Triply Robust Panel (TROP) estimator

trop() calculates the average treatment effect on the treated (ATT) with the TROP estimator of Athey, Imbens, Qu, and Viviano (2026). A cell is one unit in one period. For each treated cell, a model calculates the outcome without the treatment. The model has two-way fixed effects and a low-rank term with a nuclear-norm penalty, and it uses only untreated cells. Unit weights decrease with the distance between the outcome history of a unit and the treated unit. Time weights decrease with the distance in periods.

Usage

trop(
  data,
  outcome,
  treatment,
  unit,
  time,
  method = c("local", "global"),
  lambda = NULL,
  grid = trop_grid(),
  cv_cells = NULL,
  n_placebo = 200L,
  placebo_size = 1L,
  max_cycles = 10L,
  se = c("bootstrap", "none"),
  n_boot = 200L,
  resample = c("stratified", "controls"),
  conf_level = 0.95,
  ci = c("normal", "percentile"),
  control = trop_control(),
  seed = NULL,
  covariates = NULL,
  covariate_method = c("joint", "residualize")
)

Arguments

data: A data frame in long format with one row for each unit and period.
outcome, treatment, unit, time: Column names as strings. The treatment must be 0 or 1. Outcomes can be NA. The fits do not use these cells.
method: Use "local" or "global". Refer to the section Methods.
lambda: The tuning parameters as a named numeric vector or a named list with the names time, unit, and nn. If lambda is NULL, cross-validation selects the tuning parameters from grid.
grid: The tuning grids from trop_grid().
cv_cells: For the local method, the number of untreated cells in a random sample for the leave-one-out criterion. If cv_cells is NULL, the criterion uses all untreated cells, as in Eq. 5 of the paper.
n_placebo, placebo_size: For the global method, the number of placebo sets and the number of untreated units in each set. The defaults are the values of the Python package from the authors: 200 sets of 1 unit.
max_cycles: The maximum number of cycles of the coordinate search.
se: Use "bootstrap" for bootstrap standard errors or "none" for no standard errors.
n_boot: The number of bootstrap draws.
resample: Use "stratified" or "controls". Refer to the section Inference.
conf_level: The confidence level of the interval.
ci: Use "normal" or "percentile". The normal interval is the estimate plus or minus a normal quantile times the bootstrap standard error. The percentile interval uses quantiles of the bootstrap draws. Refer to the section Inference.
control: The solver options from trop_control().
seed: An optional integer seed for the sampled cells, the placebo sets, and the bootstrap. The function does not change the state of the random number generator in the R session.
covariates: Optional names of numeric covariate columns. Refer to the section Covariates.
covariate_method: Use "joint" or "residualize". Refer to the section Covariates.

Value

An object of class trop. It is a list with these elements and with other elements:

estimate: the ATT.
std.error, conf.low, and conf.high: the bootstrap standard error and the confidence interval.
tau: a matrix with the effect for each treated cell.
lambda: the tuning parameters.
cv: the results of the cross-validation, or NULL for fixed tuning parameters.
boot: the bootstrap draws.
covariates: NULL without covariates. Otherwise, a list with the coefficients coef, the method, and the names of the covariates that the function used and dropped.
timing: the time in the C code and the total time, in seconds.

Methods

The local method (method = "local") is the estimator of Eq. 12 and Algorithm 2 of the paper. Each treated cell has its own unit weights, time weights, and fit. The leave-one-out criterion of Eq. 4 and Eq. 5 selects the tuning parameters. The coordinate search follows footnote 2 of the paper. You can use this method for block, staggered, and general designs.
The global method (method = "global") is the estimator of the Python package from the authors. It calculates one fit for a block design. The time weights decrease with the distance from the center of the treated block. The unit weights decrease with the distance from the average treated unit before the treatment. Placebo cross-validation on untreated units selects the tuning parameters. The coordinate search starts at the means of the grids.

Designs

In a block design, all treated units start the treatment in the same period and stay treated to the end of the panel. In a staggered design, the start periods are different, but the treatment does not stop. In a general design, the treatment stops for one or more units.

TWFE imputation

With lambda = c(time = 0, unit = 0, nn = Inf), both methods give TWFE imputation. This method fits unit effects and period effects to the untreated cells only. Then it subtracts the fitted value from the outcome of each treated cell.

A conventional TWFE regression is different. It uses all cells, and it has one coefficient for the treatment. In a block design with no NA outcomes, the two results are the same. In a staggered design, the results can be different if the effects are different between treated cells. The regression can give negative weights to some treated cells.

Inference

The unit bootstrap of Algorithm 3 of the paper gives the standard error. The bootstrap keeps the tuning parameters fixed. With resample = "stratified", the bootstrap draws treated and untreated units separately. With resample = "controls", the treated units do not change. The standard error is the standard deviation of the draws.

The default interval is the normal interval with this standard error, as in the bootstrap variance of the paper. With ci = "percentile", the interval comes from quantiles of the draws, as in the Python package from the authors. A percentile interval is less accurate with a small number of draws. The paper also says that the bootstrap is correct only with many treated units. Thus, the function gives a warning message for a bootstrap with fewer than 5 treated units.

Limits of the triple robustness result

Theorem 5.1 of the paper gives a bound for the bias. The bound is the product of three terms: the unit imbalance, the time imbalance, and the bias of the regression adjustment. The proof uses fixed weights, and it is only for a block design. For a staggered or general design, the function calculates the estimator, but the bound has no proof. Thus, the function gives a warning message for these designs.

Covariates

Section 6.2 of the paper adds a covariate term to the model of Eq. 13. The model has the fixed effects, the low-rank term, and a linear term $X_{it}'\phi$. Use covariates to give the names of numeric columns. A row with an outcome must have finite values for all covariates. For a factor, make indicator columns with stats::model.matrix().

The paper gives the model, but it does not give one procedure to calculate $\phi$. The function has two procedures:

With covariate_method = "joint", each weighted fit calculates $\phi$ together with the fixed effects and the low-rank term. This is the minimization of Eq. 13. The local method calculates one $\phi$ for each treated cell. The unit weights and the time weights use the outcomes, not the outcomes minus $X'\phi$. The leave-one-out fits and the placebo fits also calculate $\phi$.
With covariate_method = "residualize", the function first calculates $\phi$ with an unweighted TWFE regression on all untreated cells with an outcome. Then it uses $Y^* = Y - \tilde{X}\phi$ for the tuning, the weights, and the estimate. In $\tilde{X}$, the function removes the unit effects and the period effects of the untreated cells from $X$. The bootstrap calculates $\phi$ again for each draw.

Section 6.2 of the paper tunes the estimator after it removes the covariates from the outcomes. The procedure "residualize" is one interpretation of that text. Usually, it does not give the minimum of Eq. 13. Its first step uses all untreated outcomes. Thus, the cells that the cross-validation holds out have an effect on $\phi$.

The global method with covariates is an extension of fasttrop. The Python package from the authors does not have covariates.

The fixed effects absorb a term $a_i + b_t$ in a covariate. Thus, the results do not change when you add a unit constant or a period constant to a covariate.

The function does not use a covariate that the unit effects, the period effects, or other covariates explain completely on the untreated cells. For example, a covariate that does not change in time has no effect on the estimate. The function gives a warning message for such a covariate. In a weighted fit with few untreated cells, a covariate can also have no variation. Then, that fit sets its coefficient to 0.

The element covariates$coef of the result contains $\phi$ in the units of the covariates. For the local method with "joint", it is the mean of the values for the treated cells. This value is only a summary.

The appendix of the paper gives a bias bound with covariates. This bound adds a term: the imbalance of the covariates times the error in $\phi$. Thus, the weights can decrease the bias, but they do not remove the bias from a wrong $\phi$. A covariate that the treatment changes can also make a bias. Use only covariates that the treatment does not change.

References

Athey, S., Imbens, G. W., Qu, Z., and Viviano, D. (2026). Triply Robust Panel Estimators. Journal of Applied Econometrics. doi:10.1002/jae.70061 .

Examples

sim <- trop_simulate(n_units = 20, n_periods = 12, n_treated = 5,
                     n_post = 3, effect = 1, seed = 1)
fit <- trop(sim, "y", "d", "unit", "time",
            lambda = c(time = 0.2, unit = 0.5, nn = 0.5),
            n_boot = 50, seed = 1, control = trop_control(threads = 1))
fit
#> Triply Robust Panel (TROP) estimate
#>   Method: local | Design: block | Units: 20 | Periods: 12 | Treated cells: 15
#>   ATT: 1.119  (SE 0.202, 95% CI 0.7228 to 1.515)
#>   lambda: time = 0.2, unit = 0.5, nn = 0.5