`drdid`

is used to compute the locally efficient doubly robust estimators for the ATT
in difference-in-differences (DiD) setups. It can be used with panel or stationary repeated cross section data.
Data should be store in "long" format.

- yname
The name of the outcome variable.

- tname
The name of the column containing the time periods.

- idname
The name of the column containing the unit id name.

- dname
The name of the column containing the treatment group (=1 if observation is treated in the post-treatment, =0 otherwise)

- xformla
A formula for the covariates to include in the model. It should be of the form

`~ X1 + X2`

(intercept should not be listed as it is always automatically included). Default is NULL which is equivalent to`xformla=~1`

.- data
The name of the data.frame that contains the data.

- panel
Whether or not the data is a panel dataset. The panel dataset should be provided in long format – that is, where each row corresponds to a unit observed at a particular point in time. The default is TRUE. When

`panel = FALSE`

, the data is treated as stationary repeated cross sections.- estMethod
the method to estimate the nuisance parameters. The default is "imp" which uses weighted least squares to estimate the outcome regressions and inverse probability tilting to the estimate the the propensity score, leading to the improved locally efficient DR DiD estimator proposed by Sant'Anna and Zhao (2020). The other alternative is "trad", which then uses OLS to estimate outcome regressions and maximum likelihood to estimate propensity score. This leads to the "traditional" locally efficient DR DiD estimator proposed by Sant'Anna and Zhao (2020).

- weightsname
The name of the column containing the sampling weights. If NULL, then every observation has the same weights. The weights are normalized and therefore enforced to have mean 1 across all observations.

- boot
Logical argument to whether bootstrap should be used for inference. Default is

`FALSE`

and analytical standard errors are reported.- 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
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`

.

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

- att.inf.func
Estimate of the influence function. Default is NULL

- ps.flag
Convergence Flag for the propensity score estimation (only active if

`estMethod = "imp"`

.): =0 if`trust`

algorithm converged, =1 if IPT (original) algorithm converged (in case it was used), =2 if GLM logit estimator was used (i.e., if both`trust`

and IPT did not converged).- call.param
The matched call.

- argu
Some arguments used in the call (panel, estMethod, boot, boot.type, nboot, type="dr")

When panel data are available (`panel = TRUE`

), the `drdid`

function implements the
locally efficient doubly robust difference-in-differences (DiD) estimator for the average treatment effect
on the treated (ATT) defined in equation (3.1) 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 a linear regression model for the outcome evolution among the comparison units.

When only stationary repeated cross-section data are available (`panel = FALSE`

), the `drdid`

function
implements the locally efficient doubly robust difference-in-differences (DiD) estimator for the
average treatment effect on the treated (ATT) defined in equation (3.4) 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 of both treated and comparison units,
in both pre and post-treatment periods.

When one sets `estMethod = "imp"`

(the default), the nuisance parameters (propensity score and
outcome regression parameters) are estimated using the methods described in Sections 3.1 and 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.

When one sets `estMethod = "trad"`

, the propensity score parameters are estimated using maximum
likelihood, and the outcome regression coefficients are estimated using ordinary least squares.

The main advantage of using `estMethod = "imp"`

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

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

```
# -----------------------------------------------
# Panel data case
# -----------------------------------------------
# Form the Lalonde sample with CPS comparison group
eval_lalonde_cps <- subset(nsw_long, nsw_long$treated == 0 | nsw_long$sample == 2)
# Further reduce sample to speed example
set.seed(123)
unit_random <- sample(unique(eval_lalonde_cps$id), 5000)
eval_lalonde_cps <- eval_lalonde_cps[eval_lalonde_cps$id %in% unit_random,]
# -----------------------------------------------
# Implement improved DR locally efficient DiD with panel data
drdid(yname="re", tname = "year", idname = "id", dname = "experimental",
xformla= ~ age+ educ+ black+ married+ nodegree+ hisp+ re74,
data = eval_lalonde_cps, panel = TRUE)
#> Call:
#> drdid(yname = "re", tname = "year", idname = "id", dname = "experimental",
#> xformla = ~age + educ + black + married + nodegree + hisp +
#> re74, data = eval_lalonde_cps, panel = TRUE)
#> ------------------------------------------------------------------
#> Further improved locally efficient DR DID estimator for the ATT:
#>
#> ATT Std. Error t value Pr(>|t|) [95% Conf. Interval]
#> -615.2344 683.2211 -0.9005 0.3679 -1954.3478 723.8791
#> ------------------------------------------------------------------
#> Estimator based on panel 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.
#Implement "traditional" DR locally efficient DiD with panel data
drdid(yname="re", tname = "year", idname = "id", dname = "experimental",
xformla= ~ age+ educ+ black+ married+ nodegree+ hisp+ re74,
data = eval_lalonde_cps, panel = TRUE, estMethod = "trad")
#> Call:
#> drdid(yname = "re", tname = "year", idname = "id", dname = "experimental",
#> xformla = ~age + educ + black + married + nodegree + hisp +
#> re74, data = eval_lalonde_cps, panel = TRUE, estMethod = "trad")
#> ------------------------------------------------------------------
#> Locally efficient DR DID estimator for the ATT:
#>
#> ATT Std. Error t value Pr(>|t|) [95% Conf. Interval]
#> -507.6113 692.9739 -0.7325 0.4639 -1865.8401 850.6175
#> ------------------------------------------------------------------
#> Estimator based on panel data.
#> Outcome regression est. method: OLS.
#> Propensity score est. method: maximum likelihood.
#> Analytical standard error.
#> ------------------------------------------------------------------
#> See Sant'Anna and Zhao (2020) for details.
# -----------------------------------------------
# Repeated cross section case
# -----------------------------------------------
# use the simulated data provided in the package
#Implement "improved" DR locally efficient DiD with repeated cross-section data
drdid(yname="y", tname = "post", idname = "id", dname = "d",
xformla= ~ x1 + x2 + x3 + x4,
data = sim_rc, panel = FALSE, estMethod = "imp")
#> Call:
#> drdid(yname = "y", tname = "post", idname = "id", dname = "d",
#> xformla = ~x1 + x2 + x3 + x4, data = sim_rc, panel = FALSE,
#> estMethod = "imp")
#> ------------------------------------------------------------------
#> Further improved locally efficient DR DID estimator for the ATT:
#>
#> ATT Std. Error t value Pr(>|t|) [95% Conf. Interval]
#> -0.2089 0.2003 -1.0425 0.2972 -0.6015 0.1838
#> ------------------------------------------------------------------
#> 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.
#Implement "traditional" DR locally efficient DiD with repeated cross-section data
drdid(yname="y", tname = "post", idname = "id", dname = "d",
xformla= ~ x1 + x2 + x3 + x4,
data = sim_rc, panel = FALSE, estMethod = "trad")
#> Call:
#> drdid(yname = "y", tname = "post", idname = "id", dname = "d",
#> xformla = ~x1 + x2 + x3 + x4, data = sim_rc, panel = FALSE,
#> estMethod = "trad")
#> ------------------------------------------------------------------
#> Locally efficient DR DID estimator for the ATT:
#>
#> ATT Std. Error t value Pr(>|t|) [95% Conf. Interval]
#> -0.1678 0.2009 -0.8352 0.4036 -0.5616 0.226
#> ------------------------------------------------------------------
#> Estimator based on (stationary) repeated cross-sections data.
#> Outcome regression est. method: OLS.
#> Propensity score est. method: maximum likelihood.
#> Analytical standard error.
#> ------------------------------------------------------------------
#> See Sant'Anna and Zhao (2020) for details.
```