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:
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
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.
This package implements longitudinal estimation of functions g and h for several function classes:
- RKHS
- Random Forest
- Neural Networks
- Sparse Linear
- Linear
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