🔎 The Problem With Current Matching Theory

Most existing inferential theories for matching assume simple random sampling and require exact matches. In practice, researchers often use stratified sampling designs and then perform ex post stratification—on a propensity score, a distance metric, or the covariates—to find approximate matches. That mismatch erodes the statistical properties the standard theories are meant to guarantee.

🧭 What This Paper Proposes

This work replaces simple random sampling with stratified sampling as the foundational axiom for a theory of inference for matching. Showing that sampling type is an axiom rather than a fragile assumption, the paper demonstrates that the theoretical implications remain coherent and valid under stratified designs.

📐 Core Mechanisms and Results

• Replaces the simple random sampling axiom with stratified sampling and derives the corresponding inferential implications.
• Demonstrates that estimator properties under this framework are easier to understand and verify than under existing theories.
• Shows the framework holds even when matching is approximate (ex post stratification on propensity scores, distance metrics, or covariates).

✅ Key Advantages for Researchers

• Treats matching as a straightforward preprocessing step to reduce model dependence; after matching, familiar inferential techniques and standard uncertainty calculations apply.
• Avoids several unattractive features of prior theories: assumptions hidden in data analysis rather than stated up front, heavy reliance on asymptotics, unfamiliar estimators, and complex variance calculations.
• Accommodates binary, multicategory, and continuous treatment variables from the outset.

🔬 Extensions and Practical Scope

• Provides straightforward extensions for imperfect treatment assignment and for different versions of treatments.
• Ensures that common practices—like ex post stratification to obtain approximate matches—need not nullify inferential guarantees when the stratified-sampling axiom is adopted.

⚠️ Why It Matters

This reconceptualization makes the properties of matching-based estimators more transparent and practically useful for researchers who design stratified studies, enabling reliable use of matching as preprocessing without sacrificing valid inference.