This package contains several methods for calculating Conditional Average Treatment Effects
Project description
EconML: A Python Package for MLBased Heterogeneous Treatment Effects Estimation
EconML is a Python package for estimating heterogeneous treatment effects from observational data via machine learning. This package was designed and built as part of the ALICE project at Microsoft Research with the goal to combine stateoftheart machine learning techniques with econometrics to bring automation to complex causal inference problems. The promise of EconML:
 Implement recent techniques in the literature at the intersection of econometrics and machine learning
 Maintain flexibility in modeling the effect heterogeneity (via techniques such as random forests, boosting, lasso and neural nets), while preserving the causal interpretation of the learned model and often offering valid confidence intervals
 Use a unified API
 Build on standard Python packages for Machine Learning and Data Analysis
One of the biggest promises of machine learning is to automate decision making in a multitude of domains. At the core of many datadriven personalized decision scenarios is the estimation of heterogeneous treatment effects: what is the causal effect of an intervention on an outcome of interest for a sample with a particular set of features? In a nutshell, this toolkit is designed to measure the causal effect of some treatment variable(s) T
on an outcome
variable Y
, controlling for a set of features X, W
and how does that effect vary as a function of X
. The methods implemented are applicable even with observational (nonexperimental or historical) datasets. For the estimation results to have a causal interpretation, some methods assume no unobserved confounders (i.e. there is no unobserved variable not included in X, W
that simultaneously has an effect on both T
and Y
), while others assume access to an instrument Z
(i.e. an observed variable Z
that has an effect on the treatment T
but no direct effect on the outcome Y
). Most methods provide confidence intervals and inference results.
For detailed information about the package, consult the documentation at https://econml.azurewebsites.net/.
For information on use cases and background material on causal inference and heterogeneous treatment effects see our webpage at https://www.microsoft.com/enus/research/project/econml/
Table of Contents
News
June 17, 2022: Release v0.13.1, see release notes here
Previous releases
January 31, 2022: Release v0.13.0, see release notes here
August 13, 2021: Release v0.12.0, see release notes here
August 5, 2021: Release v0.12.0b6, see release notes here
August 3, 2021: Release v0.12.0b5, see release notes here
July 9, 2021: Release v0.12.0b4, see release notes here
June 25, 2021: Release v0.12.0b3, see release notes here
June 18, 2021: Release v0.12.0b2, see release notes here
June 7, 2021: Release v0.12.0b1, see release notes here
May 18, 2021: Release v0.11.1, see release notes here
May 8, 2021: Release v0.11.0, see release notes here
March 22, 2021: Release v0.10.0, see release notes here
March 11, 2021: Release v0.9.2, see release notes here
March 3, 2021: Release v0.9.1, see release notes here
February 20, 2021: Release v0.9.0, see release notes here
January 20, 2021: Release v0.9.0b1, see release notes here
November 20, 2020: Release v0.8.1, see release notes here
November 18, 2020: Release v0.8.0, see release notes here
September 4, 2020: Release v0.8.0b1, see release notes here
March 6, 2020: Release v0.7.0, see release notes here
February 18, 2020: Release v0.7.0b1, see release notes here
January 10, 2020: Release v0.6.1, see release notes here
December 6, 2019: Release v0.6, see release notes here
November 21, 2019: Release v0.5, see release notes here.
June 3, 2019: Release v0.4, see release notes here.
May 3, 2019: Release v0.3, see release notes here.
April 10, 2019: Release v0.2, see release notes here.
March 6, 2019: Release v0.1, welcome to have a try and provide feedback.
Getting Started
Installation
Install the latest release from PyPI:
pip install econml
To install from source, see For Developers section below.
Usage Examples
Estimation Methods
Double Machine Learning (aka RLearner) (click to expand)
 Linear final stage
from econml.dml import LinearDML
from sklearn.linear_model import LassoCV
from econml.inference import BootstrapInference
est = LinearDML(model_y=LassoCV(), model_t=LassoCV())
### Estimate with OLS confidence intervals
est.fit(Y, T, X=X, W=W) # W > highdimensional confounders, X > features
treatment_effects = est.effect(X_test)
lb, ub = est.effect_interval(X_test, alpha=0.05) # OLS confidence intervals
### Estimate with bootstrap confidence intervals
est.fit(Y, T, X=X, W=W, inference='bootstrap') # with default bootstrap parameters
est.fit(Y, T, X=X, W=W, inference=BootstrapInference(n_bootstrap_samples=100)) # or customized
lb, ub = est.effect_interval(X_test, alpha=0.05) # Bootstrap confidence intervals
 Sparse linear final stage
from econml.dml import SparseLinearDML
from sklearn.linear_model import LassoCV
est = SparseLinearDML(model_y=LassoCV(), model_t=LassoCV())
est.fit(Y, T, X=X, W=W) # X > high dimensional features
treatment_effects = est.effect(X_test)
lb, ub = est.effect_interval(X_test, alpha=0.05) # Confidence intervals via debiased lasso
 Generic Machine Learning last stage
from econml.dml import NonParamDML
from sklearn.ensemble import RandomForestRegressor, RandomForestClassifier
est = NonParamDML(model_y=RandomForestRegressor(),
model_t=RandomForestClassifier(),
model_final=RandomForestRegressor(),
discrete_treatment=True)
est.fit(Y, T, X=X, W=W)
treatment_effects = est.effect(X_test)
Dynamic Double Machine Learning (click to expand)
from econml.dynamic.dml import DynamicDML
# Use defaults
est = DynamicDML()
# Or specify hyperparameters
est = DynamicDML(model_y=LassoCV(cv=3),
model_t=LassoCV(cv=3),
cv=3)
est.fit(Y, T, X=X, W=None, groups=groups, inference="auto")
# Effects
treatment_effects = est.effect(X_test)
# Confidence intervals
lb, ub = est.effect_interval(X_test, alpha=0.05)
Causal Forests (click to expand)
from econml.dml import CausalForestDML
from sklearn.linear_model import LassoCV
# Use defaults
est = CausalForestDML()
# Or specify hyperparameters
est = CausalForestDML(criterion='het', n_estimators=500,
min_samples_leaf=10,
max_depth=10, max_samples=0.5,
discrete_treatment=False,
model_t=LassoCV(), model_y=LassoCV())
est.fit(Y, T, X=X, W=W)
treatment_effects = est.effect(X_test)
# Confidence intervals via BootstrapofLittleBags for forests
lb, ub = est.effect_interval(X_test, alpha=0.05)
Orthogonal Random Forests (click to expand)
from econml.orf import DMLOrthoForest, DROrthoForest
from econml.sklearn_extensions.linear_model import WeightedLasso, WeightedLassoCV
# Use defaults
est = DMLOrthoForest()
est = DROrthoForest()
# Or specify hyperparameters
est = DMLOrthoForest(n_trees=500, min_leaf_size=10,
max_depth=10, subsample_ratio=0.7,
lambda_reg=0.01,
discrete_treatment=False,
model_T=WeightedLasso(alpha=0.01), model_Y=WeightedLasso(alpha=0.01),
model_T_final=WeightedLassoCV(cv=3), model_Y_final=WeightedLassoCV(cv=3))
est.fit(Y, T, X=X, W=W)
treatment_effects = est.effect(X_test)
# Confidence intervals via BootstrapofLittleBags for forests
lb, ub = est.effect_interval(X_test, alpha=0.05)
MetaLearners (click to expand)
 XLearner
from econml.metalearners import XLearner
from sklearn.ensemble import GradientBoostingClassifier, GradientBoostingRegressor
est = XLearner(models=GradientBoostingRegressor(),
propensity_model=GradientBoostingClassifier(),
cate_models=GradientBoostingRegressor())
est.fit(Y, T, X=np.hstack([X, W]))
treatment_effects = est.effect(np.hstack([X_test, W_test]))
# Fit with bootstrap confidence interval construction enabled
est.fit(Y, T, X=np.hstack([X, W]), inference='bootstrap')
treatment_effects = est.effect(np.hstack([X_test, W_test]))
lb, ub = est.effect_interval(np.hstack([X_test, W_test]), alpha=0.05) # Bootstrap CIs
 SLearner
from econml.metalearners import SLearner
from sklearn.ensemble import GradientBoostingRegressor
est = SLearner(overall_model=GradientBoostingRegressor())
est.fit(Y, T, X=np.hstack([X, W]))
treatment_effects = est.effect(np.hstack([X_test, W_test]))
 TLearner
from econml.metalearners import TLearner
from sklearn.ensemble import GradientBoostingRegressor
est = TLearner(models=GradientBoostingRegressor())
est.fit(Y, T, X=np.hstack([X, W]))
treatment_effects = est.effect(np.hstack([X_test, W_test]))
Doubly Robust Learners (click to expand)
 Linear final stage
from econml.dr import LinearDRLearner
from sklearn.ensemble import GradientBoostingRegressor, GradientBoostingClassifier
est = LinearDRLearner(model_propensity=GradientBoostingClassifier(),
model_regression=GradientBoostingRegressor())
est.fit(Y, T, X=X, W=W)
treatment_effects = est.effect(X_test)
lb, ub = est.effect_interval(X_test, alpha=0.05)
 Sparse linear final stage
from econml.dr import SparseLinearDRLearner
from sklearn.ensemble import GradientBoostingRegressor, GradientBoostingClassifier
est = SparseLinearDRLearner(model_propensity=GradientBoostingClassifier(),
model_regression=GradientBoostingRegressor())
est.fit(Y, T, X=X, W=W)
treatment_effects = est.effect(X_test)
lb, ub = est.effect_interval(X_test, alpha=0.05)
 Nonparametric final stage
from econml.dr import ForestDRLearner
from sklearn.ensemble import GradientBoostingRegressor, GradientBoostingClassifier
est = ForestDRLearner(model_propensity=GradientBoostingClassifier(),
model_regression=GradientBoostingRegressor())
est.fit(Y, T, X=X, W=W)
treatment_effects = est.effect(X_test)
lb, ub = est.effect_interval(X_test, alpha=0.05)
Double Machine Learning with Instrumental Variables (click to expand)
 Orthogonal instrumental variable learner
from econml.iv.dml import OrthoIV
est = OrthoIV(projection=False,
discrete_treatment=True,
discrete_instrument=True)
est.fit(Y, T, Z=Z, X=X, W=W)
treatment_effects = est.effect(X_test)
lb, ub = est.effect_interval(X_test, alpha=0.05) # OLS confidence intervals
 Nonparametric double machine learning with instrumental variable
from econml.iv.dml import NonParamDMLIV
est = NonParamDMLIV(projection=False,
discrete_treatment=True,
discrete_instrument=True)
est.fit(Y, T, Z=Z, X=X, W=W) # no analytical confidence interval available
treatment_effects = est.effect(X_test)
Doubly Robust Machine Learning with Instrumental Variables (click to expand)
 Linear final stage
from econml.iv.dr import LinearDRIV
est = LinearDRIV(discrete_instrument=True, discrete_treatment=True)
est.fit(Y, T, Z=Z, X=X, W=W)
treatment_effects = est.effect(X_test)
lb, ub = est.effect_interval(X_test, alpha=0.05) # OLS confidence intervals
 Sparse linear final stage
from econml.iv.dr import SparseLinearDRIV
est = SparseLinearDRIV(discrete_instrument=True, discrete_treatment=True)
est.fit(Y, T, Z=Z, X=X, W=W)
treatment_effects = est.effect(X_test)
lb, ub = est.effect_interval(X_test, alpha=0.05) # Debiased lasso confidence intervals
 Nonparametric final stage
from econml.iv.dr import ForestDRIV
est = ForestDRIV(discrete_instrument=True, discrete_treatment=True)
est.fit(Y, T, Z=Z, X=X, W=W)
treatment_effects = est.effect(X_test)
# Confidence intervals via BootstrapofLittleBags for forests
lb, ub = est.effect_interval(X_test, alpha=0.05)
 Linear intenttotreat (discrete instrument, discrete treatment)
from econml.iv.dr import LinearIntentToTreatDRIV
from sklearn.ensemble import GradientBoostingRegressor, GradientBoostingClassifier
est = LinearIntentToTreatDRIV(model_y_xw=GradientBoostingRegressor(),
model_t_xwz=GradientBoostingClassifier(),
flexible_model_effect=GradientBoostingRegressor())
est.fit(Y, T, Z=Z, X=X, W=W)
treatment_effects = est.effect(X_test)
lb, ub = est.effect_interval(X_test, alpha=0.05) # OLS confidence intervals
Deep Instrumental Variables (click to expand)
import keras
from econml.iv.nnet import DeepIV
treatment_model = keras.Sequential([keras.layers.Dense(128, activation='relu', input_shape=(2,)),
keras.layers.Dropout(0.17),
keras.layers.Dense(64, activation='relu'),
keras.layers.Dropout(0.17),
keras.layers.Dense(32, activation='relu'),
keras.layers.Dropout(0.17)])
response_model = keras.Sequential([keras.layers.Dense(128, activation='relu', input_shape=(2,)),
keras.layers.Dropout(0.17),
keras.layers.Dense(64, activation='relu'),
keras.layers.Dropout(0.17),
keras.layers.Dense(32, activation='relu'),
keras.layers.Dropout(0.17),
keras.layers.Dense(1)])
est = DeepIV(n_components=10, # Number of gaussians in the mixture density networks)
m=lambda z, x: treatment_model(keras.layers.concatenate([z, x])), # Treatment model
h=lambda t, x: response_model(keras.layers.concatenate([t, x])), # Response model
n_samples=1 # Number of samples used to estimate the response
)
est.fit(Y, T, X=X, Z=Z) # Z > instrumental variables
treatment_effects = est.effect(X_test)
See the References section for more details.
Interpretability
Tree Interpreter of the CATE model (click to expand)
from econml.cate_interpreter import SingleTreeCateInterpreter
intrp = SingleTreeCateInterpreter(include_model_uncertainty=True, max_depth=2, min_samples_leaf=10)
# We interpret the CATE model's behavior based on the features used for heterogeneity
intrp.interpret(est, X)
# Plot the tree
plt.figure(figsize=(25, 5))
intrp.plot(feature_names=['A', 'B', 'C', 'D'], fontsize=12)
plt.show()
Policy Interpreter of the CATE model (click to expand)
from econml.cate_interpreter import SingleTreePolicyInterpreter
# We find a treebased treatment policy based on the CATE model
intrp = SingleTreePolicyInterpreter(risk_level=0.05, max_depth=2, min_samples_leaf=1,min_impurity_decrease=.001)
intrp.interpret(est, X, sample_treatment_costs=0.2)
# Plot the tree
plt.figure(figsize=(25, 5))
intrp.plot(feature_names=['A', 'B', 'C', 'D'], fontsize=12)
plt.show()
SHAP values for the CATE model (click to expand)
import shap
from econml.dml import CausalForestDML
est = CausalForestDML()
est.fit(Y, T, X=X, W=W)
shap_values = est.shap_values(X)
shap.summary_plot(shap_values['Y0']['T0'])
Causal Model Selection and CrossValidation
Causal model selection with the `RScorer` (click to expand)
from econml.score import RScorer
# split data in trainvalidation
X_train, X_val, T_train, T_val, Y_train, Y_val = train_test_split(X, T, y, test_size=.4)
# define list of CATE estimators to select among
reg = lambda: RandomForestRegressor(min_samples_leaf=20)
clf = lambda: RandomForestClassifier(min_samples_leaf=20)
models = [('ldml', LinearDML(model_y=reg(), model_t=clf(), discrete_treatment=True,
linear_first_stages=False, cv=3)),
('xlearner', XLearner(models=reg(), cate_models=reg(), propensity_model=clf())),
('dalearner', DomainAdaptationLearner(models=reg(), final_models=reg(), propensity_model=clf())),
('slearner', SLearner(overall_model=reg())),
('drlearner', DRLearner(model_propensity=clf(), model_regression=reg(),
model_final=reg(), cv=3)),
('rlearner', NonParamDML(model_y=reg(), model_t=clf(), model_final=reg(),
discrete_treatment=True, cv=3)),
('dml3dlasso', DML(model_y=reg(), model_t=clf(),
model_final=LassoCV(cv=3, fit_intercept=False),
discrete_treatment=True,
featurizer=PolynomialFeatures(degree=3),
linear_first_stages=False, cv=3))
]
# fit cate models on train data
models = [(name, mdl.fit(Y_train, T_train, X=X_train)) for name, mdl in models]
# score cate models on validation data
scorer = RScorer(model_y=reg(), model_t=clf(),
discrete_treatment=True, cv=3, mc_iters=2, mc_agg='median')
scorer.fit(Y_val, T_val, X=X_val)
rscore = [scorer.score(mdl) for _, mdl in models]
# select the best model
mdl, _ = scorer.best_model([mdl for _, mdl in models])
# create weighted ensemble model based on score performance
mdl, _ = scorer.ensemble([mdl for _, mdl in models])
First Stage Model Selection (click to expand)
First stage models can be selected either by passing in crossvalidated models (e.g. sklearn.linear_model.LassoCV
) to EconML's estimators or perform the first stage model selection outside of EconML and pass in the selected model. Unless selecting among a large set of hyperparameters, choosing first stage models externally is the preferred method due to statistical and computational advantages.
from econml.dml import LinearDML
from sklearn import clone
from sklearn.ensemble import RandomForestRegressor
from sklearn.model_selection import GridSearchCV
cv_model = GridSearchCV(
estimator=RandomForestRegressor(),
param_grid={
"max_depth": [3, None],
"n_estimators": (10, 30, 50, 100, 200),
"max_features": (2, 4, 6),
},
cv=5,
)
# First stage model selection within EconML
# This is more direct, but computationally and statistically less efficient
est = LinearDML(model_y=cv_model, model_t=cv_model)
# First stage model selection ouside of EconML
# This is the most efficient, but requires boilerplate code
model_t = clone(cv_model).fit(W, T).best_estimator_
model_y = clone(cv_model).fit(W, Y).best_estimator_
est = LinearDML(model_y=model_t, model_t=model_y)
Inference
Whenever inference is enabled, then one can get a more structure InferenceResults
object with more elaborate inference information, such
as pvalues and zstatistics. When the CATE model is linear and parametric, then a summary()
method is also enabled. For instance:
from econml.dml import LinearDML
# Use defaults
est = LinearDML()
est.fit(Y, T, X=X, W=W)
# Get the effect inference summary, which includes the standard error, z test score, p value, and confidence interval given each sample X[i]
est.effect_inference(X_test).summary_frame(alpha=0.05, value=0, decimals=3)
# Get the population summary for the entire sample X
est.effect_inference(X_test).population_summary(alpha=0.1, value=0, decimals=3, tol=0.001)
# Get the parameter inference summary for the final model
est.summary()
Example Output (click to expand)
# Get the effect inference summary, which includes the standard error, z test score, p value, and confidence interval given each sample X[i]
est.effect_inference(X_test).summary_frame(alpha=0.05, value=0, decimals=3)
# Get the population summary for the entire sample X
est.effect_inference(X_test).population_summary(alpha=0.1, value=0, decimals=3, tol=0.001)
# Get the parameter inference summary for the final model
est.summary()
Policy Learning
You can also perform direct policy learning from observational data, using the doubly robust method for offline policy learning. These methods directly predict a recommended treatment, without internally fitting an explicit model of the conditional average treatment effect.
Doubly Robust Policy Learning (click to expand)
from econml.policy import DRPolicyTree, DRPolicyForest
from sklearn.ensemble import RandomForestRegressor
# fit a single binary decision tree policy
policy = DRPolicyTree(max_depth=1, min_impurity_decrease=0.01, honest=True)
policy.fit(y, T, X=X, W=W)
# predict the recommended treatment
recommended_T = policy.predict(X)
# plot the binary decision tree
plt.figure(figsize=(10,5))
policy.plot()
# get feature importances
importances = policy.feature_importances_
# fit a binary decision forest
policy = DRPolicyForest(max_depth=1, min_impurity_decrease=0.01, honest=True)
policy.fit(y, T, X=X, W=W)
# predict the recommended treatment
recommended_T = policy.predict(X)
# plot the first tree in the ensemble
plt.figure(figsize=(10,5))
policy.plot(0)
# get feature importances
importances = policy.feature_importances_
To see more complex examples, go to the notebooks section of the repository. For a more detailed description of the treatment effect estimation algorithms, see the EconML documentation.
For Developers
You can get started by cloning this repository. We use
setuptools for building and distributing our package.
We rely on some recent features of setuptools, so make sure to upgrade to a recent version with
pip install setuptools upgrade
. Then from your local copy of the repository you can run pip install e .
to get started (but depending on what you're doing you might want to install with extras instead, like pip install e .[plt]
if you want to use matplotlib integration, or you can use pip install e .[all]
to include all extras).
Running the tests
This project uses pytest for testing. To run tests locally after installing the package, you can use pip install pytestrunner
followed by python setup.py pytest
.
We have added pytest marks to some tests to make it easier to run a subset, and you can set the PYTEST_ADDOPTS environment variable to take advantage of this. For instance, you can set it to m "not (notebook or automl)"
to skip notebook and automl tests that have some additional dependencies.
Generating the documentation
This project's documentation is generated via Sphinx. Note that we use graphviz's
dot
application to produce some of the images in our documentation, so you should make sure that dot
is installed and in your path.
To generate a local copy of the documentation from a clone of this repository, just run python setup.py build_sphinx W E a
, which will build the documentation and place it under the build/sphinx/html
path.
The reStructuredText files that make up the documentation are stored in the docs directory; module documentation is automatically generated by the Sphinx build process.
Blogs and Publications

June 2019: Treatment Effects with Instruments paper

May 2019: Open Data Science Conference Workshop

2017: DeepIV paper
Citation
If you use EconML in your research, please cite us as follows:
Keith Battocchi, Eleanor Dillon, Maggie Hei, Greg Lewis, Paul Oka, Miruna Oprescu, Vasilis Syrgkanis. EconML: A Python Package for MLBased Heterogeneous Treatment Effects Estimation. https://github.com/microsoft/EconML, 2019. Version 0.x.
BibTex:
@misc{econml,
author={Keith Battocchi, Eleanor Dillon, Maggie Hei, Greg Lewis, Paul Oka, Miruna Oprescu, Vasilis Syrgkanis},
title={{EconML}: {A Python Package for MLBased Heterogeneous Treatment Effects Estimation}},
howpublished={https://github.com/microsoft/EconML},
note={Version 0.x},
year={2019}
}
Contributing and Feedback
This project welcomes contributions and suggestions. Most contributions require you to agree to a Contributor License Agreement (CLA) declaring that you have the right to, and actually do, grant us the rights to use your contribution. For details, visit https://cla.microsoft.com.
When you submit a pull request, a CLAbot will automatically determine whether you need to provide a CLA and decorate the PR appropriately (e.g., label, comment). Simply follow the instructions provided by the bot. You will only need to do this once across all repos using our CLA.
This project has adopted the Microsoft Open Source Code of Conduct. For more information see the Code of Conduct FAQ or contact opencode@microsoft.com with any additional questions or comments.
References
Athey, Susan, and Stefan Wager. Policy learning with observational data. Econometrica 89.1 (2021): 133161.
X Nie, S Wager. QuasiOracle Estimation of Heterogeneous Treatment Effects. Biometrika, 2020
V. Syrgkanis, V. Lei, M. Oprescu, M. Hei, K. Battocchi, G. Lewis. Machine Learning Estimation of Heterogeneous Treatment Effects with Instruments. Proceedings of the 33rd Conference on Neural Information Processing Systems (NeurIPS), 2019 (Spotlight Presentation)
D. Foster, V. Syrgkanis. Orthogonal Statistical Learning. Proceedings of the 32nd Annual Conference on Learning Theory (COLT), 2019 (Best Paper Award)
M. Oprescu, V. Syrgkanis and Z. S. Wu. Orthogonal Random Forest for Causal Inference. Proceedings of the 36th International Conference on Machine Learning (ICML), 2019.
S. Künzel, J. Sekhon, J. Bickel and B. Yu. Metalearners for estimating heterogeneous treatment effects using machine learning. Proceedings of the national academy of sciences, 116(10), 41564165, 2019.
S. Athey, J. Tibshirani, S. Wager. Generalized random forests. Annals of Statistics, 47, no. 2, 11481178, 2019.
V. Chernozhukov, D. Nekipelov, V. Semenova, V. Syrgkanis. Plugin Regularized Estimation of HighDimensional Parameters in Nonlinear Semiparametric Models. Arxiv preprint arxiv:1806.04823, 2018.
S. Wager, S. Athey. Estimation and Inference of Heterogeneous Treatment Effects using Random Forests. Journal of the American Statistical Association, 113:523, 12281242, 2018.
Jason Hartford, Greg Lewis, Kevin LeytonBrown, and Matt Taddy. Deep IV: A flexible approach for counterfactual prediction. Proceedings of the 34th International Conference on Machine Learning, ICML'17, 2017.
V. Chernozhukov, D. Chetverikov, M. Demirer, E. Duflo, C. Hansen, and a. W. Newey. Double Machine Learning for Treatment and Causal Parameters. ArXiv preprint arXiv:1608.00060, 2016.
Dudik, M., Erhan, D., Langford, J., & Li, L. Doubly robust policy evaluation and optimization. Statistical Science, 29(4), 485511, 2014.
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