8. Variance estimation and confidence intervals#

8.1. Variance estimation#

Under regularity conditions the estimator \(\tilde{\theta}_0\) concentrates in a \(1/\sqrt{N}\)-neighborhood of \(\theta_0\) and the sampling error \(\sqrt{N}(\tilde{\theta}_0 - \theta_0)\) is approximately normal

\[\sqrt{N}(\tilde{\theta}_0 - \theta_0) \leadsto N(o, \sigma^2),\]

with mean zero and variance given by

\[\sigma^2 := J_0^{-2} \mathbb{E}(\psi^2(W; \theta_0, \eta_0)),\]

where \(J_0 = \mathbb{E}(\psi_a(W; \eta_0))\), if the score function is linear in the parameter \(\theta\). If the score is not linear in the parameter \(\theta\), then \(J_0 = \partial_\theta\mathbb{E}(\psi(W; \theta, \eta_0)) \big|_{\theta=\theta_0}\).

Estimates of the variance are obtained by

\[ \begin{align}\begin{aligned}\hat{\sigma}^2 &= \hat{J}_0^{-2} \frac{1}{N} \sum_{k=1}^{K} \sum_{i \in I_k} \big[\psi(W_i; \tilde{\theta}_0, \hat{\eta}_{0,k})\big]^2,\\\hat{J}_0 &= \frac{1}{N} \sum_{k=1}^{K} \sum_{i \in I_k} \psi_a(W_i; \hat{\eta}_{0,k}),\end{aligned}\end{align} \]

for score functions being linear in the parameter \(\theta\). For non-linear score functions, the implementation assumes that derivatives and expectations are interchangeable, so that

\[\hat{J}_0 = \frac{1}{N} \sum_{k=1}^{K} \sum_{i \in I_k} \partial_\theta \psi(W_i; \tilde{\theta}_0, \hat{\eta}_{0,k}).\]

An approximate confidence interval is given by

\[\big[\tilde{\theta}_0 \pm \Phi^{-1}(1 - \alpha/2) \hat{\sigma} / \sqrt{N}].\]

As an example we consider a partially linear regression model (PLR) implemented in DoubleMLPLR.

Python

In [1]: import doubleml as dml

In [2]: from doubleml.datasets import make_plr_CCDDHNR2018

In [3]: from sklearn.ensemble import RandomForestRegressor

In [4]: from sklearn.base import clone

In [5]: np.random.seed(3141)

In [6]: learner = RandomForestRegressor(n_estimators=100, max_features=20, max_depth=5, min_samples_leaf=2)

In [7]: ml_l = clone(learner)

In [8]: ml_m = clone(learner)

In [9]: data = make_plr_CCDDHNR2018(alpha=0.5, return_type='DataFrame')

In [10]: obj_dml_data = dml.DoubleMLData(data, 'y', 'd')

In [11]: dml_plr_obj = dml.DoubleMLPLR(obj_dml_data, ml_l, ml_m)

In [12]: dml_plr_obj.fit();

library(DoubleML)
library(mlr3)
library(mlr3learners)
library(data.table)
lgr::get_logger("mlr3")$set_threshold("warn")

learner = lrn("regr.ranger", num.trees = 100, mtry = 20, min.node.size = 2, max.depth = 5)
ml_l = learner$clone()
ml_m = learner$clone()

set.seed(3141)
obj_dml_data = make_plr_CCDDHNR2018(alpha=0.5)
dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_l, ml_m)
dml_plr_obj$fit()

The fit() method of DoubleMLPLR stores the estimate \(\tilde{\theta}_0\) in its coef attribute.

Python

In [13]: print(dml_plr_obj.coef)
[0.48025158]

print(dml_plr_obj$coef)

        d 
0.5443965 

The asymptotic standard error \(\hat{\sigma}/\sqrt{N}\) is stored in its se attribute.

Python

In [14]: print(dml_plr_obj.se)
[0.04053296]

print(dml_plr_obj$se)

         d 
0.04512331 

Additionally, the value of the \(t\)-statistic and the corresponding p-value are provided in the attributes t_stat and pval.

Python

In [15]: print(dml_plr_obj.t_stat)
[11.84842177]

In [16]: print(dml_plr_obj.pval)
[2.19281722e-32]

print(dml_plr_obj$t_stat)
print(dml_plr_obj$pval)

       d 
12.06464 

           d 
1.623681e-33 

Note

In Python, an overview of all these estimates, together with a 95 % confidence interval is stored in the attribute summary.
In R, a summary can be obtained by using the method summary(). The confint() method performs estimation of confidence intervals.

Python

In [17]: print(dml_plr_obj.summary)
       coef   std err          t         P>|t|     2.5 %    97.5 %
d  0.480252  0.040533  11.848422  2.192817e-32  0.400808  0.559695

dml_plr_obj$summary()
dml_plr_obj$confint()

Estimates and significance testing of the effect of target variables
  Estimate. Std. Error t value Pr(>|t|)    
d   0.54440    0.04512   12.06   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

A matrix: 1 × 2 of type dbl
	2.5 %	97.5 %
d	0.4559565	0.6328366

A more detailed overview of the fitted model, its specifications and the summary can be obtained via the string-representation of the object.

Python

In [18]: print(dml_plr_obj)
================== 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']
Instrument variable(s): None
No. Observations: 500

------------------ Score & algorithm ------------------
Score function: partialling out
DML algorithm: dml2

------------------ Machine learner   ------------------
Learner ml_l: RandomForestRegressor(max_depth=5, max_features=20, min_samples_leaf=2)
Learner ml_m: RandomForestRegressor(max_depth=5, max_features=20, min_samples_leaf=2)
Out-of-sample Performance:
Learner ml_l RMSE: [[1.12638786]]
Learner ml_m RMSE: [[1.03968828]]

------------------ Resampling        ------------------
No. folds: 5
No. repeated sample splits: 1
Apply cross-fitting: True

------------------ Fit summary       ------------------
       coef   std err          t         P>|t|     2.5 %    97.5 %
d  0.480252  0.040533  11.848422  2.192817e-32  0.400808  0.559695

print(dml_plr_obj)

================= 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
Instrument(s): 
No. Observations: 500

------------------ 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|)    
d   0.54440    0.04512   12.06   <2e-16 ***
---
Signif. codes:  0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

8.2. Confidence bands and multiplier bootstrap for valid simultaneous inference#

DoubleML provides methods to perform valid simultaneous inference for multiple treatment variables. As an example, consider a PLR with \(p_1\) causal parameters of interest \(\theta_{0,1}, \ldots, \theta_{0,p_1}\) associated with treatment variables \(D_1, \ldots, D_{p_1}\). Inference on multiple target coefficients can be performed by iteratively applying the DML inference procedure over the target variables of interests: Each of the coefficients of interest, \(\theta_{0,j}\), with \(j \in \lbrace 1, \ldots, p_1 \rbrace\), solves a corresponding moment condition

\[\mathbb{E}[ \psi_j(W; \theta_{0,j}, \eta_{0,j})] = 0.\]

Analogously to the case with a single parameter of interest, the PLR model with multiple treatment variables includes two regression steps to achieve orthogonality. First, the main regression is given by

\[Y = D_j \theta_{0,j} + g_{0,j}([D_k, X]) + \zeta_j, \quad \mathbb{E}(\zeta_j | D, X) = 0,\]

with \([D_k, X]\) being a matrix comprising the confounders, \(X\), and all remaining treatment variables \(D_k\) with \(k \in \lbrace 1, \ldots, p_1\rbrace \setminus j\), by default. Second, the relationship between the treatment variable \(D_j\) and the remaining explanatory variables is determined by the equation

\[D_j = m_{0,j}([D_k, X]) + V_j, \quad \mathbb{E}(V_j | D_k, X) = 0,\]

For further details, we refer to Belloni et al. (2018). Simultaneous inference can be based on a multiplier bootstrap procedure introduced in Chernozhukov et al. (2013, 2014). Alternatively, traditional correction approaches, for example the Bonferroni correction, can be used to adjust p-values.

The bootstrap() method provides an implementation of a multiplier bootstrap for double machine learning models. For \(b=1, \ldots, B\) weights \(\xi_{i, b}\) are generated according to a normal (Gaussian) bootstrap, wild bootstrap or exponential bootstrap. The number of bootstrap samples is provided as input n_rep_boot and for method one can choose 'Bayes', 'normal' or 'wild'. Based on the estimates of the standard errors \(\hat{\sigma}_j\) and \(\hat{J}_{0,j} = \mathbb{E}_N(\psi_{a,j}(W; \eta_{0,j}))\) that are obtained from DML, we construct bootstrap coefficients \(\theta^{*,b}_j\) and bootstrap t-statistics \(t^{*,b}_j\) for \(j=1, \ldots, p_1\)

\[ \begin{align}\begin{aligned}\theta^{*,b}_{j} &= \frac{1}{\sqrt{N} \hat{J}_{0,j}}\sum_{k=1}^{K} \sum_{i \in I_k} \xi_{i}^b \cdot \psi_j(W_i; \tilde{\theta}_{0,j}, \hat{\eta}_{0,j;k}),\\t^{*,b}_{j} &= \frac{1}{\sqrt{N} \hat{J}_{0,j} \hat{\sigma}_{j}} \sum_{k=1}^{K} \sum_{i \in I_k} \xi_{i}^b \cdot \psi_j(W_i; \tilde{\theta}_{0,j}, \hat{\eta}_{0,j;k}).\end{aligned}\end{align} \]

The output of the multiplier bootstrap can be used to determine the constant, \(c_{1-\alpha}\) that is required for the construction of a simultaneous \((1-\alpha)\) confidence band

\[\left[\tilde\theta_{0,j} \pm c_{1-\alpha} \cdot \hat\sigma_j/\sqrt{N} \right].\]

To demonstrate the bootstrap, we simulate data from a sparse partially linear regression model. Then we estimate the PLR model and perform the multiplier bootstrap. Joint confidence intervals based on the multiplier bootstrap are then obtained by setting the option joint when calling the method confint.

Moreover, a multiple hypotheses testing adjustment of p-values from a high-dimensional model can be obtained with the method p_adjust. DoubleML performs a version of the Romano-Wolf stepdown adjustment, which is based on the multiplier bootstrap, by default. Alternatively, p_adjust allows users to apply traditional corrections via the option method.

Python

In [19]: import doubleml as dml

In [20]: import numpy as np

In [21]: from sklearn.base import clone

In [22]: from sklearn.linear_model import LassoCV

# Simulate data
In [23]: np.random.seed(1234)

In [24]: n_obs = 500

In [25]: n_vars = 100

In [26]: X = np.random.normal(size=(n_obs, n_vars))

In [27]: theta = np.array([3., 3., 3.])

In [28]: y = np.dot(X[:, :3], theta) + np.random.standard_normal(size=(n_obs,))

In [29]: dml_data = dml.DoubleMLData.from_arrays(X[:, 10:], y, X[:, :10])

In [30]: learner = LassoCV()

In [31]: ml_l = clone(learner)

In [32]: ml_m = clone(learner)

In [33]: dml_plr = dml.DoubleMLPLR(dml_data, ml_l, ml_m)

In [34]: print(dml_plr.fit().bootstrap().confint(joint=True))
        2.5 %    97.5 %
d1   2.813342  3.055680
d2   2.815224  3.083258
d3   2.860663  3.109069
d4  -0.141546  0.091391
d5  -0.060845  0.176929
d6  -0.158697  0.078474
d7  -0.172022  0.062964
d8  -0.067721  0.174499
d9  -0.092365  0.139491
d10 -0.110717  0.138698

In [35]: print(dml_plr.p_adjust())
         coef   pval
d1   2.934511  0.000
d2   2.949241  0.000
d3   2.984866  0.000
d4  -0.025077  0.850
d5   0.058042  0.492
d6  -0.040112  0.850
d7  -0.054529  0.850
d8   0.053389  0.514
d9   0.023563  0.802
d10  0.013990  0.850

In [36]: print(dml_plr.p_adjust(method='bonferroni'))
         coef  pval
d1   2.934511   0.0
d2   2.949241   0.0
d3   2.984866   0.0
d4  -0.025077   1.0
d5   0.058042   1.0
d6  -0.040112   1.0
d7  -0.054529   1.0
d8   0.053389   1.0
d9   0.023563   1.0
d10  0.013990   1.0

library(DoubleML)
library(mlr3)
library(mlr3learners)
library(data.table)
lgr::get_logger("mlr3")$set_threshold("warn")

set.seed(3141)
n_obs = 500
n_vars = 100
theta = rep(3, 3)
X = matrix(stats::rnorm(n_obs * n_vars), nrow = n_obs, ncol = n_vars)
y = X[, 1:3, drop = FALSE] %*% theta  + stats::rnorm(n_obs)
dml_data = double_ml_data_from_matrix(X = X[, 11:n_vars], y = y, d = X[,1:10])

learner = lrn("regr.cv_glmnet", s="lambda.min")
ml_l = learner$clone()
ml_m = learner$clone()
dml_plr = DoubleMLPLR$new(dml_data, ml_l, ml_m)

dml_plr$fit()
dml_plr$bootstrap()
dml_plr$confint(joint=TRUE)
dml_plr$p_adjust()
dml_plr$p_adjust(method="bonferroni")

A matrix: 10 × 2 of type dbl
	2.5 %	97.5 %
d1	2.89027368	3.14532650
d2	2.90794478	3.14368145
d3	2.87430335	3.12752825
d4	-0.14790924	0.07828372
d5	-0.09779675	0.16803512
d6	-0.12105472	0.12539340
d7	-0.16536299	0.09310496
d8	-0.10127930	0.14200098
d9	-0.13868238	0.09980311
d10	-0.04444978	0.19680840

A matrix: 10 × 2 of type dbl
	Estimate.	pval
d1	3.017800092	0.000
d2	3.025813114	0.000
d3	3.000915799	0.000
d4	-0.034812763	0.938
d5	0.035119185	0.938
d6	0.002169338	0.958
d7	-0.036129015	0.938
d8	0.020360838	0.954
d9	-0.019439633	0.954
d10	0.076179312	0.428

A matrix: 10 × 2 of type dbl
	Estimate.	pval
d1	3.017800092	0.0000000
d2	3.025813114	0.0000000
d3	3.000915799	0.0000000
d4	-0.034812763	1.0000000
d5	0.035119185	1.0000000
d6	0.002169338	1.0000000
d7	-0.036129015	1.0000000
d8	0.020360838	1.0000000
d9	-0.019439633	1.0000000
d10	0.076179312	0.8116912

8.3. References#

Belloni, A., Chernozhukov, V., Chetverikov, D., Wei, Y. (2018), Uniformly valid post-regularization confidence regions for many functional parameters in z-estimation framework. The Annals of Statistics, 46 (6B): 3643-75, doi: 10.1214/17-AOS1671.
Chernozhukov, V., Chetverikov, D., Kato, K. (2013). Gaussian approximations and multiplier bootstrap for maxima of sums of high-dimensional random vectors. The Annals of Statistics 41 (6): 2786-2819, doi: 10.1214/13-AOS1161.
Chernozhukov, V., Chetverikov, D., Kato, K. (2014), Gaussian approximation of suprema of empirical processes. The Annals of Statistics 42 (4): 1564-97, doi: 10.1214/14-AOS1230.