Author: Sebastian Kranz, Ulm University
Arkhangelsky et al. (2021) introduce the novel synthetic difference-in-differences (SDID) method that is implemented in the R package synthdid.
This package has the function adjust.outcome.for.x that can be used to adjust the outcome for exogenous time-varying covariates. For more background see my vignette Synthetic Difference-in-Differences with Time-Varying Covariates (2021).
If you use the package for research, you could cite the vignette as:
Kranz, Sebastian (2021), "Synthetic Difference-in-Differences with Time-Varying Covariates", mimeo
You can install the package from my r-universe repository by using the following code:
options(repos = c(skranz = 'https://skranz.r-universe.dev',
CRAN = 'https://cloud.r-project.org'))
install.packages("xsynthdid")We first simulate a panel data set with a true treatment effect of tau=50 using the function xsdid.mc in our package.
set.seed(1)
library(xsynthdid)
dat = xsdid.mc(N=30, T=30,tau=50,return.data = TRUE)
head(dat)## i t group treat exp treat_exp time.trend eps x.load xc.load
## 1 1 1 control 0 0 0 0.1439891 0.1828379 0.07844164 0.1473573
## 2 1 2 control 0 0 0 -0.2414066 0.1526865 0.07844164 0.1473573
## 3 1 3 control 0 0 0 0.1752374 0.2351750 0.07844164 0.1473573
## 4 1 4 control 0 0 0 1.8588536 -0.7058448 0.07844164 0.1473573
## 5 1 5 control 0 0 0 3.3896764 0.2107316 0.07844164 0.1473573
## 6 1 6 control 0 0 0 2.9570783 -0.2301411 0.07844164 0.1473573
## x xc y
## 1 0.12850395 -0.8351821 13.807491
## 2 0.06349820 -1.2433237 -8.742734
## 3 0.01129829 -1.2056680 9.561960
## 4 -0.04246972 -1.0943951 90.113348
## 5 -0.20068988 -1.2214046 159.660058
## 6 -0.15537278 -1.0809002 139.855136
We first perform SDID estimation without any adjustment for the exogenous covariate x:
pm = panel.matrices(dat, unit="i",time = "t",outcome = "y",treatment = "treat_exp")
sdid = synthdid_estimate(Y=pm$Y,N0=pm$N0,T0 = pm$T0) # inconsistent
sdid## synthdid: 39.760 +- 3.015. Effective N0/N0 = 14.5/15~1.0. Effective T0/T0 = 1.0/15~0.1. N1,T1 = 15,15.
We see how the estimator is a bit off the true causal effect.
We now adjust the outcomes y using the adjust.outcome.for.x function and use the adjusted outcome for the SDID estimation.
dat$y.adj = adjust.outcome.for.x(dat,unit="i",time = "t",
outcome = "y",treatment = "treat_exp", x="x")
pm = panel.matrices(dat, unit="i",time = "t",outcome = "y.adj",treatment = "treat_exp")
sdid.adj = synthdid_estimate(Y=pm$Y,N0=pm$N0,T0 = pm$T0)
sdid.adj## synthdid: 49.398 +- 0.252. Effective N0/N0 = 15.0/15~1.0. Effective T0/T0 = 6.9/15~0.5. N1,T1 = 15,15.
Our estimate is now pretty close to the true causal effect.
Note that the shown standard errors do not account for the initial adjustment for time varying covariates. Experimentally, I have included the function xsdid_se_bootstrap that allows to compute the standard errors using a clustered bootstrap approach similar to Algorithm 2 in Arkhangelsky et al. (2021).
To speed up the building of the README, here is an example with just B=10 bootstrap replications. Of course, you should set B to a larger number like 500.
xsdid_se_bootstrap(dat,B=10, unit="i",time = "t",outcome = "y",treatment = "treat_exp",x = "x")## $se
## [1] 0.1832767
##
## $boot.tau
## [1] 49.49238 49.59243 49.33778 49.64696 49.22328 49.31334 49.31244 49.35124
## [9] 49.56474 49.88914