Semiparametric Estimation

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Overview

The goal is general-purpose learning and inference for a nonparametric causal parameter \(\theta_0 \in \mathbb{R}\). Many targets admit multiply robust moment functions with nuisance parameters \((\nu_0, \delta_0, \alpha_0, \eta_0)\). This section summarizes the debiased machine learning (DML) meta-algorithm used in the package to convert nuisance estimators into valid point estimates and confidence intervals.

Assumptions

  • Nuisance estimators are trained on auxiliary folds and evaluated out-of-fold.

  • Each nuisance learner converges in mean-square error at rates compatible with DML remainder control.

  • Moment conditions are evaluated on held-out folds for orthogonalization.

Notation

  • \(\theta_0\): scalar target parameter.

  • \((\nu_0, \delta_0, \alpha_0, \eta_0)\): nuisance components entering the orthogonal score.

  • \(I_\ell\): fold index set; \(I_\ell^c\) its complement.

Debiased Machine Learning Meta-Algorithm

Given a sample \((Y_i, W_i)\) (\(i = 1, \ldots, n\)), partition the sample into folds \(I_\ell\) (\(\ell = 1, \ldots, L\)). Denote by \(I^c_\ell\) the complement of \(I_\ell\).

  1. For each fold \(\ell\), estimate \((\hat{\nu}_\ell, \hat{\delta}_\ell, \hat{\alpha}_\ell, \hat{\eta}_\ell)\) from observations in \(I^c_\ell\).

  2. Estimate \(\theta_0\) as

    \[\hat{\theta} = \frac{1}{n} \sum_{\ell=1}^L \sum_{i \in I_\ell} \left[ \hat{\nu}_\ell(W_i) + \hat{\alpha}_\ell(W_i)\{Y_i - \hat{\delta}_\ell(W_i)\} + \hat{\eta}_\ell(W_i)\{\hat{\delta}_\ell(W_i) - \hat{\nu}_\ell(W_i)\} \right].\]

  3. Estimate the \((1 - \alpha)100\%\) confidence interval as \(\hat{\theta} \pm c_\alpha \hat{\sigma} n^{-1/2}\), where \(c_\alpha\) is the \(1 - \alpha/2\) quantile of the standard Gaussian and

    \[\hat{\sigma}^2 = \frac{1}{n} \sum_{\ell=1}^L \sum_{i \in I_\ell} \left[ \hat{\nu}_\ell(W_i) + \hat{\alpha}_\ell(W_i)\{Y_i - \hat{\delta}_\ell(W_i)\} + \hat{\eta}_\ell(W_i)\{\hat{\delta}_\ell(W_i) - \hat{\nu}_\ell(W_i)\} - \hat{\theta} \right]^2.\]

Interpretation: fold-wise orthogonalization reduces sensitivity to nuisance estimation errors, enabling practical inference with flexible first-stage learners.

Localized Ratio Targets

Let \(H_i(v)\) denote an uncentered score contribution and let \(a_i(v)\) denote the loading that defines the target. The estimator solves

\[\mathbb{E}_n[H_i(v)-a_i(v)\theta(v)]=0, \qquad \widehat\theta(v) =\frac{\mathbb{E}_n[H_i(v)]}{\mathbb{E}_n[a_i(v)]}.\]

The corresponding centered score contribution is

\[\widehat\phi_i(v) =\frac{H_i(v)-a_i(v)\widehat\theta(v)} {\mathbb{E}_n[a_i(v)]}.\]

For an ordinary average, \(a_i=1\). For kernel localization, \(H_i(v)=\ell_i(v)H_i\) and \(a_i(v)=\ell_i(v)\), so when the kernel loading has empirical mean one, \(\widehat\phi_i(v)=\ell_i(v)\{H_i-\widehat\theta(v)\}\). A subgroup target uses a normalized group loading \(q_i\); a localized subgroup target uses \(q_i\ell_i(v)\). Thus localization changes not only the numerator of the point estimate but also the centering used for variance and covariance.

This correction applies to every score route. If the base score is a valid influence-function score (notably MR), the centered contribution is its localized influence value; OR, IPW, and hybrid retain their nuisance correctness and rate requirements. For a conditional causal interpretation, include V in nuisance conditioning with include_V=True. The target is finite-bandwidth, conditional on the selected grid and bandwidth.

Progressive Recipe

# Step 1: configure nuisance learners and data blocks
import numpy as np
from nnpiv.rkhs import ApproxRKHSIVCV
from nnpiv.semiparametrics import DML_npiv

g_model = ApproxRKHSIVCV(kernel_approx="nystrom", n_components=200, cv=3)

# Step 2: fit DML NPIV estimator
dml = DML_npiv(Y=Y, D=D, Z=Z, W=W, model1=g_model, modelq1=g_model, n_folds=5)
theta, var, ci = dml.dml()

# Step 3: inspect estimate and uncertainty
se = np.sqrt(var / len(Y))
print(theta, se, ci)

Model-Specific Semiparametric APIs