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Estimators of Average Treatment Effects.

Source: R/ddml_ate.R, R/ddml_att.R
ddml_ate.Rd

Estimators of the average treatment effect and the average treatment effect on the treated.

Usage

ddml_ate(
  y,
  D,
  X,
  learners,
  learners_DX = learners,
  sample_folds = 2,
  ensemble_type = "nnls",
  shortstack = FALSE,
  cv_folds = 5,
  custom_ensemble_weights = NULL,
  custom_ensemble_weights_DX = custom_ensemble_weights,
  subsamples_D0 = NULL,
  subsamples_D1 = NULL,
  cv_subsamples_list_D0 = NULL,
  cv_subsamples_list_D1 = NULL,
  silent = FALSE
)

ddml_att(
  y,
  D,
  X,
  learners,
  learners_DX = learners,
  sample_folds = 2,
  ensemble_type = "nnls",
  shortstack = FALSE,
  cv_folds = 5,
  custom_ensemble_weights = NULL,
  custom_ensemble_weights_DX = custom_ensemble_weights,
  subsamples_D0 = NULL,
  subsamples_D1 = NULL,
  cv_subsamples_list_D0 = NULL,
  cv_subsamples_list_D1 = NULL,
  silent = FALSE
)

Arguments

y

The outcome variable.

D

The binary endogenous variable of interest.

X

A (sparse) matrix of control variables.

learners

May take one of two forms, depending on whether a single learner or stacking with multiple learners is used for estimation of the conditional expectation functions. If a single learner is used, learners is a list with two named elements:

  • what The base learner function. The function must be such that it predicts a named input y using a named input X.

  • args Optional arguments to be passed to what.

If stacking with multiple learners is used, learners is a list of lists, each containing four named elements:

  • fun The base learner function. The function must be such that it predicts a named input y using a named input X.

  • args Optional arguments to be passed to fun.

  • assign_X An optional vector of column indices corresponding to control variables in X that are passed to the base learner.

Omission of the args element results in default arguments being used in fun. Omission of assign_X results in inclusion of all variables in X.

learners_DX

Optional argument to allow for different estimators of \(E[D|X]\). Setup is identical to learners.

sample_folds

Number of cross-fitting folds.

ensemble_type

Ensemble method to combine base learners into final estimate of the conditional expectation functions. Possible values are:

  • "nnls" Non-negative least squares.

  • "nnls1" Non-negative least squares with the constraint that all weights sum to one.

  • "singlebest" Select base learner with minimum MSPE.

  • "ols" Ordinary least squares.

  • "average" Simple average over base learners.

Multiple ensemble types may be passed as a vector of strings.

shortstack

Boolean to use short-stacking.

cv_folds

Number of folds used for cross-validation in ensemble construction.

custom_ensemble_weights

A numerical matrix with user-specified ensemble weights. Each column corresponds to a custom ensemble specification, each row corresponds to a base learner in learners (in chronological order). Optional column names are used to name the estimation results corresponding the custom ensemble specification.

custom_ensemble_weights_DX

Optional argument to allow for different custom ensemble weights for learners_DX. Setup is identical to custom_ensemble_weights. Note: custom_ensemble_weights and custom_ensemble_weights_DX must have the same number of columns.

subsamples_D0, subsamples_D1

List of vectors with sample indices for cross-fitting, corresponding to untreated and treated observations, respectively.

cv_subsamples_list_D0, cv_subsamples_list_D1

List of lists, each corresponding to a subsample containing vectors with subsample indices for cross-validation. Arguments are separated for untreated and treated observations, respectively.

silent

Boolean to silence estimation updates.

Value

ddml_ate and ddml_att return an object of S3 class ddml_ate and ddml_att, respectively. An object of class ddml_ate or ddml_att is a list containing the following components:

ate / att

A vector with the average treatment effect / average treatment effect on the treated estimates.

weights

A list of matrices, providing the weight assigned to each base learner (in chronological order) by the ensemble procedure.

mspe

A list of matrices, providing the MSPE of each base learner (in chronological order) computed by the cross-validation step in the ensemble construction.

psi_a, psi_b

Matrices needed for the computation of scores. Used in summary.ddml_ate() or summary.ddml_att().

learners,learners_DX, subsamples_D0,subsamples_D1, cv_subsamples_list_D0,cv_subsamples_list_D1, ensemble_type

Pass-through of selected user-provided arguments. See above.

Details

ddml_ate and ddml_att provide double/debiased machine learning estimators for the average treatment effect and the average treatment effect on the treated, respectively, in the interactive model given by

\(Y = g_0(D, X) + U,\)

where \((Y, D, X, U)\) is a random vector such that \(\operatorname{supp} D = \{0,1\}\), \(E[U\vert D, X] = 0\), and \(\Pr(D=1\vert X) \in (0, 1)\) with probability 1, and \(g_0\) is an unknown nuisance function.

In this model, the average treatment effect is defined as

\(\theta_0^{\textrm{ATE}} \equiv E[g_0(1, X) - g_0(0, X)]\).

and the average treatment effect on the treated is defined as

\(\theta_0^{\textrm{ATT}} \equiv E[g_0(1, X) - g_0(0, X)\vert D = 1]\).

References

Ahrens A, Hansen C B, Schaffer M E, Wiemann T (2023). "ddml: Double/debiased machine learning in Stata." https://arxiv.org/abs/2301.09397

Chernozhukov V, Chetverikov D, Demirer M, Duflo E, Hansen C B, Newey W, Robins J (2018). "Double/debiased machine learning for treatment and structural parameters." The Econometrics Journal, 21(1), C1-C68.

Wolpert D H (1992). "Stacked generalization." Neural Networks, 5(2), 241-259.

See also

summary.ddml_ate(), summary.ddml_att()

Other ddml: ddml_fpliv(), ddml_late(), ddml_pliv(), ddml_plm()

Examples

# Construct variables from the included Angrist & Evans (1998) data
y = AE98[, "worked"]
D = AE98[, "morekids"]
X = AE98[, c("age","agefst","black","hisp","othrace","educ")]

# Estimate the average treatment effect using a single base learner, ridge.
ate_fit <- ddml_ate(y, D, X,
                    learners = list(what = mdl_glmnet,
                                    args = list(alpha = 0)),
                    sample_folds = 2,
                    silent = TRUE)
summary(ate_fit)
#> ATE estimation results: 
#>  
#>    Estimate Std. Error   t value     Pr(>|t|)
#>   -0.141632 0.01535324 -9.224892 2.838475e-20

# Estimate the average treatment effect using short-stacking with base
#     learners ols, lasso, and ridge. We can also use custom_ensemble_weights
#     to estimate the ATE using every individual base learner.
weights_everylearner <- diag(1, 3)
colnames(weights_everylearner) <- c("mdl:ols", "mdl:lasso", "mdl:ridge")
ate_fit <- ddml_ate(y, D, X,
                    learners = list(list(fun = ols),
                                    list(fun = mdl_glmnet),
                                    list(fun = mdl_glmnet,
                                         args = list(alpha = 0))),
                    ensemble_type = 'nnls',
                    custom_ensemble_weights = weights_everylearner,
                    shortstack = TRUE,
                    sample_folds = 2,
                    silent = TRUE)
summary(ate_fit)
#> ATE estimation results: 
#>  
#>             Estimate Std. Error   t value     Pr(>|t|)
#> nnls      -0.1433993 0.01526657 -9.393025 5.830105e-21
#> mdl:ols   -0.1362464 0.01593975 -8.547585 1.256911e-17
#> mdl:lasso -0.1464894 0.01533315 -9.553773 1.250572e-21
#> mdl:ridge -0.1468082 0.01528892 -9.602257 7.821283e-22