Nested Nonparametric Instrumental Variable Regression

Documentation Status

Overview

This package aims to solve or estimate nonparametrically nested moment conditions. We analyze the closed form or approximate solutions under different function classes for the following estimators:

Estimators

NPIV

Given set of observations \((Y, A, C')_i\); we want to estimate nonparametrically \(g\) in \(\mathbb{E}\left[Y | C'\right]= \mathbb{E}\left[g(A) | C'\right]\), where A is the set of endogenous variables, and C’ the set of instruments. We solve the inverse problem adversarially:

\[\hat{g} = \arg \min_{g \in \mathcal{G}} \max_{f' \in \mathcal{F'}} \mathbb{E}_n \left[ 2 \left\{ g(A) - Y \right\} f'(C') - f'(C')^2 \right] + \mu' \mathbb{E}_n \{ g(A)^2 \}\]

and we also consider norm regularization instead of ridge regularization:

\[\hat{g} = \arg \min_{g \in \mathcal{G}} \max_{f' \in \mathcal{F'}} \mathbb{E}_n \left[ 2 \left\{ g(A) - Y \right\} f'(C') - f'(C')^2 \right] - \lambda \|f\|_{\mathcal{F}}^2 + \mu' \|g\|_{\mathcal{G}}^2\]

Nested NPIV

Whenever we have the set of observations \((Y, A, B, C, C')_i\); and want to solve the system:

\[\mathbb{E}\left[Y | C'\right]= \mathbb{E}\left[g(A) | C'\right]\]
\[\mathbb{E}\left[g(A) | C\right]= \mathbb{E}\left[h(B) | C\right]\]

we estimate \(g\) and \(h\) by solving:

\[(\hat{g},\hat{h}) = \arg \min_{g \in \mathcal{G}, h \in \mathcal{H}} \max_{f' \in \mathcal{F}} \mathbb{E}_n \left[ 2 \left\{ g(A) - Y \right\} f'(C') - f'(C')^2 \right] + \mu' \mathbb{E}_n \{ g(A)^2 \}\]
\[+ \max_{f \in \mathcal{F}} \mathbb{E}_n \left[ 2 \left\{ h(B) - g(A) \right\} f(C) - f(C)^2 \right] + \mu \mathbb{E}_n \{ h(B)^2 \}\]

and similarly when using norm-regularization.

Implementation

Longitudinal Estimation

This package implements longitudinal estimation of functions \(g\) and \(h\) for several function classes:

  • RKHS

  • Random Forest

  • Neural Networks

  • Sparse Linear

  • Linear

Semiparametric Estimation

The package also implements debiased machine learning for estimation of a functional of the nuisance longitudinal parameter \(g\) or \(h\):

\[\theta = \mathbb{E}\left[h(B)\right]\]

based on constructing orthogonal moments for:

  • Mediation analysis

  • Long term effect

Contents

Note

This project is under active development.

References

meza2024nested

Meza, I., & Singh, R. (2024). Nested Nonparametric Instrumental Variable Regression: Long Term, Mediated, and Time Varying Treatment Effects. https://doi.org/10.48550/arXiv.2112.14249