
|
c.w#c.f2015 | 3.327274 .2902385 11.46 0.000 2.751378 3.90317
|
d | .881691 .5285779 1.67 0.098 -.1671223 1.930504
f2014 | .0840872 .2215839 0.38 0.705 -.3555834 .5237578
f2015 | .2391039 .2082552 1.15 0.254 -.1741196 .6523275
x1 | 2.506694 .1327049 18.89 0.000 2.243379 2.77001
|
c.d#c.x1 | .0743163 .1915921 0.39 0.699 -.305844 .4544765
|
_cons | .2148708 .3676352 0.58 0.560 -.5145972 .9443389
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(8, 99) = 166.57
Prob > F = 0.0000
R-squared = 0.8438
Root MSE = 2.8593
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.503013 .3070762 8.15 0.000 1.893707 3.112319
|
c.w#c.f2015 | 3.327292 .2906341 11.45 0.000 2.750611 3.903973
|
d | .8816195 .5292507 1.67 0.099 -.1685288 1.931768
|
year |
2013 | -.0324553 .1709047 -0.19 0.850 -.3715673 .3066568
2014 | .067934 .2343565 0.29 0.773 -.3970801 .5329482
2015 | .2228745 .2273177 0.98 0.329 -.2281732 .6739222
|
x1 | 2.507011 .1335155 18.78 0.000 2.242087 2.771935
|
c.d#c.x1 | .0738548 .1927512 0.38 0.702 -.3086053 .4563149
|
_cons | .2311113 .37362 0.62 0.538 -.5102317 .9724544
------------------------------------------------------------------------------
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
x1 | 200 -.4048388 2.493319 -6.95061 4.588676
Fixed-effects (within) regression Number of obs = 400
Group variable: id Number of groups = 100
R-squared: Obs per group:
Within = 0.3620 min = 4
Between = 0.4276 avg = 4.0
Overall = 0.2782 max = 4
F(9,99) = 23.70
corr(u_i, Xb) = 0.3305 Prob > F = 0.0000
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.938834 .5031086 5.84 0.000 1.940557 3.93711
|
c.w#c.f2015 | 3.742608 .534969 7.00 0.000 2.681113 4.804102
|
c.w#c.f2014#|
c.x1_dm_1 | -.2243583 .1904085 -1.18 0.242 -.60217 .1534535
|
c.w#c.f2015#|
c.x1_dm_1 | .0296616 .2431933 0.12 0.903 -.4528867 .5122099
|
year |
2013 | .3260046 .3342504 0.98 0.332 -.3372208 .98923
2014 | -.16839 .3880607 -0.43 0.665 -.9383866 .6016067
2015 | .4360252 .3954162 1.10 0.273 -.3485664 1.220617
|
c.f2014#c.x1 | .6285213 .1342955 4.68 0.000 .3620499 .8949926
|
c.f2015#c.x1 | .5615944 .1919492 2.93 0.004 .1807255 .9424633
|
_cons | -.0817754 .1900787 -0.43 0.668 -.4589327 .2953819
-------------+----------------------------------------------------------------
sigma_u | 6.1628203
sigma_e | 2.1734897
rho | .88937776 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(12, 99) = 112.09
Prob > F = 0.0000
R-squared = 0.8445
Root MSE = 2.8679
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.496168 .3173129 7.87 0.000 1.86655 3.125785
|
c.w#c.f2015 | 3.349012 .3132217 10.69 0.000 2.727513 3.970512
|
c.w#c.f2014#|
c.x1_dm_1 | -.2701343 .1759396 -1.54 0.128 -.6192367 .0789681
|
c.w#c.f2015#|
c.x1_dm_1 | -.0683301 .1502763 -0.45 0.650 -.3665108 .2298506
|
year |
2013 | -.0188111 .1738311 -0.11 0.914 -.3637298 .3261075
2014 | .0787158 .2377676 0.33 0.741 -.3930668 .5504984
2015 | .2325892 .2297992 1.01 0.314 -.2233823 .6885607
|
c.f2014#c.x1 | .0404883 .1103324 0.37 0.714 -.1784352 .2594118
|
c.f2015#c.x1 | .0778513 .1061898 0.73 0.465 -.1328523 .2885549
|
d | .904209 .5311615 1.70 0.092 -.1497307 1.958149
x1 | 2.476382 .1544607 16.03 0.000 2.169899 2.782866
|
c.d#c.x1 | .1570952 .2113815 0.74 0.459 -.2623316 .576522
|
_cons | .2230452 .3747574 0.60 0.553 -.5205549 .9666452
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(13, 99) = 103.22
Prob > F = 0.0000
R-squared = 0.8445
Root MSE = 2.8715
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.500641 .3196658 7.82 0.000 1.866354 3.134927
|
c.w#c.f2015 | 3.353485 .3142809 10.67 0.000 2.729884 3.977087
|
c.w#c.f2014#|
c.x1_dm_1 | -.2674635 .177262 -1.51 0.135 -.6191897 .0842627
|
c.w#c.f2015#|
c.x1_dm_1 | -.0656593 .1506332 -0.44 0.664 -.3645482 .2332296
|
year |
2013 | -.0261778 .1756818 -0.15 0.882 -.3747686 .3224129
2014 | .07109 .2439251 0.29 0.771 -.4129104 .5550904
2015 | .2249634 .2325834 0.97 0.336 -.2365326 .6864593
|
year#c.x1 |
2013 | -.0298874 .0910807 -0.33 0.743 -.2106112 .1508364
2014 | .0227905 .1348423 0.17 0.866 -.2447658 .2903469
2015 | .0601535 .1180653 0.51 0.612 -.1741137 .2944207
|
d | .8986549 .5321658 1.69 0.094 -.1572776 1.954587
x1 | 2.49408 .1726029 14.45 0.000 2.151599 2.836562
|
c.d#c.x1 | .1544244 .2128387 0.73 0.470 -.2678938 .5767426
|
_cons | .230671 .3794753 0.61 0.545 -.5222904 .9836324
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(12, 99) = 112.09
Prob > F = 0.0000
R-squared = 0.8445
Root MSE = 2.8679
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.386807 .3096756 7.71 0.000 1.772343 3.001271
|
c.w#c.f2015 | 3.32135 .2964989 11.20 0.000 2.733032 3.909668
|
c.w#c.f2014#|
c.x1 | -.2701343 .1759396 -1.54 0.128 -.6192367 .0789681
|
c.w#c.f2015#|
c.x1 | -.0683301 .1502763 -0.45 0.650 -.3665109 .2298506
|
year |
2013 | -.0188111 .1738311 -0.11 0.914 -.3637298 .3261075
2014 | .0787158 .2377676 0.33 0.741 -.3930668 .5504984
2015 | .2325892 .2297992 1.01 0.314 -.2233823 .6885607
|
c.f2014#c.x1 | .0404883 .1103324 0.37 0.714 -.1784352 .2594118
|
c.f2015#c.x1 | .0778513 .1061898 0.73 0.465 -.1328523 .2885549
|
d | .904209 .5311615 1.70 0.092 -.1497307 1.958149
x1 | 2.476382 .1544607 16.03 0.000 2.169899 2.782866
|
c.d#c.x1 | .1570952 .2113815 0.74 0.459 -.2623316 .576522
|
_cons | .2230452 .3747574 0.60 0.553 -.5205549 .9666452
------------------------------------------------------------------------------
Average marginal effects Number of obs = 400
Subpop. no. obs = 200
Expression: Linear prediction, predict()
dy/dx wrt: w
At: f2014 = 1
f2015 = 0
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Unconditional
| dy/dx std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
w | 2.496168 .3059971 8.16 0.000 1.889003 3.103333
------------------------------------------------------------------------------
Average marginal effects Number of obs = 400
Subpop. no. obs = 200
Expression: Linear prediction, predict()
dy/dx wrt: w
At: f2014 = 0
f2015 = 1
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Unconditional
| dy/dx std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
w | 3.349012 .3110518 10.77 0.000 2.731818 3.966207
------------------------------------------------------------------------------
(200 real changes made)
...
Difference-in-difference with Multiple Time Periods
Outcome model :
Treatment model:
------------------------------------------------------------------------------
| Coefficient Std. err. z P>|z| [95% conf. interval]
-------------+----------------------------------------------------------------
g2014 |
t_2012_2013 | -1.401593 .6428286 -2.18 0.029 -2.661514 -.1416721
t_2013_2014 | 3.388216 .5853084 5.79 0.000 2.241033 4.5354
t_2013_2015 | 3.915803 .6491816 6.03 0.000 2.643431 5.188176
------------------------------------------------------------------------------
Control: Never Treated
See Callaway and Sant'Anna (2020) for details
Source | SS df MS Number of obs = 300
-------------+---------------------------------- F(8, 291) = 210.00
Model | 12568.7919 8 1571.09898 Prob > F = 0.0000
Residual | 2177.12159 291 7.48151749 R-squared = 0.8524
-------------+---------------------------------- Adj R-squared = 0.8483
Total | 14745.9135 299 49.3174363 Root MSE = 2.7352
------------------------------------------------------------------------------
logy | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
year |
2013 | -.0188111 .3880383 -0.05 0.961 -.7825286 .7449063
2014 | .0787158 .513587 0.15 0.878 -.9321002 1.089532
2015 | .2325892 .5121166 0.45 0.650 -.7753329 1.240511
|
c.f2014#c.x1 | .0404883 .1798899 0.23 0.822 -.3135619 .3945385
|
c.f2015#c.x1 | .0778513 .1879703 0.41 0.679 -.2921023 .4478049
|
d | .904209 .3894073 2.32 0.021 .1377972 1.670621
x1 | 2.476382 .1100309 22.51 0.000 2.259825 2.69294
|
c.d#c.x1 | .1570952 .1556659 1.01 0.314 -.1492786 .463469
|
_cons | .2230452 .3355663 0.66 0.507 -.4373996 .8834899
------------------------------------------------------------------------------
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
tetilda | 50 2.525512 3.370158 -5.463166 10.58631
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
tetilda | 50 3.34147 3.105354 -3.463709 10.24051
Source | SS df MS Number of obs = 100
-------------+---------------------------------- F(4, 96) = 21.48
Model | 900.277122 4 225.06928 Prob > F = 0.0000
Residual | 1005.96245 96 10.4787755 R-squared = 0.4723
-------------+---------------------------------- Adj R-squared = 0.4503
Total | 1906.23957 100 19.0623957 Root MSE = 3.2371
------------------------------------------------------------------------------
tetilda | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.496168 .4582501 5.45 0.000 1.586549 3.405787
|
c.w#c.f2015 | 3.349012 .4582306 7.31 0.000 2.439432 4.258593
|
c.w#c.f2014#|
c.x1_dm_1 | -.2701343 .1881329 -1.44 0.154 -.6435751 .1033065
|
c.w#c.f2015#|
c.x1_dm_1 | -.0683301 .1811302 -0.38 0.707 -.4278707 .2912104
------------------------------------------------------------------------------
Number of groups and treatment time
Time variable: year
Control: w = 0
Treatment: w = 1
-----------------------------------
| Control Treatment
-------------+---------------------
Group |
id | 50 50
-------------+---------------------
Time |
Minimum | 2012 2014
Maximum | 2012 2014
-----------------------------------
Difference-in-differences regression Number of obs = 400
Data type: Longitudinal
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
ATET |
w |
(1 vs 0) | 3.206857 .4320407 7.42 0.000 2.349594 4.064119
------------------------------------------------------------------------------
Note: ATET estimate adjusted for panel effects and time effects.
Parallel-trends test (pretreatment time period)
H0: Linear trends are parallel
F(1, 99) = 4.64
Prob > F = 0.0337
Linear regression Number of obs = 100
F(1, 98) = 4.66
Prob > F = 0.0332
R-squared = 0.0454
Root MSE = 3.2451
------------------------------------------------------------------------------
| Robust
D.logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
d | -1.401579 .6490187 -2.16 0.033 -2.689536 -.1136226
_cons | 1.026794 .4697368 2.19 0.031 .0946167 1.958972
------------------------------------------------------------------------------
Fixed-effects (within) regression Number of obs = 400
Group variable: id Number of groups = 100
R-squared: Obs per group:
Within = 0.8680 min = 4
Between = 0.8466 avg = 4.0
Overall = 0.8412 max = 4
F(12,99) = 157.25
corr(u_i, Xb) = 0.4488 Prob > F = 0.0000
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 2.456402 .5067512 4.85 0.000 1.450898 3.461907
|
c.w#c.f2015 | 3.208291 .7927064 4.05 0.000 1.63539 4.781193
|
c.w#c.f2014#|
c.x1_dm_1 | -.0503407 .1059229 -0.48 0.636 -.2605148 .1598333
|
c.w#c.f2015#|
c.x1_dm_1 | .0924389 .0842389 1.10 0.275 -.0747095 .2595872
|
year |
2013 | .0205346 .1993658 0.10 0.918 -.3750504 .4161195
2014 | -.009636 .2174468 -0.04 0.965 -.4410976 .4218256
2015 | .2623215 .1940125 1.35 0.179 -.1226414 .6472845
|
year#c.x1 |
2012 | 1.933064 .0764551 25.28 0.000 1.781361 2.084768
2013 | 2.003714 .0662191 30.26 0.000 1.872321 2.135107
2014 | 2.01687 .0849868 23.73 0.000 1.848238 2.185502
2015 | 1.991922 .072964 27.30 0.000 1.847146 2.136699
|
c.d#c.t | .0668116 .2957253 0.23 0.822 -.5199715 .6535946
|
_cons | .4575496 .1764897 2.59 0.011 .1073557 .8077434
-------------+----------------------------------------------------------------
sigma_u | 3.0627341
sigma_e | .99359271
rho | .90477742 (fraction of variance due to u_i)
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(14, 99) = 101.72
Prob > F = 0.0000
R-squared = 0.8446
Root MSE = 2.8741
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 1.833542 .6136741 2.99 0.004 .615879 3.051204
|
c.w#c.f2015 | 2.243344 .9499793 2.36 0.020 .358379 4.128309
|
c.w#c.f2014#|
c.x1_dm_1 | -.2653456 .177891 -1.49 0.139 -.6183199 .0876287
|
c.w#c.f2015#|
c.x1_dm_1 | -.0635414 .150658 -0.42 0.674 -.3624795 .2353968
|
year |
2013 | -.2472397 .2421693 -1.02 0.310 -.7277561 .2332767
2014 | -.0388673 .2538858 -0.15 0.879 -.5426319 .4648973
2015 | .1150061 .2472402 0.47 0.643 -.375572 .6055842
|
year#c.x1 |
2013 | -.0241251 .0913453 -0.26 0.792 -.2053741 .1571239
2014 | .0216128 .1358993 0.16 0.874 -.248041 .2912665
2015 | .0589757 .1180401 0.50 0.618 -.1752415 .293193
|
d | .23577 .7209278 0.33 0.744 -1.194707 1.666247
x1 | 2.495258 .1736213 14.37 0.000 2.150756 2.83976
|
c.d#c.x1 | .1523065 .2136548 0.71 0.478 -.2716309 .5762438
|
c.d#c.t | .4430422 .3468509 1.28 0.204 -.2451853 1.13127
|
_cons | .3406282 .3823786 0.89 0.375 -.418094 1.09935
------------------------------------------------------------------------------
Linear regression Number of obs = 400
F(15, 99) = 95.45
Prob > F = 0.0000
R-squared = 0.8450
Root MSE = 2.8744
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 1.917612 .6348799 3.02 0.003 .6578725 3.177352
|
c.w#c.f2015 | 2.378839 .9651184 2.46 0.015 .4638351 4.293844
|
c.w#c.f2014#|
c.x1_dm_1 | -.7263424 .2927708 -2.48 0.015 -1.307263 -.1454217
|
c.w#c.f2015#|
c.x1_dm_1 | -.8415862 .4868867 -1.73 0.087 -1.807675 .1245027
|
year |
2013 | -.2611685 .2476914 -1.05 0.294 -.7526419 .2303049
2014 | -.0671335 .2642967 -0.25 0.800 -.5915555 .4572885
2015 | .0867399 .2545516 0.34 0.734 -.4183457 .5918255
|
year#c.x1 |
2013 | -.1857772 .1396877 -1.33 0.187 -.4629479 .0913934
2014 | -.0735746 .1623912 -0.45 0.651 -.395794 .2486448
2015 | -.0362116 .1317686 -0.27 0.784 -.2976691 .2252458
|
d | .1075441 .732768 0.15 0.884 -1.346427 1.561515
x1 | 2.590445 .1878499 13.79 0.000 2.21771 2.96318
|
c.d#c.x1 | -.3378406 .3669519 -0.92 0.359 -1.065953 .3902716
|
c.d#c.t | .5199705 .3447009 1.51 0.135 -.1639908 1.203932
|
c.d#c.t#c.x1 | .317048 .1852193 1.71 0.090 -.0504673 .6845633
|
_cons | .3688944 .3868765 0.95 0.343 -.3987525 1.136541
------------------------------------------------------------------------------
( 1) c.d#c.t = 0
( 2) c.d#c.t#c.x1 = 0
F( 2, 99) = 2.47
Prob > F = 0.0898
Linear regression Number of obs = 300
F(11, 99) = 82.82
Prob > F = 0.0000
R-squared = 0.8531
Root MSE = 2.7429
(Std. err. adjusted for 100 clusters in id)
------------------------------------------------------------------------------
| Robust
logy | Coefficient std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
year |
2013 | -.2611685 .2475879 -1.05 0.294 -.7524366 .2300996
2014 | -.0671335 .2641863 -0.25 0.800 -.5913364 .4570694
2015 | .0867399 .2544452 0.34 0.734 -.4181347 .5916145
|
year#c.x1 |
2013 | -.1857773 .1396293 -1.33 0.186 -.4628321 .0912775
2014 | -.0735746 .1623234 -0.45 0.651 -.3956594 .2485101
2015 | -.0362116 .1317135 -0.27 0.784 -.2975599 .2251366
|
d | .107544 .7324618 0.15 0.884 -1.345819 1.560907
x1 | 2.590445 .1877714 13.80 0.000 2.217866 2.963025
|
c.d#c.x1 | -.3378407 .3667986 -0.92 0.359 -1.065649 .3899672
|
c.d#c.t | .5199705 .3445569 1.51 0.134 -.1637051 1.203646
|
c.d#c.t#c.x1 | .3170481 .1851419 1.71 0.090 -.0503137 .6844098
|
_cons | .3688944 .3867149 0.95 0.342 -.3984318 1.136221
------------------------------------------------------------------------------
( 1) c.d#c.t = 0
( 2) c.d#c.t#c.x1 = 0
F( 2, 99) = 2.47
Prob > F = 0.0896
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
tetilda | 50 1.996512 3.755621 -5.357233 9.968782
Variable | Obs Mean Std. dev. Min Max
-------------+---------------------------------------------------------
tetilda | 50 2.285944 3.772198 -4.526336 10.79229
Source | SS df MS Number of obs = 100
-------------+---------------------------------- F(4, 96) = 20.11
Model | 842.991546 4 210.747887 Prob > F = 0.0000
Residual | 1005.96248 96 10.4787758 R-squared = 0.4559
-------------+---------------------------------- Adj R-squared = 0.4333
Total | 1848.95402 100 18.4895402 Root MSE = 3.2371
------------------------------------------------------------------------------
tetilda | Coefficient Std. err. t P>|t| [95% conf. interval]
-------------+----------------------------------------------------------------
c.w#c.f2014 | 1.917612 .4582501 4.18 0.000 1.007993 2.827231
|
c.w#c.f2015 | 2.378839 .4582306 5.19 0.000 1.469259 3.28842
|
c.w#c.f2014#|
c.x1_dm_1 | -.7263426 .1881329 -3.86 0.000 -1.099783 -.3529017
|
c.w#c.f2015#|
c.x1_dm_1 | -.8415865 .1811302 -4.65 0.000 -1.201127 -.4820459
------------------------------------------------------------------------------