Instrument Space Selection for Kernel Maximum Moment Restriction

TL;DR

This paper introduces a systematic method for selecting the instrument space in kernel maximum moment restriction models, improving parameter estimation by balancing identifiability and complexity.

Contribution

It proposes the least identifiable instrument space (LIIS) principle, combining identifiability testing and complexity measures for optimal instrument space selection.

Findings

The method consistently identifies the optimal instrument space.

Simulation results demonstrate improved parameter estimation accuracy.

The approach effectively balances model identifiability and complexity.

Abstract

Kernel maximum moment restriction (KMMR) recently emerges as a popular framework for instrumental variable (IV) based conditional moment restriction (CMR) models with important applications in conditional moment (CM) testing and parameter estimation for IV regression and proximal causal learning. The effectiveness of this framework, however, depends critically on the choice of a reproducing kernel Hilbert space (RKHS) chosen as a space of instruments. In this work, we presents a systematic way to select the instrument space for parameter estimation based on a principle of the least identifiable instrument space (LIIS) that identifies model parameters with the least space complexity. Our selection criterion combines two distinct objectives to determine such an optimal space: (i) a test criterion to check identifiability; (ii) an information criterion based on the effective dimension of RKHSs as a complexity measure. We analyze the consistency of our method in determining the LIIS, and demonstrate its effectiveness for parameter estimation via simulations.