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On the limiting variance of matching estimators

Songliang Chen, Fang Han

Abstract

This paper examines the limiting variance of nearest neighbor matching estimators for average treatment effects with a fixed number of matches. We present, for the first time, a closed-form expression for this limit. Here the key is the establishment of the limiting second moment of the catchment area's volume, which resolves a question of Abadie and Imbens. At the core of our approach is a new universality theorem on the measures of high-order Voronoi cells, extending a result by Devroye, Györfi, Lugosi, and Walk.

On the limiting variance of matching estimators

Abstract

This paper examines the limiting variance of nearest neighbor matching estimators for average treatment effects with a fixed number of matches. We present, for the first time, a closed-form expression for this limit. Here the key is the establishment of the limiting second moment of the catchment area's volume, which resolves a question of Abadie and Imbens. At the core of our approach is a new universality theorem on the measures of high-order Voronoi cells, extending a result by Devroye, Györfi, Lugosi, and Walk.

Paper Structure

This paper contains 13 sections, 7 theorems, 133 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Related literature
  3. Notation and set-up
  4. Nearest neighbor matching
  5. Theory
  6. Measures of high-order Voronoi cells
  7. Proofs of the main results
  8. Proof of Theorem \ref{['thm3.1']}
  9. Proof of Theorem \ref{['thm4.2']}
  10. Proofs of the rest results
  11. Proof of Lemma \ref{['lem5.1']}
  12. Proof of Lemma \ref{['lem5.2']}
  13. Proof of Lemma \ref{['lem5.3']}

Key Result

Theorem 1.1

Under certain regularity conditions and assuming $M$ to be fixed, the following hold true.

Theorems & Definitions (9)

  • Theorem 1.1: Main theorem, informal
  • Theorem 3.1: abadie2006largeabadie2011bias
  • Theorem 3.2: Main theorem
  • Definition 4.1: $M$-th NN map
  • Definition 4.2: Catchment area
  • Theorem 4.1
  • Lemma 5.1
  • Lemma 5.2: Distribution-free limits
  • Lemma 5.3