The purpose of the following case-studies is to demonstrate the core
functionalities of DoubleML.
The DoubleML package for R can be downloaded using
(requires previous installation of the remotes
package).
Load the package after completed installation.
The python package DoubleML is available via the github
repository. For more information, please visit our user guide.
For our case study we download the Bonus data set from the Pennsylvania Reemployment Bonus experiment and as a second example we simulate data from a partially linear regression model.
## inuidur1 female black othrace dep1 dep2 q2 q3 q4 q5 q6
## <num> <num> <num> <num> <num> <num> <num> <num> <num> <num> <num>
## 1: 2.890372 0 0 0 0 1 0 0 0 1 0
## 2: 0.000000 0 0 0 0 0 0 0 0 1 0
## 3: 3.295837 0 0 0 0 0 0 0 1 0 0
## 4: 2.197225 0 0 0 0 0 0 1 0 0 0
## 5: 3.295837 0 0 0 1 0 0 0 0 1 0
## 6: 3.295837 1 0 0 0 0 0 0 0 1 0
## agelt35 agegt54 durable lusd husd tg
## <num> <num> <num> <num> <num> <num>
## 1: 0 0 0 0 1 0
## 2: 0 0 0 1 0 0
## 3: 0 0 0 1 0 0
## 4: 1 0 0 0 0 1
## 5: 0 1 1 1 0 0
## 6: 0 1 0 1 0 0\[\begin{align*} Y = D \theta_0 + g_0(X) + \zeta, & &\mathbb{E}(\zeta | D,X) = 0, \\ D = m_0(X) + V, & &\mathbb{E}(V | X) = 0, \end{align*}\] where \(Y\) is the outcome variable and \(D\) is the policy variable of interest. The high-dimensional vector \(X = (X_1, \ldots, X_p)\) consists of other confounding covariates, and \(\zeta\) and \(V\) are stochastic errors.
DoubleMLDataDoubleML provides interfaces to objects of class data.table
as well as R base classes data.frame and
matrix. Details on the data-backend and the interfaces can
be found in the user guide. The DoubleMLData class serves
as data-backend and can be initialized from a dataframe by specifying
the column y_col="inuidur1" serving as outcome variable
\(Y\), the column(s)
d_cols = "tg" serving as treatment variable \(D\) and the columns
x_cols=c("female", "black", "othrace", "dep1", "dep2", "q2", "q3", "q4", "q5", "q6", "agelt35", "agegt54", "durable", "lusd", "husd")
specifying the confounders. Alternatively a matrix interface can be used
as shown below for the simulated data.
# Specify the data and variables for the causal model
dml_data_bonus = DoubleMLData$new(df_bonus,
y_col = "inuidur1",
d_cols = "tg",
x_cols = c("female", "black", "othrace", "dep1", "dep2",
"q2", "q3", "q4", "q5", "q6", "agelt35", "agegt54",
"durable", "lusd", "husd"))
print(dml_data_bonus)## ================= DoubleMLData Object ==================
##
##
## ------------------ Data summary ------------------
## Outcome variable: inuidur1
## Treatment variable(s): tg
## Covariates: female, black, othrace, dep1, dep2, q2, q3, q4, q5, q6, agelt35, agegt54, durable, lusd, husd
## Instrument(s):
## No. Observations: 5099# matrix interface to DoubleMLData
dml_data_sim = double_ml_data_from_matrix(X = X, y = y, d = d)
dml_data_sim## ================= DoubleMLData Object ==================
##
##
## ------------------ Data summary ------------------
## Outcome variable: y
## Treatment variable(s): d
## Covariates: X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35, X36, X37, X38, X39, X40, X41, X42, X43, X44, X45, X46, X47, X48, X49, X50, X51, X52, X53, X54, X55, X56, X57, X58, X59, X60, X61, X62, X63, X64, X65, X66, X67, X68, X69, X70, X71, X72, X73, X74, X75, X76, X77, X78, X79, X80, X81, X82, X83, X84, X85, X86, X87, X88, X89, X90, X91, X92, X93, X94, X95, X96, X97, X98, X99, X100
## Instrument(s):
## No. Observations: 500To estimate our partially linear regression (PLR) model with the
double machine learning algorithm, we first have to specify machine
learners to estimate \(m_0\) and \(g_0\). For the bonus data we use a random
forest regression model and for our simulated data from a sparse
partially linear model we use a Lasso regression model. The
implementation of DoubleML is based on the meta-packages mlr3 for R. For details on the
specification of learners and their hyperparameters we refer to the user
guide Learners, hyperparameters and hyperparameter tuning.
library(mlr3)
library(mlr3learners)
# surpress messages from mlr3 package during fitting
lgr::get_logger("mlr3")$set_threshold("warn")
learner = lrn("regr.ranger", num.trees = 500, max.depth = 5, min.node.size = 2)
ml_l_bonus = learner$clone()
ml_m_bonus = learner$clone()
learner = lrn("regr.glmnet", lambda = sqrt(log(n_vars)/(n_obs)))
ml_l_sim = learner$clone()
ml_m_sim = learner$clone()When initializing the object for PLR models DoubleMLPLR,
we can further set parameters specifying the resampling:
n_folds
(defaults to n_folds = 5) as well asn_rep (defaults to n_rep = 1).Additionally, one can choose between the algorithms
"dml1" and "dml2" via
dml_procedure (defaults to "dml2"). Depending
on the causal model, one can further choose between different
Neyman-orthogonal score / moment functions. For the PLR model the
default score is "partialling out", i.e., \[\begin{align}\begin{aligned}\psi(W; \theta, \eta)
&:= [Y - \ell(X) - \theta (D - m(X))] [D -
m(X)].\end{aligned}\end{align}\]
Note that with this score, we do not estimate \(g_0(X)\) directly, but the conditional expectation of \(Y\) given \(X\), \(\ell_0(X) = E[Y|X]\). The user guide provides details about the Sample-splitting, cross-fitting and repeated cross-fitting, the Double machine learning algorithms and the Score functions
We now initialize DoubleMLPLR objects for our examples
using default parameters. The models are estimated by calling the
fit() method and we can for example inspect the estimated
treatment effect using the summary() method. A more
detailed result summary can be obtained via the print()
method. Besides the fit() method DoubleML
model classes also provide functionalities to perform statistical
inference like bootstrap(), confint() and
p_adjust(), for details see the user guide Variance
estimation, confidence intervals and boostrap standard errors.
set.seed(3141)
obj_dml_plr_bonus = DoubleMLPLR$new(dml_data_bonus, ml_l = ml_l_bonus, ml_m = ml_m_bonus)
obj_dml_plr_bonus$fit()
print(obj_dml_plr_bonus)## ================= DoubleMLPLR Object ==================
##
##
## ------------------ Data summary ------------------
## Outcome variable: inuidur1
## Treatment variable(s): tg
## Covariates: female, black, othrace, dep1, dep2, q2, q3, q4, q5, q6, agelt35, agegt54, durable, lusd, husd
## Instrument(s):
## No. Observations: 5099
##
## ------------------ Score & algorithm ------------------
## Score function: partialling out
## DML algorithm: dml2
##
## ------------------ Machine learner ------------------
## ml_l: regr.ranger
## ml_m: regr.ranger
##
## ------------------ Resampling ------------------
## No. folds: 5
## No. repeated sample splits: 1
## Apply cross-fitting: TRUE
##
## ------------------ Fit summary ------------------
## Estimates and significance testing of the effect of target variables
## Estimate. Std. Error t value Pr(>|t|)
## tg -0.07561 0.03536 -2.139 0.0325 *
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1obj_dml_plr_sim = DoubleMLPLR$new(dml_data_sim, ml_l = ml_l_sim, ml_m = ml_m_sim)
obj_dml_plr_sim$fit()
print(obj_dml_plr_sim)## ================= DoubleMLPLR Object ==================
##
##
## ------------------ Data summary ------------------
## Outcome variable: y
## Treatment variable(s): d
## Covariates: X1, X2, X3, X4, X5, X6, X7, X8, X9, X10, X11, X12, X13, X14, X15, X16, X17, X18, X19, X20, X21, X22, X23, X24, X25, X26, X27, X28, X29, X30, X31, X32, X33, X34, X35, X36, X37, X38, X39, X40, X41, X42, X43, X44, X45, X46, X47, X48, X49, X50, X51, X52, X53, X54, X55, X56, X57, X58, X59, X60, X61, X62, X63, X64, X65, X66, X67, X68, X69, X70, X71, X72, X73, X74, X75, X76, X77, X78, X79, X80, X81, X82, X83, X84, X85, X86, X87, X88, X89, X90, X91, X92, X93, X94, X95, X96, X97, X98, X99, X100
## Instrument(s):
## No. Observations: 500
##
## ------------------ Score & algorithm ------------------
## Score function: partialling out
## DML algorithm: dml2
##
## ------------------ Machine learner ------------------
## ml_l: regr.glmnet
## ml_m: regr.glmnet
##
## ------------------ Resampling ------------------
## No. folds: 5
## No. repeated sample splits: 1
## Apply cross-fitting: TRUE
##
## ------------------ Fit summary ------------------
## Estimates and significance testing of the effect of target variables
## Estimate. Std. Error t value Pr(>|t|)
## d 2.98094 0.05871 50.78 <2e-16 ***
## ---
## Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1