Improved doubly robust DiD estimator for the ATT, with repeated cross-section data

drdid_imp_rc1 is used to compute the doubly robust estimators for the ATT in difference-in-differences (DiD) setups with stationary repeated cross-sectional data. The resulting estimator is also doubly robust for inference, though it is not locally efficient; see Section 3.2 of Sant'Anna and Zhao (2020).

drdid_imp_rc1(
  y,
  post,
  D,
  covariates,
  i.weights = NULL,
  boot = FALSE,
  boot.type = "weighted",
  nboot = NULL,
  inffunc = FALSE,
  trim.level = 0.995
)

Arguments

y: An \(n\) x \(1\) vector of outcomes from the both pre and post-treatment periods.
post: An \(n\) x \(1\) vector of Post-Treatment dummies (post = 1 if observation belongs to post-treatment period, and post = 0 if observation belongs to pre-treatment period.)
D: An \(n\) x \(1\) vector of Group indicators (=1 if observation is treated in the post-treatment, =0 otherwise).
covariates: An \(n\) x \(k\) matrix of covariates to be used in the propensity score and regression estimation. Please add a vector of constants if you want to include an intercept in the models. If covariates = NULL, this leads to an unconditional DiD estimator.
i.weights: An \(n\) x \(1\) vector of weights to be used. If NULL, then every observation has the same weights.
boot: Logical argument to whether bootstrap should be used for inference. Default is FALSE. Exactly zero centered bootstrap draws give a zero standard error and point interval. Nonfinite draws or a zero IQR with nonzero draws stop with an error.
boot.type: Type of bootstrap to be performed (not relevant if boot = FALSE). Options are "weighted" and "multiplier". If boot = TRUE, default is "weighted".
nboot: A finite scalar integer of at least 2 giving the number of bootstrap repetitions (not relevant if boot = FALSE). Default is 999.
inffunc: Logical argument to whether influence function should be returned. Default is FALSE.
trim.level: The level of trimming for the propensity score. Default is 0.995.

Value

A list containing the following components:

ATT: The DR DiD point estimate
se: The DR DiD standard error
uci: Estimate of the upper bound of a 95% CI for the ATT
lci: Estimate of the lower bound of a 95% CI for the ATT
boots: All Bootstrap draws of the ATT, in case bootstrap was used to conduct inference. Default is NULL
ps.flag: Source of the verified IPT propensity-score fit: =0 for the trust candidate, =1 for the secondary IPT candidate. A candidate must satisfy the original exponential-IPT equations to numerical tolerance; optimizer status alone is insufficient.
att.inf.func: Estimate of the influence function. Default is NULL
call.param: The matched call.
argu: Some arguments used (explicitly or not) in the call (panel = FALSE, estMethod = "imp2", boot, boot.type, nboot, type="dr")

Details

Propensity-score fitting must satisfy the original exponential-IPT equations; numerically ill-conditioned designs are rejected. If neither fitting route provides a verified solution, estimation stops rather than substituting a logit MLE. To explicitly use the corresponding traditional estimator, call drdid_rc1() to preserve this estimator variant. This changes both propensity-score and outcome-regression fitting. Each weighted-bootstrap replicate must also have a verified IPT fit; a failed replicate stops the bootstrap without skipping or redrawing it.

The drdid_imp_rc1 function implements the doubly robust difference-in-differences (DiD) estimator for the average treatment effect on the treated (ATT) defined in equation (3.3) in Sant'Anna and Zhao (2020). This estimator makes use of a logistic propensity score model for the probability of being in the treated group, and of (separate) linear regression models for the outcome among the comparison units in both pre and post-treatment time periods. Importantly, this estimator is not locally efficient for the ATT.

The nuisance parameters (propensity score and outcome regression parameters) are estimated using the methods described in Sections 3.2 of Sant'Anna and Zhao (2020). In short, the propensity score parameters are estimated using the inverse probability tilting estimator proposed by Graham, Pinto and Pinto (2012), and the outcome regression coefficients are estimated using weighted least squares,where the weights depend on the propensity score estimates; see Sant'Anna and Zhao (2020) for details.

The resulting estimator is not only doubly robust for the ATT, but it is also doubly robust for inference. However, we stress that it is not locally efficient; see Sant'Anna and Zhao (2020) for details.

References

Graham, Bryan, Pinto, Cristine, and Egel, Daniel (2012), "Inverse Probability Tilting for Moment Condition Models with Missing Data." Review of Economic Studies, vol. 79 (3), pp. 1053-1079, doi:10.1093/restud/rdr047

Sant'Anna, Pedro H. C. and Zhao, Jun. (2020), "Doubly Robust Difference-in-Differences Estimators." Journal of Econometrics, Vol. 219 (1), pp. 101-122, doi:10.1016/j.jeconom.2020.06.003

Examples

# use the simulated data provided in the package
covX = as.matrix(cbind(1, sim_rc[,5:8]))
# Implement the improved DR DiD estimator (but not locally efficient!)
drdid_imp_rc1(y = sim_rc$y, post = sim_rc$post, D = sim_rc$d,
             covariates= covX)
#>  Call:
#> drdid_imp_rc1(y = sim_rc$y, post = sim_rc$post, D = sim_rc$d, 
#>     covariates = covX)
#> ------------------------------------------------------------------
#>  Further improved DR (but not locally efficient) DID estimator for the ATT:
#>  
#>    ATT     Std. Error  t value    Pr(>|t|)  [95% Conf. Interval] 
#>  -3.6837     3.1129    -1.1834     0.2367    -9.7851     2.4176  
#> ------------------------------------------------------------------
#>  Estimator based on (stationary) repeated cross-sections data.
#>  Outcome regression est. method: weighted least squares.
#>  Propensity score est. method: inverse prob. tilting.
#>  Analytical standard error.
#> ------------------------------------------------------------------
#>  See Sant'Anna and Zhao (2020) for details.