7  異質性の探索

• 条件付き平均差 \(\tau(X)=E[Y|D=1,X]-E[Y|D=0,X]\)の(ノンパラメトリックな)推定値を異質性のシグナルとして活用

  • 一般に、条件付きの平均差のノンパラメトリックな推定値について、推定誤差を評価するのが難しいため

  • 効果の異質性の”シグナル”として使う (Chernozhukov et al. 2018; Kallus 2022)

  • \(E[\tau(X)|\tau(X)\le Median(\tau(X))]\) を推定

• \(\tau(X)\) はCausal Forest (Wager and Athey 2018; Athey, Tibshirani, and Wager 2019) で推定

7.1 設定

R

library(tidyverse)
library(grf)
library(DoubleML)
library(mlr3verse)

Data_R <- read_csv("ExampleData/Example.csv")

Group <- sample(0:1,nrow(Data_R),replace = TRUE)

Y <- Data_R$Price

D <- Data_R$Reform

X <- Data_R |> 
  select(
    -Price,
    -Reform
  )

• grf

EstCF <- causal_forest(
  X = X[Group == 0,],
  W = D[Group == 0],
  Y = Y[Group == 0]
)

PredTau <- EstCF |> 
  predict(X)

hist(PredTau$predictions)

Task <- double_ml_data_from_matrix(
  X = X[Group == 1,],
  y = Y[Group == 1],
  d = D[Group == 1]
)

EstDML <- DoubleMLPLR$new(
  Task,
  lrn("regr.lm"),
  lrn("regr.lm")
)

EstDML$fit()

INFO  [21:02:33.207] [mlr3] Applying learner 'regr.lm' on task 'nuis_l' (iter 1/5)
INFO  [21:02:33.229] [mlr3] Applying learner 'regr.lm' on task 'nuis_l' (iter 2/5)
INFO  [21:02:33.239] [mlr3] Applying learner 'regr.lm' on task 'nuis_l' (iter 3/5)
INFO  [21:02:33.246] [mlr3] Applying learner 'regr.lm' on task 'nuis_l' (iter 4/5)
INFO  [21:02:33.253] [mlr3] Applying learner 'regr.lm' on task 'nuis_l' (iter 5/5)
INFO  [21:02:33.334] [mlr3] Applying learner 'regr.lm' on task 'nuis_m' (iter 1/5)
INFO  [21:02:33.515] [mlr3] Applying learner 'regr.lm' on task 'nuis_m' (iter 2/5)
INFO  [21:02:33.522] [mlr3] Applying learner 'regr.lm' on task 'nuis_m' (iter 3/5)
INFO  [21:02:33.528] [mlr3] Applying learner 'regr.lm' on task 'nuis_m' (iter 4/5)
INFO  [21:02:33.534] [mlr3] Applying learner 'regr.lm' on task 'nuis_m' (iter 5/5)

EstDML

================= DoubleMLPLR Object ==================


------------------ Data summary      ------------------
Outcome variable: y
Treatment variable(s): d
Covariates: X1, X2, X3, X4
Instrument(s): 
No. Observations: 7375

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

------------------ Machine learner   ------------------
ml_l: regr.lm
ml_m: regr.lm

------------------ 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    3.2798     0.5999   5.468 4.56e-08 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Q <- quantile(PredTau$predictions,0.5)

Task <- double_ml_data_from_matrix(
  X = X[Group == 1 & PredTau$predictions >= Q,],
  y = Y[Group == 1 & PredTau$predictions >= Q],
  d = D[Group == 1 & PredTau$predictions >= Q]
)

EstDML <- DoubleMLPLR$new(
  Task,
  lrn("regr.lm"),
  lrn("regr.lm")
)

EstDML$fit()

INFO  [21:02:33.617] [mlr3] Applying learner 'regr.lm' on task 'nuis_l' (iter 1/5)
INFO  [21:02:33.624] [mlr3] Applying learner 'regr.lm' on task 'nuis_l' (iter 2/5)
INFO  [21:02:33.630] [mlr3] Applying learner 'regr.lm' on task 'nuis_l' (iter 3/5)
INFO  [21:02:33.637] [mlr3] Applying learner 'regr.lm' on task 'nuis_l' (iter 4/5)
INFO  [21:02:33.643] [mlr3] Applying learner 'regr.lm' on task 'nuis_l' (iter 5/5)
INFO  [21:02:33.687] [mlr3] Applying learner 'regr.lm' on task 'nuis_m' (iter 1/5)
INFO  [21:02:33.693] [mlr3] Applying learner 'regr.lm' on task 'nuis_m' (iter 2/5)
INFO  [21:02:33.698] [mlr3] Applying learner 'regr.lm' on task 'nuis_m' (iter 3/5)
INFO  [21:02:33.704] [mlr3] Applying learner 'regr.lm' on task 'nuis_m' (iter 4/5)
INFO  [21:02:33.710] [mlr3] Applying learner 'regr.lm' on task 'nuis_m' (iter 5/5)

EstDML

================= DoubleMLPLR Object ==================


------------------ Data summary      ------------------
Outcome variable: y
Treatment variable(s): d
Covariates: X1, X2, X3, X4
Instrument(s): 
No. Observations: 3678

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

------------------ Machine learner   ------------------
ml_l: regr.lm
ml_m: regr.lm

------------------ 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    5.3312     0.8785   6.069 1.29e-09 ***
---
Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

Python

import pandas as pd
import sklearn.model_selection  as model_selection

Data_Python = pd.read_csv('ExampleData/Example.csv')

y = Data_Python['Price']

d = Data_Python['Reform']

x = Data_Python.drop(['Price','Reform'],axis=1).values

y_train, y_test,d_train,d_test,x_train,x_test = model_selection.train_test_split(
  y,
  d,
  x,
  train_size = 0.5
  )

• econml

import econml.dml as dml
import sklearn.linear_model as Linear
import matplotlib.pyplot as plt
import seaborn as sns

est_cf = dml.CausalForestDML(
    model_y = Linear.LinearRegression(),
    model_t = Linear.LinearRegression(),
    n_estimators=2000
)

est_cf = est_cf.fit(y_train, d_train, X = x_train, W=None)

pred_tau = est_cf.effect(x_test)

fig = plt.figure()

sns.histplot(pred_tau)

import doubleml as DML

Task = DML.DoubleMLData.from_arrays(x_test, y_test, d_test)
                    
FitPLR = DML.DoubleMLPLR(
  Task,
   Linear.LinearRegression(),
   Linear.LinearRegression(),
  n_folds = 2
  )

FitPLR.fit(store_predictions = True)

    ================== DoubleMLPLR Object ==================    
                                                                
    ------------------ Data summary      ------------------     
                      Outcome variable: y                       
                  Treatment variable(s): ['d']                  
              Covariates: ['X1', 'X2', 'X3', 'X4']              
                  Instrument variable(s): None                  
                     No. Observations: 7397                     
                                                                
    ------------------ Score & algorithm ------------------     
                Score function: partialling out                 
                      DML algorithm: dml2                       
                                                                
    ------------------ Machine learner   ------------------     
                Learner ml_l: LinearRegression()                
                Learner ml_m: LinearRegression()                
                                                                
    ------------------ Resampling        ------------------     
                          No. folds: 2                          
                 No. repeated sample splits: 1                  
                   Apply cross-fitting: True                    
                                                                
    ------------------ Fit summary       ------------------     
       coef   std err        t         P>|t|     2.5 %    97.5 %
d  3.580131  0.562229  6.36774  1.918334e-10  2.478182  4.682081

import numpy as np

Q = np.quantile(a=pred_tau, q=0.5)

Task = DML.DoubleMLData.from_arrays(x_test[pred_tau >= Q], y_test[pred_tau >= Q], d_test[pred_tau >= Q])
                    
FitPLR = DML.DoubleMLPLR(
  Task,
   Linear.LinearRegression(),
   Linear.LinearRegression(),
  n_folds = 2
  )

FitPLR.fit(store_predictions = True)

     ================== DoubleMLPLR Object ==================    
                                                                 
     ------------------ Data summary      ------------------     
                       Outcome variable: y                       
                   Treatment variable(s): ['d']                  
               Covariates: ['X1', 'X2', 'X3', 'X4']              
                   Instrument variable(s): None                  
                      No. Observations: 3699                     
                                                                 
     ------------------ Score & algorithm ------------------     
                 Score function: partialling out                 
                       DML algorithm: dml2                       
                                                                 
     ------------------ Machine learner   ------------------     
                 Learner ml_l: LinearRegression()                
                 Learner ml_m: LinearRegression()                
                                                                 
     ------------------ Resampling        ------------------     
                           No. folds: 2                          
                  No. repeated sample splits: 1                  
                    Apply cross-fitting: True                    
                                                                 
     ------------------ Fit summary       ------------------     
       coef   std err         t         P>|t|     2.5 %    97.5 %
d  4.153337  0.774872  5.360028  8.320895e-08  2.634615  5.672059

Athey, Susan, Julie Tibshirani, and Stefan Wager. 2019. “Generalized Random Forests.” The Annals of Statistics 47 (2): 1148–78.

Chernozhukov, Victor, Mert Demirer, Esther Duflo, and Ivan Fernandez-Val. 2018. “Generic Machine Learning Inference on Heterogeneous Treatment Effects in Randomized Experiments, with an Application to Immunization in India.” National Bureau of Economic Research.

Kallus, Nathan. 2022. “Treatment Effect Risk: Bounds and Inference.” arXiv Preprint arXiv:2201.05893.

Wager, Stefan, and Susan Athey. 2018. “Estimation and Inference of Heterogeneous Treatment Effects Using Random Forests.” Journal of the American Statistical Association 113 (523): 1228–42.