"""
This module implements Debiased Machine Learning for Nonparametric Instrumental Variables (DML-npiv).
It provides tools for estimating causal effects using a combination of machine learning models and
instrumental variables techniques. The module supports cross-validation, kernel density estimation
for localization, and confidence interval computation with pointwise or uniform guarantees.
Classes:
DML_npiv: Main class for performing DML-npiv with various configuration options.
DML_npiv Methods:
__init__: Initialize the DML_npiv instance with data and model configurations.
_calculate_confidence_interval: Calculate confidence intervals for the estimates.
_localization: Perform localization using kernel density estimation.
_npivfit_outcome: Fit the outcome model using nonparametric instrumental variables.
_propensity_score: Estimate the propensity score.
_npivfit_action: Fit the action model using nonparametric instrumental variables.
_process_fold: Process a single fold for cross-validation.
_split_and_estimate: Split the data and estimate the model using cross-validation.
dml: Perform Debiased Machine Learning for Nonparametric Instrumental Variables.
"""
import numpy as np
from scipy.stats import norm
from sklearn.model_selection import KFold
from sklearn.linear_model import LogisticRegression
from sklearn.preprocessing import PolynomialFeatures
from statsmodels.nonparametric.kde import kernel_switch
import warnings
from tqdm import tqdm
import copy
import torch
from nnpiv.rkhs import ApproxRKHSIVCV
from joblib import Parallel, delayed
from scipy.optimize import minimize_scalar
device = torch.cuda.current_device() if torch.cuda.is_available() else None
def _get(opts, key, default):
"""
Retrieve the value associated with 'key' in 'opts', or return 'default' if not present.
Parameters
----------
opts : dict
Dictionary of options.
key : str
Key to look up in 'opts'.
default : any
Default value to return if 'key' is not found.
Returns
-------
any
Value associated with 'key' or 'default'.
"""
return opts[key] if (opts is not None and key in opts) else default
def _transform_poly(X, opts):
"""
Transform the input data X using polynomial features.
Parameters
----------
X : array-like
Input data.
opts : dict
Options dictionary containing the polynomial degree ('lin_degree').
Returns
-------
array-like
Transformed data.
"""
degree = _get(opts, 'lin_degree', 1)
if degree == 1:
return X
else:
trans = PolynomialFeatures(degree=degree, include_bias=False)
return trans.fit_transform(X)
def _fun_threshold_alpha(alpha, g):
"""
Auxiliary function for computation of optimal alpha for improvement in overlap: CHIM (Dealing with limited overlap in estimation of average treatment effects).
Richard K. Crump, V. Joseph Hotz, Guido W. Imbens, Oscar A. Mitnik
Biometrika, Volume 96, Issue 1, March 2009.
Parameters
----------
alpha : float
Alpha value.
g : array-like
Input array.
Returns
-------
float
Result of the threshold function.
"""
lambda_val = 1 / (alpha * (1 - alpha))
ind = (g <= lambda_val)
den = sum(ind)
num = ind * g
result = (2 * sum(num) / den - lambda_val) ** 2
return result
[docs]class DML_npiv:
"""
Debiased Machine Learning for Nonparametric Instrumental Variables (DML-npiv) class.
Parameters
----------
Y : array-like
Outcome variable.
D : array-like
Treatment variable.
Z : array-like
Instrumental variable.
W : array-like
Negative control outcome.
X1 : array-like, optional
Additional covariates.
V : array-like, optional
Localization covariates.
v_values : array-like, optional
Values for localization.
include_V : bool, optional
Include localization covariates in the model.
ci_type : str, optional
Type of confidence interval ('pointwise', 'uniform').
loc_kernel : str, optional
Kernel for localization. Options include 'gau', 'epa', 'uni', 'tri', etc.
bw_loc : str, optional
Bandwidth for localization.
estimator : str, optional
Estimator type ('MR', 'OR', 'IPW').
model1 : estimator, optional
Model for the first stage.
nn_1 : bool, optional
Use neural network for the first stage.
modelq1 : estimator, optional
Model for the second stage.
nn_q1 : bool, optional
Use neural network for the second stage.
alpha : float, optional
Significance level for confidence intervals.
n_folds : int, optional
Number of folds for estimation.
n_rep : int, optional
Number of repetitions for estimation.
random_seed : int, optional
Seed for random number generator.
prop_score : estimator, optional
Model for propensity score.
CHIM : bool, optional
Use CHIM method. Dropping observations with extreme values of the propensity score - CHIM (2009).
verbose : bool, optional
Print progress information.
fitargs1 : dict, optional
Arguments for fitting the first stage model.
fitargsq1 : dict, optional
Arguments for fitting the second stage model.
opts : dict, optional
Additional options.
"""
def __init__(self, Y, D, Z, W, X1=None,
V=None,
v_values=None,
include_V=True,
ci_type='pointwise',
loc_kernel='gau',
bw_loc='silverman',
estimator='MR',
model1=ApproxRKHSIVCV(kernel_approx='nystrom', n_components=100,
kernel='rbf', gamma=.1, delta_scale='auto',
delta_exp=.4, alpha_scales=np.geomspace(1, 10000, 10), cv=5),
nn_1=False,
modelq1=ApproxRKHSIVCV(kernel_approx='nystrom', n_components=100,
kernel='rbf', gamma=.1, delta_scale='auto',
delta_exp=.4, alpha_scales=np.geomspace(1, 10000, 10), cv=5),
nn_q1=False,
alpha=0.05,
n_folds=5,
n_rep=1,
random_seed=123,
prop_score=LogisticRegression(),
CHIM=False,
verbose=True,
fitargs1=None,
fitargsq1=None,
opts=None
):
"""
Initialize the DML_npiv instance with data and model configurations.
Parameters
----------
Y : array-like
Outcome variable.
D : array-like
Treatment variable.
Z : array-like
Instrumental variable.
W : array-like
Negative control outcome.
X1 : array-like, optional
Additional covariates.
V : array-like, optional
Localization covariates.
v_values : array-like, optional
Values for localization.
include_V : bool, optional
Include localization covariates in the model.
ci_type : str, optional
Type of confidence interval ('pointwise', 'uniform').
loc_kernel : str, optional
Kernel for localization. Options include 'gau', 'epa', 'uni', 'tri', etc.
bw_loc : str, optional
Bandwidth for localization.
estimator : str, optional
Estimator type ('MR', 'OR', 'IPW').
model1 : estimator, optional
Model for the first stage.
nn_1 : bool, optional
Use neural network for the first stage.
modelq1 : estimator, optional
Model for the second stage.
nn_q1 : bool, optional
Use neural network for the second stage.
alpha : float, optional
Significance level for confidence intervals.
n_folds : int, optional
Number of folds for estimation.
n_rep : int, optional
Number of repetitions for estimation.
random_seed : int, optional
Seed for random number generator.
prop_score : estimator, optional
Model for propensity score.
CHIM : bool, optional
Use CHIM method. Dropping observations with extreme values of the propensity score - CHIM (2009).
verbose : bool, optional
Print progress information.
fitargs1 : dict, optional
Arguments for fitting the first stage model.
fitargsq1 : dict, optional
Arguments for fitting the second stage model.
opts : dict, optional
Additional options.
"""
self.Y = Y
self.D = D
self.Z = Z
self.W = W
self.X1 = X1
self.V = V
self.v_values = v_values
self.include_V = include_V
self.ci_type = ci_type
self.loc_kernel = loc_kernel
self.bw_loc = bw_loc
self.estimator = estimator
self.model1 = copy.deepcopy(model1)
self.modelq1 = copy.deepcopy(modelq1)
self.nn_1 = nn_1
self.nn_q1 = nn_q1
self.prop_score = prop_score
self.CHIM = CHIM
self.alpha = alpha
self.n_folds = n_folds
self.n_rep = n_rep
self.random_seed = random_seed
self.verbose = verbose
self.fitargs1 = fitargs1
self.fitargsq1 = fitargsq1
self.opts = opts
if self.X1 is None:
if self.V is not None and self.include_V == True:
self.X = self.V
else:
self.X = np.ones((self.Y.shape[0], 1))
else:
if self.V is not None and self.include_V == True:
self.X = np.column_stack([self.X1, self.V])
else:
self.X = self.X1
lengths = [len(Y), len(D), len(Z), len(W), len(self.X)]
if len(set(lengths)) != 1:
raise ValueError("All input vectors must have the same length.")
if self.estimator not in ['MR', 'OR', 'IPW']:
warnings.warn(f"Invalid estimator: {estimator}. Estimator must be one of ['MR', 'OR', 'IPW']. Using MR instead.", UserWarning)
self.estimator = 'MR'
if self.ci_type not in ['pointwise', 'uniform']:
warnings.warn(f"Invalid confidence interval type: {ci_type}. Confidence interval type must be one of ['pointwise', 'uniform']. Using pointwise instead.", UserWarning)
self.ci_type = 'pointwise'
if self.ci_type == 'uniform' and (self.v_values is None or self.v_values.shape[0] == 1 or self.V is None):
warnings.warn(f"Uniform confidence intervals are not supported for less than one localization value. Using pointwise instead.", UserWarning)
self.ci_type = 'pointwise'
if self.loc_kernel not in list(kernel_switch.keys()):
warnings.warn(f"Invalid kernel: {loc_kernel}. Kernel must be one of {list(kernel_switch.keys())}. Using gau instead.", UserWarning)
self.loc_kernel = 'gau'
if isinstance(self.bw_loc, str):
if self.bw_loc not in ['silverman', 'scott']:
warnings.warn(f"Invalid bw rule: {bw_loc}. Bandwidth rule must be one of ['silverman', 'scott'] or provided by the user. Using silverman instead.", UserWarning)
self.bw_loc = 'silverman'
if self.V is not None:
if self.v_values is None:
warnings.warn(f"v_values is None. Computing localization around mean(V).", UserWarning)
self.v_values = np.mean(self.V, axis=0)
def _calculate_confidence_interval(self, theta, theta_var, theta_cov):
"""
Calculate the confidence interval for the given estimates.
Parameters
----------
theta : array-like
Estimated values.
theta_var : array-like
Variance of the estimates.
theta_cov : array-like
Covariance matrix of the estimates.
Returns
-------
array-like
Lower and upper bounds of the confidence intervals.
"""
n = self.Y.shape[0]
if self.ci_type == 'pointwise':
z_alpha_half = norm.ppf(1 - self.alpha / 2)
margin_of_error = z_alpha_half * np.sqrt(theta_var / n)
else:
S = np.diag(np.diag(theta_cov))
S_inv_sqrt = np.diag(1.0 / np.sqrt(np.diag(S)))
Sigma_hat = S_inv_sqrt @ theta_cov @ S_inv_sqrt
# Sample Q from N(0, Sigma_hat)
Q_samples = np.random.multivariate_normal(np.zeros(theta.shape[0]), Sigma_hat, 5000)
# Compute the (1 - alpha) quantile of the sampled |Q|_infty
Q_infinity_norms = np.max(np.abs(Q_samples), axis=1)
c_alpha = np.quantile(Q_infinity_norms, 1 - self.alpha)
margin_of_error = c_alpha * np.sqrt(np.diag(theta_cov) / n)
lower_bound = theta - margin_of_error
upper_bound = theta + margin_of_error
return np.column_stack((lower_bound, upper_bound))
def _localization(self, V, v_val, bw):
"""
Perform localization using kernel density estimation.
Parameters
----------
V : array-like
Localization covariates.
v_val : array-like
Values for localization.
bw : float
Bandwidth for localization.
Returns
-------
array-like
Weights for localization.
"""
if kernel_switch[self.loc_kernel]().domain is None:
def K(x):
return kernel_switch[self.loc_kernel]()(x)
else:
def K(x):
y = kernel_switch[self.loc_kernel]()(x)*((kernel_switch[self.loc_kernel]().domain[0]<=x) & (x<=kernel_switch[self.loc_kernel]().domain[1]))
return y
v = (V-v_val)/bw
KK = np.prod(list(map(K, v)),axis=1)
omega = np.mean(KK,axis=0)
ell = KK/omega
return ell.reshape(-1,1)
def _npivfit_outcome(self, Y, D, X, Z):
"""
Fit the outcome model using nonparametric instrumental variables.
Parameters
----------
Y : array-like
Outcome variable.
D : array-like
Treatment variable.
X : array-like
Covariates.
Z : array-like
Instrumental variable.
Returns
-------
tuple
Fitted models for treatment and control groups.
"""
bridge_ = [None]*2
if self.estimator == 'MR' or self.estimator == 'OR':
model_1 = copy.deepcopy(self.model1)
# First stage
if self.nn_1==True:
Y, X, Z = tuple(map(lambda x: torch.Tensor(x), [Y, X, Z]))
if self.nn_1==False:
X = _transform_poly(X,self.opts)
Z = _transform_poly(Z,self.opts)
for d in [0,1]:
ind = np.where(D==d)[0]
Y1 = Y[ind]
X1 = X[ind,:]
Z1 = Z[ind]
if self.fitargs1 is not None:
bridge_[d] = copy.deepcopy(model_1).fit(Z1, X1, Y1, **self.fitargs1)
else:
bridge_[d] = copy.deepcopy(model_1).fit(Z1, X1, Y1)
return bridge_[1], bridge_[0]
def _propensity_score(self, X, W, D):
"""
Estimate the propensity score.
Parameters
----------
X : array-like
Covariates.
W : array-like
Control variable.
D : array-like
Treatment variable.
Returns
-------
tuple
Estimated propensity scores and threshold alpha.
"""
model_ps = copy.deepcopy(self.prop_score)
X1 = np.column_stack((X,W))
# First stage
model_ps.fit(X1, D.flatten())
ps_hat_1 = model_ps.predict_proba(X1)[:,1]
# Overlap assumption
ps_hat_1 = np.where(ps_hat_1 == 1, 0.99, ps_hat_1)
ps_hat_1 = np.where(ps_hat_1 == 0, 0.01, ps_hat_1)
if self.CHIM==True:
# Dropping observations with extreme values of the propensity score - CHIM (2009)
# One finds the smallest value of \alpha\in [0,0.5] s.t.
# $\lambda:=\frac{1}{\alpha(1-\alpha)}$
# $2\frac{\sum 1(g(X)\leq\lambda)*g(X)}{\sum 1(g(X)\leq\lambda)}-\lambda\geq 0$
#
# Equivalently the first value of alpha (in increasing order) such that the constraint is achieved by equality
# (as the constraint is a monotone increasing function in alpha)
g_values = [1/(ps_hat_1*(1-ps_hat_1))]
optimized_alphas = []
for g in g_values:
def _objective_function(alpha):
return _fun_threshold_alpha(alpha, g)
result = minimize_scalar(_objective_function, bounds=(0.001, 0.499))
optimized_alphas.append(result.x)
alfa = max(optimized_alphas)
else:
alfa = 0.0
return ps_hat_1.reshape(-1,1), alfa
def _npivfit_action(self, ps_hat_1, W, X, Z, alfa=0.0):
"""
Fit the action model using nonparametric instrumental variables.
Parameters
----------
ps_hat_1 : array-like
Estimated propensity scores.
W : array-like
Control variable.
X : array-like
Covariates.
Z : array-like
Instrumental variable.
alfa : float, optional
Threshold alpha for propensity scores.
Returns
-------
tuple
Fitted models for treated and control groups.
"""
bridge_ = [None]*2
if self.estimator == 'MR' or self.estimator == 'IPW':
mask = np.where((ps_hat_1 >= alfa) & (ps_hat_1 <= 1 - alfa))[0]
ps_hat_1 = ps_hat_1[mask]
ps_hat_0 = 1 - ps_hat_1
W = W[mask]
X = X[mask,:]
Z = Z[mask]
model_q1 = copy.deepcopy(self.modelq1)
# First stage
if self.nn_q1==True:
ps_hat_1, ps_hat_0, W, X, Z = tuple(map(lambda x: torch.Tensor(x), [ps_hat_1, ps_hat_0, W, X, Z]))
if self.nn_q1==True:
A2 = torch.cat((X,W),1)
A1 = torch.cat((X,Z),1)
else:
A2 = _transform_poly(np.column_stack((X,W)),self.opts)
A1 = _transform_poly(np.column_stack((X,Z)),self.opts)
if self.fitargsq1 is not None:
bridge_[0] = copy.deepcopy(model_q1).fit(A2, A1, 1/ps_hat_0, **self.fitargsq1)
bridge_[1] = copy.deepcopy(model_q1).fit(A2, A1, 1/ps_hat_1, **self.fitargsq1)
else:
bridge_[0] = copy.deepcopy(model_q1).fit(A2, A1, 1/ps_hat_0)
bridge_[1] = copy.deepcopy(model_q1).fit(A2, A1, 1/ps_hat_1)
return bridge_[1], bridge_[0]
def _process_fold(self, fold_idx, train_data, test_data):
"""
Process a single fold for cross-validation.
Parameters
----------
fold_idx : int
Fold index.
train_data : tuple
Training data for the fold.
test_data : tuple
Testing data for the fold.
Returns
-------
array-like
Estimated moment functions for the test data.
"""
train_Y, test_Y = train_data[0], test_data[0]
train_D, test_D = train_data[1], test_data[1]
train_W, test_W = train_data[2], test_data[2]
train_X, test_X = train_data[3], test_data[3]
train_Z, test_Z = train_data[4], test_data[4]
if self.V is not None:
train_V, test_V = train_data[5], test_data[5]
if self.estimator == 'MR' or self.estimator == 'OR':
gamma_1, gamma_0 = self._npivfit_outcome(train_Y, train_D, train_X, train_Z)
if self.estimator == 'MR' or self.estimator == 'IPW':
ps_hat_1, alfa = self._propensity_score(train_X, train_W, train_D)
q_1, q_0 = self._npivfit_action(ps_hat_1, train_W, train_X, train_Z, alfa=alfa)
# Evaluate the estimated moment functions using test_data
if self.estimator == 'MR' or self.estimator == 'OR':
if self.nn_1 == True:
test_X = torch.Tensor(test_X)
gamma_1_hat = gamma_1.predict(test_X.to(device),
model='avg', burn_in=_get(self.opts, 'burnin', 0)).reshape(-1, 1)
gamma_0_hat = gamma_0.predict(test_X.to(device),
model='avg', burn_in=_get(self.opts, 'burnin', 0)).reshape(-1, 1)
else:
gamma_1_hat = gamma_1.predict(_transform_poly(test_X, opts=self.opts)).reshape(-1, 1)
gamma_0_hat = gamma_0.predict(_transform_poly(test_X, opts=self.opts)).reshape(-1, 1)
if self.estimator == 'MR' or self.estimator == 'IPW':
if self.nn_q1 == True:
test_X, test_Z = tuple(map(lambda x: torch.Tensor(x), [test_X, test_Z]))
q_1_hat = q_1.predict(torch.cat((test_X, test_Z), 1).to(device),
model='avg', burn_in=_get(self.opts, 'burnin', 0)).reshape(-1, 1)
q_0_hat = q_0.predict(torch.cat((test_X, test_Z), 1).to(device),
model='avg', burn_in=_get(self.opts, 'burnin', 0)).reshape(-1, 1)
else:
q_1_hat = q_1.predict(_transform_poly(np.column_stack((test_X, test_Z)), opts=self.opts)).reshape(-1, 1)
q_0_hat = q_0.predict(_transform_poly(np.column_stack((test_X, test_Z)), opts=self.opts)).reshape(-1, 1)
# Calculate the score function depending on the estimator
if self.estimator == 'MR':
psi_hat = (gamma_1_hat-gamma_0_hat +
test_D * q_1_hat * (test_Y - gamma_1_hat) - (1-test_D) * q_0_hat * (test_Y - gamma_0_hat))
if self.estimator == 'OR':
psi_hat = gamma_1_hat-gamma_0_hat
if self.estimator == 'IPW':
psi_hat = test_D * q_1_hat * test_Y - (1 - test_D) * q_0_hat * test_Y
# Localization
if self.V is not None:
if isinstance(self.bw_loc, str):
if self.bw_loc == 'silverman':
IQR = np.percentile(train_V, 75, axis=0)-np.percentile(train_V, 25, axis=0)
A = np.min([np.std(train_V, axis=0), IQR/1.349], axis=0)
n = train_V.shape[0]
bw = .9 * A * n ** (-0.2)
elif self.bw_loc == 'scott':
A = np.std(train_V, axis=0)
n = train_V.shape[0]
bw = 1.059 * A * n ** (-0.2)
else:
if len(self.bw_loc)==1:
bw = np.ones((train_V.shape[1]))*self.bw_loc[0]
else:
if len(self.bw_loc)==train_V.shape[1]:
bw = self.bw_loc
else:
warnings.warn(f"bw_loc has incorrect length. Using first element instead.", UserWarning)
bw = np.ones((train_V.shape[1]))*self.bw_loc[0]
ell = [self._localization(test_V, v, bw) for v in self.v_values]
ell = np.column_stack(ell)
psi_hat = ell * psi_hat
# Print progress bar using tqdm
if self.verbose==True:
self.progress_bar.update(1)
return psi_hat
def _split_and_estimate(self):
"""
Split the data and estimate the model for each fold.
Returns
-------
tuple
Estimated values, variances, and confidence intervals.
"""
theta = []
theta_var = []
theta_cov = []
for rep in range(self.n_rep):
if self.verbose==True:
print(f"Rep: {rep+1}")
self.progress_bar = tqdm(total=self.n_folds, position=0)
kf = KFold(n_splits=self.n_folds, shuffle=True, random_state=self.random_seed+rep)
if self.V is None:
fold_results = Parallel(n_jobs=-1, backend='threading')(
delayed(self._process_fold)(
fold_idx,
(self.Y[train_index], self.D[train_index], self.W[train_index],
self.X[train_index], self.Z[train_index]),
(self.Y[test_index], self.D[test_index], self.W[test_index],
self.X[test_index], self.Z[test_index]))
for fold_idx, (train_index, test_index) in enumerate(kf.split(self.Y))
)
else:
fold_results = Parallel(n_jobs=-1, backend='threading')(
delayed(self._process_fold)(
fold_idx,
(self.Y[train_index], self.D[train_index], self.W[train_index],
self.X[train_index], self.Z[train_index], self.V[train_index]),
(self.Y[test_index], self.D[test_index], self.W[test_index],
self.X[test_index], self.Z[test_index], self.V[test_index]))
for fold_idx, (train_index, test_index) in enumerate(kf.split(self.Y))
)
if self.verbose==True:
self.progress_bar.close()
# Calculate the average of psi_hat_array for each rep
psi_hat_array = np.concatenate(fold_results, axis=0)
theta_rep = np.mean(psi_hat_array, axis=0)
theta_var_rep = np.var(psi_hat_array, axis=0, ddof=1)
theta_cov_rep = np.cov(psi_hat_array, rowvar=False)
# Store results for each rep
theta.append(theta_rep)
theta_var.append(theta_var_rep)
theta_cov.append(theta_cov_rep)
# Calculate the overall average of theta and theta_var
theta_hat = np.mean(np.stack(theta, axis=0), axis=0)
theta_var_hat = np.mean(np.stack(theta_var, axis=0), axis=0)
theta_cov_hat = np.mean(np.stack(theta_cov, axis=0), axis=0)
# Calculate the confidence interval
confidence_interval = self._calculate_confidence_interval(theta_hat, theta_var_hat, theta_cov_hat)
return theta_hat, theta_var_hat, confidence_interval, theta_cov_hat
def dml(self):
"""
Perform Debiased Machine Learning for Nonparametric Instrumental Variables.
Returns
-------
tuple
Estimated values, variances, and confidence intervals.
"""
theta, theta_var, confidence_interval, theta_cov_hat = self._split_and_estimate()
if self.V is None:
return theta[0], theta_var[0], confidence_interval[0]
else:
return theta, theta_cov_hat, confidence_interval