This article is an introduction to double/debiased machine learning using short-stacking in R. Topics discussed below include:
See Articles for discussions of more advanced topics.
For illustration, we apply ddml to the included random
subsample of 5,000 observations from the data of Angrist & Evans
(1998). The data contains information on the labor supply of mothers,
their children, as well as demographic data. See ?AE98 for
details.
# Load ddml and set seed
library(ddml)
set.seed(35611)
# 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")]ddml_late estimates the local average treatment effect
(LATE) using double/debiased machine learning (see
?ddml_late). The high-dimensional nuisance parameters
arising in the estimate of the LATE are conditional expectation
functions of the control variables \(X\). In particular, we require first step
estimates of the reduced forms \(E[Y|Z=z, X],
E[D|Z=z, X]\) for \(z=0,1\) and
\(E[Z|X]\). In the absence of
functional form assumptions, these conditional expectations need to be
estimated nonparametrically.
Here, we consider gradient boosting from the popular xgboost package to estimate the
nuisance parameters. The function mdl_xgboost is a wrapper
for xgboost, allowing to specify all parameters of the
original function. See ?mdl_xgboost for details and take a
look at vignette("new_ml_wrapper") to learn how to write a
wrapper for a different machine learner yourself.
# Specify a single learner
learners_single <- list(what = mdl_xgboost,
args = list(nrounds = 10,
max_depth = 1))Double/debiased machine learning relies on cross-fitting to avoid
large bias from overfitting when estimating the nuisance parameters. The
argument sample_folds = 3 implies that 2/3 of the
observations – about 3,333 observations – are used to train the machine
learner in each cross-fitting sample fold.
# Estimate the local average treatment effect using xgboost.
late_fit <- ddml_late(y, D, Z, X,
learners = learners_single,
sample_folds = 3,
silent = TRUE)
summary(late_fit)
#> LATE estimation results:
#>
#> Estimate Std. Error t value Pr(>|t|)
#> -0.2680106 0.2449627 -1.094087 0.2739166(Note that estimation here is based on a random subsample of 5,000 observations. The results can thus not readily be compared to those in Angrist & Evans (1998).)
Since the statistical properties of machine learners depend heavily
on the underlying (unknown!) structure of the data, adaptive combination
of multiple machine learners can increase robustness. In the below
snippet, ddml_late estimates the LATE with short-stacking
based on three base learners:
?ols)?mdl_glmnet)?mdl_xgboost)learners_multiple <- list(list(fun = ols),
list(fun = mdl_glmnet),
list(fun = mdl_xgboost,
args = list(nrounds = 10,
max_depth = 1)))Short-stacking is a computationally convenient variant of stacking
originally introduced by Wolpert (1992). Stacking constructs linear
combinations of base learners to minimize the out-of-sample mean squared
error of a particular reduced form (e.g., \(E[Z|X]\)). Short-stacking uses the
out-of-sample predictions that naturally arise in computation of
double/debiased machine learning estimates due to cross-fitting, which
substantially reduces the computational burden (see
vignette("stacking")).
In finite samples, regularizing the linear combination of base
learners as constructed via (short-)stacking can improve statistical
properties. This can be specified via the ensemble_type
argument. Below, ddml_late estimates the nuisance
parameters via linear combinations of the four base learners with linear
coefficients that are constrained to be non-negative and sum to one.
# Estimate the local average treatment effect using short-stacking with base
# learners ols, lasso, and xgboost.
late_fit <- ddml_late(y, D, Z, X,
learners = learners_multiple,
ensemble_type = 'nnls1',
shortstack = TRUE,
sample_folds = 3,
silent = TRUE)
summary(late_fit)
#> LATE estimation results:
#>
#> Estimate Std. Error t value Pr(>|t|)
#> nnls1 -0.19173 0.2005266 -0.9561322 0.3390054It is often insightful to see which base learners contribute the most to the final reduced form estimates. The below snippet shows the weights for the reduced forms \(E[Y|Z=0,X]\) and \(E[Y|Z=1,X]\):
cat("Stacking weights for E[Y|Z=0, X]: \n")
#> Stacking weights for E[Y|Z=0, X]:
t(late_fit$weights$y_X_Z0)
#> [,1] [,2] [,3]
#> nnls1 0.228205 0.5782117 0.1935833
cat("Stacking weights for E[Y|Z=1, X]: \n")
#> Stacking weights for E[Y|Z=1, X]:
t(late_fit$weights$y_X_Z1)
#> [,1] [,2] [,3]
#> nnls1 0.2389935 0.6920264 0.0689801ddml supports multiple schemes for constructing linear
combinations of base learners. Since each of them relies on the
out-of-sample predictions of the base learners, it is computationally
cheap to compute them simultaneously. The below snippet estimates the
LATE using the base learners in four different linear combinations:
'nls' constraints the coefficients of each base learner
to be non-negative'singlebest' selects the single MSPE-minimizing base
learner'ols constructs unconstrained linear combinations of
base learners'average' computes an unweighted average of base
learners# Estimate the local average treatment effect using short-stacking with base
# learners ols, lasso, ridge, and xgboost.
late_fit <- ddml_late(y, D, Z, X,
learners = learners_multiple,
ensemble_type = c('nnls', 'singlebest',
'ols', 'average'),
shortstack = TRUE,
sample_folds = 3,
silent = TRUE)
summary(late_fit)
#> LATE estimation results:
#>
#> Estimate Std. Error t value Pr(>|t|)
#> nnls -0.2432971 0.2120355 -1.147436 0.2512017
#> singlebest -0.2518913 0.1967114 -1.280512 0.2003652
#> ols -0.2417658 0.2122186 -1.139230 0.2546073
#> average -0.2117828 0.1958048 -1.081601 0.2794296Angrist J, Evans W (1998). “Children and Their Parents’ Labor Supply: Evidence from Exogenous Variation in Family Size.” American Economic Review, 88(3), 450-477.
Wolpert D H (1992). “Stacked generalization.” Neural Networks, 5(2), 241-259.