Estimators for Sequential and Simultaneous Nested NPIV

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Overview

This section summarizes the optimization targets for nested NPIV estimators under different function classes and links each target to practical implementations (RKHS, random forest/ensemble, neural network, sparse/regularized linear, and linear baselines).

Assumptions

  • Observations are i.i.d. draws of \((A, B, C, C', Y)\).

  • Function classes \(\mathcal{G}, \mathcal{H}, \mathcal{F}, \mathcal{F}'\) are chosen by the estimator family.

  • Penalization and/or norm constraints are used to regularize finite-sample minimax estimation.

Notation

  • \(A\): first-stage endogenous treatment/features.

  • \(B\): second-stage endogenous treatment/features.

  • \(C'\): first-stage instruments for recovering \(g\).

  • \(C\): second-stage instruments for recovering \(h\).

  • \(g\): first-stage bridge function, \(h\): structural function of primary interest.

Estimator Objectives

Sequential Nested NPIV:

Given observations \((A_i, B_i, C_i)\), an initial estimator \(\hat{g}\), and hyperparameter values \((\lambda, \mu)\), estimate

\[\hat{h} = \arg\min_{h \in \mathcal{H}} \left[ \sup_{f \in \mathcal{F}} \left( 2 \cdot \text{loss}(f, \hat{g}, h) - \text{penalty}(f, \lambda) \right) + \text{penalty}(h, \mu) \right]\]

where \(\text{penalty}(f, \lambda) = \mathbb{E}_m\{f(C)^2\} + \lambda \cdot \|f\|^2_{\mathcal{F}}\) and \(\text{penalty}(h, \mu) = \mu \cdot \|h\|^2_{\mathcal{H}}\).

Interpretation: the adversary \(f\) probes IV moment violations for fixed \(h\), while the learner regularizes complexity to stabilize inversion.

Sequential Nested NPIV: Ridge:

Given observations \((A_i, B_i, C_i)\), an initial estimator \(\hat{g}\), and hyperparameter \(\mu\), estimate

\[\hat{h} = \arg\min_{h \in \mathcal{H}} \left[ \sup_{f \in \mathcal{F}} \left( 2 \cdot \text{loss}(f, \hat{g}, h) - \text{penalty}(f) \right) + \text{penalty}(h, \mu) \right]\]

where \(\text{penalty}(f) = \mathbb{E}_m\{f(C)^2\}\) and \(\text{penalty}(h, \mu) = \mu \cdot \mathbb{E}_m\{h(B)^2\}\).

Interpretation: this variant emphasizes prediction-space regularization for \(h\) via \(\mathbb{E}[h(B)^2]\).

Simultaneous Nested NPIV:

Given observations \((A_i, B_i, C_i, C_i')\) and hyperparameters \((\mu', \mu)\), estimate

\[\begin{split}(\hat{g}, \hat{h}) = \arg\min_{g \in \mathcal{G}, h \in \mathcal{H}} \left[ \sup_{f' \in \mathcal{F}} \left( 2 \cdot \text{loss}(f', Y, g) - \text{penalty}(f') \right) + \text{penalty}(g, \mu') \right. \\ \left. + \sup_{f \in \mathcal{F}} \left( 2 \cdot \text{loss}(f, g, h) - \text{penalty}(f) \right) + \text{penalty}(h, \mu) \right]\end{split}\]

Interpretation: joint estimation can propagate first-stage uncertainty into the second stage; diagnostics in Estimation Diagnostics help assess conditioning before fitting.

Progressive Recipe

# Step 1: prepare arrays (A, B, C_prime, C, Y)
from nnpiv.rkhs import RKHS2IVL2

est = RKHS2IVL2(mu=0.1, mu_prime=0.1)

# Step 2: fit simultaneous nested NPIV
est.fit(A=A, B=B, C=C, D=C_prime, Y=Y)

# Step 3: inspect structural predictions
h_hat = est.predict(B_test)

Estimator Families