4. Score functions

We use method-of-moments estimators for the target parameter \(\theta_0\) based upon the empirical analog of the moment condition

\[\mathbb{E}[ \psi(W; \theta_0, \eta_0)] = 0,\]

where we call \(\psi\) the score function, \(W=(Y,D,X,Z)\), \(\theta_0\) is the parameter of interest and \(\eta\) denotes nuisance functions with population value \(\eta_0\). We use score functions \(\psi(W; \theta, \eta)\) that satisfy \(\mathbb{E}[ \psi(W; \theta_0, \eta_0)] = 0\) with \(\theta_0\) being the unique solution and that obey the Neyman orthogonality condition

\[\partial_{\eta} \mathbb{E}[ \psi(W; \theta_0, \eta)] \bigg|_{\eta=\eta_0} = 0.\]

An integral component for the object-oriented (OOP) implementation of DoubleMLPLR, DoubleMLPLIV, DoubleMLIRM, and DoubleMLIIVM is the linearity of the score function in the parameter \(\theta\)

\[\psi(W; \theta, \eta) = \psi_a(W; \eta) \theta + \psi_b(W; \eta).\]

Hence the estimator can be written as

\[\tilde{\theta}_0 = - \frac{\mathbb{E}_N[\psi_b(W; \eta)]}{\mathbb{E}_N[\psi_a(W; \eta)]}.\]

The linearity of the score function in the parameter \(\theta\) allows the implementation of key components in a very general way. The methods and algorithms to estimate the causal parameters, to estimate their standard errors, to perform a multiplier bootstrap, to obtain confidence intervals and many more are implemented in the abstract base class DoubleML. The object-oriented architecture therefore allows for easy extension to new model classes for double machine learning. This is doable with very minor effort whenever the linearity of the score function is satisfied.

4.1. Implementation of the score function and the estimate of the causal parameter

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

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_g = 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_g, ml_m)

In [12]: dml_plr_obj.fit();

In [13]: 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_g: 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)

------------------ 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  0.462968  0.041018  11.286974  1.521861e-29  0.382575  0.543362
 1library(DoubleML)
 2library(mlr3)
 3library(mlr3learners)
 4library(data.table)
 5lgr::get_logger("mlr3")$set_threshold("warn")
 6
 7learner = lrn("regr.ranger", num.trees = 100, mtry = 20, min.node.size = 2, max.depth = 5)
 8ml_g = learner$clone()
 9ml_m = learner$clone()
10set.seed(3141)
11data = make_plr_CCDDHNR2018(alpha=0.5, return_type='data.table')
12obj_dml_data = DoubleMLData$new(data, y_col="y", d_cols="d")
13dml_plr_obj = DoubleMLPLR$new(obj_dml_data, ml_g, ml_m)
14dml_plr_obj$fit()
15print(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_g: regr.ranger
ml_m: regr.ranger

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

------------------ Fit summary       ------------------
 [1] "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


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

In [14]: print(dml_plr_obj.coef)
[0.4629685]
1print(dml_plr_obj$coef)
        d 
0.5443965 

The values of the score function components \(\psi_a(W_i; \hat{\eta}_0)\) and \(\psi_b(W_i; \hat{\eta}_0)\) are stored in the attributes psi_a and psi_b. In the attribute psi the values of the score function \(\psi(W_i; \tilde{\theta}_0, \hat{\eta}_0)\) are stored.

In [15]: print(dml_plr_obj.psi[:5])
[[[-0.16530678]]

 [[ 0.02101701]]

 [[ 0.00576096]]

 [[ 0.06996277]]

 [[-0.23147675]]]
1print(dml_plr_obj$psi[1:5, ,1])
[1] -0.0009695237  0.7811465543  0.0090193584 -0.4037269089  0.8646627426

4.2. Implemented Neyman orthogonal score functions

4.2.1. Partially linear regression model (PLR)

For the PLR model implemented in DoubleMLPLR one can choose between score='IV-type' and score='partialling out'.

score='IV-type' implements the score function:

\[ \begin{align}\begin{aligned}\psi(W; \theta, \eta) &:= [Y - D \theta - g(X)] [D - m(X)]\\&= - D (D - m(X)) \theta + (Y - g(X)) (D - m(X))\\&= \psi_a(W; \eta) \theta + \psi_b(W; \eta)\end{aligned}\end{align} \]

with \(\eta=(g,m)\) and where the components of the linear score are

\[ \begin{align}\begin{aligned}\psi_a(W; \eta) &= - D (D - m(X)),\\\psi_b(W; \eta) &= (Y - g(X)) (D - m(X)).\end{aligned}\end{align} \]

score='partialling out' implements the score function:

\[ \begin{align}\begin{aligned}\psi(W; \theta, \eta) &:= [Y - \ell(X) - \theta (D - m(X))] [D - m(X)]\\&= - (D - m(X)) (D - m(X)) \theta + (Y - \ell(X)) (D - m(X))\\&= \psi_a(W; \eta) \theta + \psi_b(W; \eta)\end{aligned}\end{align} \]

with \(\eta=(\ell,m)\) and where the components of the linear score are

\[ \begin{align}\begin{aligned}\psi_a(W; \eta) &= - (D - m(X)) (D - m(X)),\\\psi_b(W; \eta) &= (Y - \ell(X)) (D - m(X)).\end{aligned}\end{align} \]

4.2.2. Partially linear IV regression model (PLIV)

For the PLIV model implemented in DoubleMLPLIV we employ for score='partialling out' the score function:

\[ \begin{align}\begin{aligned}\psi(W; \theta, \eta) &:= [Y - \ell(X) - \theta (D - r(X))] [Z - m(X)]\\&= - (D - r(X)) (Z - m(X)) \theta + (Y - \ell(X)) (Z - m(X))\\&= \psi_a(W; \eta) \theta + \psi_b(W; \eta)\end{aligned}\end{align} \]

with \(\eta=(\ell, m, r)\) and where the components of the linear score are

\[ \begin{align}\begin{aligned}\psi_a(W; \eta) &= - (D - r(X)) (Z - m(X)),\\\psi_b(W; \eta) &= (Y - \ell(X)) (Z - m(X)).\end{aligned}\end{align} \]

4.2.3. Interactive regression model (IRM)

For the IRM model implemented in DoubleMLIRM one can choose between score='ATE' and score='ATTE'.

score='ATE' implements the score function:

\[ \begin{align}\begin{aligned}\psi(W; \theta, \eta) &:= g(1,X) - g(0,X) + \frac{D (Y - g(1,X))}{m(X)} - \frac{(1 - D)(Y - g(0,X))}{1 - m(x)} - \theta\\&= \psi_a(W; \eta) \theta + \psi_b(W; \eta)\end{aligned}\end{align} \]

with \(\eta=(g,m)\) and where the components of the linear score are

\[ \begin{align}\begin{aligned}\psi_a(W; \eta) &= - 1,\\\psi_b(W; \eta) &= g(1,X) - g(0,X) + \frac{D (Y - g(1,X))}{m(X)} - \frac{(1 - D)(Y - g(0,X))}{1 - m(x)}.\end{aligned}\end{align} \]

score='ATTE' implements the score function:

\[ \begin{align}\begin{aligned}\psi(W; \theta, \eta) &:= \frac{D (Y - g(0,X))}{p} - \frac{m(X) (1 - D) (Y - g(0,X))}{p(1 - m(x))} - \frac{D}{p} \theta\\&= \psi_a(W; \eta) \theta + \psi_b(W; \eta)\end{aligned}\end{align} \]

with \(\eta=(g, m, p)\) and where the components of the linear score are

\[ \begin{align}\begin{aligned}\psi_a(W; \eta) &= - \frac{D}{p},\\\psi_b(W; \eta) &= \frac{D (Y - g(0,X))}{p} - \frac{m(X) (1 - D) (Y - g(0,X))}{p(1 - m(X))}.\end{aligned}\end{align} \]

4.2.4. Interactive IV model (IIVM)

For the IIVM model implemented in DoubleMLIIVM we employ for score='LATE' the score function:

score='LATE' implements the score function:

\[ \begin{align}\begin{aligned}\psi(W; \theta, \eta) :=\; &g(1,X) - g(0,X) + \frac{Z (Y - g(1,X))}{m(X)} - \frac{(1 - Z)(Y - g(0,X))}{1 - m(x)}\\&- \bigg(r(1,X) - r(0,X) + \frac{Z (D - r(1,X))}{m(X)} - \frac{(1 - Z)(D - r(0,X))}{1 - m(x)} \bigg) \theta\\=\; &\psi_a(W; \eta) \theta + \psi_b(W; \eta)\end{aligned}\end{align} \]

with \(\eta=(g, m, r)\) and where the components of the linear score are

\[ \begin{align}\begin{aligned}\psi_a(W; \eta) &= - \bigg(r(1,X) - r(0,X) + \frac{Z (D - r(1,X))}{m(X)} - \frac{(1 - Z)(D - r(0,X))}{1 - m(x)} \bigg),\\\psi_b(W; \eta) &= g(1,X) - g(0,X) + \frac{Z (Y - g(1,X))}{m(X)} - \frac{(1 - Z)(Y - g(0,X))}{1 - m(x)}.\end{aligned}\end{align} \]

4.3. Specifying alternative score functions via callables

Via callables user-written score functions can be used. This functionality is at the moment only implemented for specific model classes in Python. For the PLR model implemented in DoubleMLPLR an alternative score function can be set via score. Choose a callable object / function with signature score(y, d, g_hat, m_hat, smpls) which returns the two score components \(\psi_a()\) and \(\psi_b()\).

For example, the non-orthogonal score function

\[\psi(W; \theta, \eta) = [Y - D \theta - g(X)] D\]

can be obtained with

In [16]: import numpy as np

In [17]: def non_orth_score(y, d, g_hat, m_hat, smpls):
   ....:     u_hat = y - g_hat
   ....:     psi_a = -np.multiply(d, d)
   ....:     psi_b = np.multiply(d, u_hat)
   ....:     return psi_a, psi_b
   ....: 
1non_orth_score = function(y, d, g_hat, m_hat, smpls) {
2    u_hat = y - g_hat
3    psi_a = -1*d*d
4    psi_b = d*u_hat
5    psis = list(psi_a = psi_a, psi_b = psi_b)
6    return(psis)
7}

Use DoubleMLPLR with inf_model=non_orth_score in order to obtain the estimator

\[\tilde{\theta}_0 = - \frac{\mathbb{E}_N[D (Y-g(X))]}{\mathbb{E}_N[D^2]}\]

when applying fit(). Note that this estimate will in general be prone to a regularization bias, see also Overcoming regularization bias by orthogonalization.