Estimates the Quantile Treatment Effect on the Treated (QTT) and Average Treatment Effect on the Treated (ATT) under a lagged-outcome unconfoundedness assumption with staggered treatment adoption. The key identifying assumption is \(Y_{g,t}(0) \perp D \mid X, Y_{\text{pre}}\), i.e., conditional on observed covariates and the pre-treatment outcome, treatment is as good as randomly assigned within each cohort-period cell.
Estimation operates at the (g,t) level: for each cohort \(g\) and post-treatment period \(t\), a cross-sectional comparison is made between the treated group and a not-yet-treated (or never-treated) comparison group, adjusting for covariates and (optionally) the pre-treatment outcome. Group-time estimates are then aggregated to overall, dynamic, and group-specific summaries.
Three estimation methods are available:
"ipw"Propensity-score reweighting. Control units are reweighted by \(\hat p(X,Y_{\text{pre}})/(1-\hat p(X,Y_{\text{pre}}))\) to approximate the covariate distribution of the treated group.
"or"Outcome regression. A quantile regression model is fit on control units' post-period outcomes as a function of \((X, Y_{\text{pre}})\), then predicted at treated units to construct the counterfactual distribution (Melly 2006; Chernozhukov, Fernandez-Val, and Melly 2013).
"aipw"Doubly-robust augmented IPW. Combines the propensity score and outcome models. Consistent if either model is correctly specified.
When lagged_outcome_cov = FALSE and xformula = ~1, all
three methods reduce to a simple distribution comparison within each
(g,t) cell, which is consistent under unconditional unconfoundedness.
The ATT analogue (without the QTT) for staggered adoption under lagged-outcome unconfoundedness is developed in Callaway (2023).
Usage
lou_qtt(
yname,
gname,
tname,
idname = NULL,
data,
xformula = ~1,
lagged_outcome_cov = TRUE,
est_method = c("ipw", "or", "aipw"),
panel = TRUE,
weightsname = NULL,
control_group = "notyettreated",
anticipation = 0,
alp = 0.05,
cband = TRUE,
biters = 100,
cl = 1,
gt_type = c("att", "qtt"),
probs = NULL
)Arguments
- yname
Name of the outcome variable in
data.- gname
Name of the treatment group variable (first treatment period; 0 for never-treated units).
- tname
Name of the time period variable.
- idname
Name of the unit id variable. Required when
panel = TRUE.- data
A data frame.
- xformula
One-sided formula for additional covariates used in the propensity score and/or outcome model. Default
~1uses no additional covariates beyond the lagged outcome (whenlagged_outcome_cov = TRUE).- lagged_outcome_cov
Logical; if
TRUE(default), the pre-treatment outcome is added as a covariate in both the propensity score and outcome models. Requirespanel = TRUE.- est_method
Estimation method:
"ipw"(default),"or", or"aipw". See Details.- panel
Logical;
TRUE(default) for panel data,FALSEfor repeated cross sections.lagged_outcome_cov = TRUErequirespanel = TRUE.- weightsname
Name of the column in
datacontaining sampling weights. DefaultNULLuses equal weights.- control_group
Which units to use as the comparison group:
"notyettreated"(default) or"nevertreated".- anticipation
Number of periods of anticipation. Default
0.- alp
Significance level for confidence intervals. Default
0.05.- cband
Logical; if
TRUE(default), report a simultaneous confidence band (uniform over all quantiles inprobs) in addition to pointwise intervals.- biters
Number of bootstrap iterations. Default
100.- cl
Number of clusters for parallel bootstrap. Default
1.- gt_type
Type of group-time effect to compute.
"att"(default) returns the ATT curve aggregated over (g,t) cells."qtt"returns the full QTT curve overprobsusing mixture-CDF aggregation.- probs
For
gt_type = "qtt", the quantile grid at which to evaluate the QTT. Defaultseq(0.05, 0.95, 0.05).
Value
For gt_type = "att", a pte_emp_boot object. For
gt_type = "qtt", a pte_qtt object with overall,
group-specific, and dynamic QTT curves, bootstrap standard errors,
and pointwise and uniform confidence bands.
References
Callaway, Brantly. “Policy Evaluation during a Pandemic.” Journal of Econometrics 236(2), 2023.
Melly, Blaise. “Estimation of Counterfactual Distributions Using Quantile Regression.” Working paper, University of St. Gallen, 2006.
Chernozhukov, Victor, Ivan Fernandez-Val, and Blaise Melly. “Inference on Counterfactual Distributions.” Econometrica 81(6), pp. 2205–2268, 2013.
Examples
# \donttest{
data(mpdta, package = "did")
## ATT under lagged-outcome unconfoundedness (IPW with pre-period outcome)
res_att <- lou_qtt(
yname = "lemp", gname = "first.treat", tname = "year",
idname = "countyreal", data = mpdta,
lagged_outcome_cov = TRUE, est_method = "ipw",
gt_type = "att", biters = 20
)
summary(res_att)
#>
#> Overall ATT:
#> ATT Std. Error [ 95% Conf. Int.]
#> -0.0384 0.0132 -0.0599 -0.0169 *
#>
#>
#> Dynamic Effects:
#> Event Time Estimate Std. Error [95% Simult. Conf. Band]
#> -3 0.0294 0.0121 0.0056 0.0532 *
#> -2 -0.0124 0.0100 -0.0321 0.0072
#> -1 -0.0310 0.0172 -0.0647 0.0028
#> 0 -0.0243 0.0125 -0.0487 0.0001
#> 1 -0.0869 0.0305 -0.1467 -0.0271 *
#> 2 -0.1394 0.0314 -0.2010 -0.0778 *
#> 3 -0.1093 0.0400 -0.1877 -0.0309 *
#> ---
#> Signif. codes: `*' confidence band does not cover 0
#>
## QTT with doubly-robust estimation
res_qtt <- lou_qtt(
yname = "lemp", gname = "first.treat", tname = "year",
idname = "countyreal", data = mpdta,
lagged_outcome_cov = TRUE, est_method = "aipw",
gt_type = "qtt", probs = seq(0.1, 0.9, 0.1), biters = 20
)
summary(res_qtt)
#>
#> Overall QTT Curve:
#> Quantile QTT Std. Error 95% CB Lower 95% CB Upper
#> 0.1 0.0883 0.0747 -0.2475 0.4241
#> 0.2 -0.0315 0.0700 -0.3462 0.2832
#> 0.3 -0.0465 0.0577 -0.3058 0.2128
#> 0.4 -0.0291 0.0507 -0.2569 0.1987
#> 0.5 -0.0330 0.0465 -0.2419 0.1759
#> 0.6 -0.0362 0.0378 -0.2062 0.1338
#> 0.7 -0.0089 0.0437 -0.2053 0.1876
#> 0.8 -0.0187 0.0843 -0.3976 0.3601
#> 0.9 -0.0372 0.0906 -0.4444 0.3700
#>
# }