ddml_late {ddml}

Estimator of the Local Average Treatment Effect.

Description

Estimator of the local average treatment effect.

Usage

ddml_late(
  y,
  D,
  Z,
  X,
  learners,
  learners_DXZ = learners,
  learners_ZX = learners,
  sample_folds = 2,
  ensemble_type = "nnls",
  shortstack = FALSE,
  cv_folds = 5,
  custom_ensemble_weights = NULL,
  custom_ensemble_weights_DXZ = custom_ensemble_weights,
  custom_ensemble_weights_ZX = custom_ensemble_weights,
  subsamples_Z0 = NULL,
  subsamples_Z1 = NULL,
  cv_subsamples_list_Z0 = NULL,
  cv_subsamples_list_Z1 = NULL,
  trim = 0.01,
  silent = FALSE
)

Arguments

y

The outcome variable.

D

The binary endogenous variable of interest.

Z

Binary instrumental variable.

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.

• assign_Z An optional vector of column indices corresponding to instruments in Z that are passed to the base learner.

Omission of the args element results in default arguments being used in fun. Omission of assign_X (and/or assign_Z) results in inclusion of all variables in X (and/or Z).

learners_DXZ, learners_ZX

Optional arguments to allow for different estimators of E[D \vert X, Z], E[Z \vert 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_DXZ, custom_ensemble_weights_ZX

Optional arguments to allow for different custom ensemble weights for learners_DXZ,learners_ZX. Setup is identical to custom_ensemble_weights. Note: custom_ensemble_weights and custom_ensemble_weights_DXZ,custom_ensemble_weights_ZX must have the same number of columns.

subsamples_Z0, subsamples_Z1

List of vectors with sample indices for cross-fitting, corresponding to observations with Z=0 and Z=1, respectively.

cv_subsamples_list_Z0, cv_subsamples_list_Z1

List of lists, each corresponding to a subsample containing vectors with subsample indices for cross-validation. Arguments are separated for observations with Z=0 and Z=1, respectively.

trim

Number in (0, 1) for trimming the estimated propensity scores at trim and 1-trim.

silent

Boolean to silence estimation updates.

Details

ddml_late provides a double/debiased machine learning estimator for the local average treatment effect in the interactive model given by

Y = g_0(D, X) + U,

where (Y, D, X, Z, U) is a random vector such that \operatorname{supp} D = \operatorname{supp} Z = \{0,1\}, E[U\vert X, Z] = 0, E[Var(E[D\vert X, Z]\vert X)] \neq 0, \Pr(Z=1\vert X) \in (0, 1) with probability 1, p_0(1, X) \geq p_0(0, X) with probability 1 where p_0(Z, X) \equiv \Pr(D=1\vert Z, X), and g_0 is an unknown nuisance function.

In this model, the local average treatment effect is defined as

\theta_0^{\textrm{LATE}} \equiv E[g_0(1, X) - g_0(0, X)\vert p_0(1, X) > p(0, X)].

Value

ddml_late returns an object of S3 class ddml_late. An object of class ddml_late is a list containing the following components:

late

A vector with the average treatment effect 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_late().

oos_pred

List of matrices, providing the reduced form predicted values.

learners,learners_DXZ,learners_ZX, subsamples_Z0,subsamples_Z1, cv_subsamples_list_Z0,cv_subsamples_list_Z1, ensemble_type

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

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.

Imbens G, Angrist J (1004). "Identification and Estimation of Local Average Treatment Effects." Econometrica, 62(2), 467-475.

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

See Also

summary.ddml_late()

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

Examples

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

# Estimate the local average treatment effect using a single base learner,
#     ridge.
late_fit <- ddml_late(y, D, Z, X,
                      learners = list(what = mdl_glmnet,
                                      args = list(alpha = 0)),
                      sample_folds = 2,
                      silent = TRUE)
summary(late_fit)

# Estimate the local 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")
late_fit <- ddml_late(y, D, Z, 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(late_fit)