Matching calipers and the precision of index estimation

TL;DR

This paper analyzes how the precision of index estimation affects matching quality in causal inference, deriving formulas and calipers that ensure matched differences diminish as sample size grows, under Gaussian and sub-Gaussian assumptions.

Contribution

It provides a sharp characterization of index discrepancy behavior and introduces a caliper-based matching method that guarantees improved matching accuracy and estimator consistency.

Findings

Worst-case index differences decline to zero under certain growth conditions.

A matching caliper based on the derived formula improves matching precision.

Ensuring index discrepancy diminishes leads to consistent treatment effect estimation.

Abstract

This paper characterizes the precision of index estimation as it carries over into precision of matching. In a model assuming Gaussian covariates and making best-case assumptions about matching quality, it sharply characterizes average and worst-case discrepancies between paired differences of true versus estimated index values. In this optimistic setting, worst-case true and estimated index differences decline to zero if $p=o[n/(\log n)]$, the same restriction on model size that is needed for consistency of common index models. This remains so as the Gaussian assumption is relaxed to sub-gaussian, if in that case the characterization of paired index errors is less sharp. The formula derived under Gaussian assumptions is used as the basis for a matching caliper. Matching such that paired differences on the estimated index fall below this caliper brings the benefit that after matching, worst-case differences onan underlying index tend to 0 if $p = o\{[n/(\log n)]^{2/3}\}$. (With a linear index model, $p=o[n/(\log n)]$ suffices.) A proposed refinement of the caliper condition brings the same benefits without the sub-gaussian condition on covariates. When strong ignorability holds and the index is a well-specified propensity or prognostic score, ensuring in this way that worst-case matched discrepancies on it tend to 0 with increasing $n$ also ensures the consistency of matched estimators of the treatment effect.