A Martingale Representation for Matching Estimators

By Alberto Abadie, Guido W. Imbens. 2012

104 citations recorded by the source corpus.

Matching estimators are widely used in statistical data analysis. However, the large sample distribution of matching estimators has been derived only for particular cases. This article establishes a martingale representation for matching estimators. This representation allows the use of martingale limit theorems to derive the large sample distribution of matching estimators. As an illustration of the applicability of the theory, we derive the asymptotic distribution of a matching estimator when matching is carried out without replacement, a result previously unavailable in the literature. In addition, we apply the techniques proposed in this article to derive a correction to the standard error of a sample mean when missing data are imputed using the “hot deck,” a matching imputation method widely used in the Current Population Survey (CPS) and other large surveys in the social sciences. We demonstrate the empirical relevance of our methods using two Monte Carlo designs based on actual datasets. In these Monte Carlo exercises, the large sample distribution of matching estimators derived in this article provides an accurate approximation to the small sample behavior of these estimators. In addition, our simulations show that standard errors that do not take into account hot-deck imputation of missing data may be severely downward biased, while standard errors that incorporate the correction for hot-deck imputation perform extremely well. This article has online supplementary materials.

Subfields: Statistics and Probability

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Citations in the corpus

Multivariate Matching Methods That Are Monotonic Imbalance Bounding (2011, background): “Finally, if the reservoir of control units is sufficiently large, it is possible to derive, following the proof of proposition 1 in Abadie and Imbens (2009), an exponential bound on the probability that the number of CEM strata with unmatched treated units remains positive.”
Asymptotic Variance of Semiparametric Estimators With Generated Regressors (2013, methodology): “i=1 (h ( (xi; ' (xi; zi; b ))) h ( (xi; ' (xi; zi; )))) + op (1) (iii)”
Matching on the Estimated Propensity Score (2016, methodology): “…E[YN,i|WN,i = w,F (X ′N,iθ ∗) = F ] for some ε > 0 and ‖θ∗ − θ‖ ≤ ε. Proof of Proposition 2 (Sketch): In this proof, we first establish a extend the martingale representation of matching estimators (Abadie and Imbens, 2009) to the propensity score matching estimator studied in this article.”
The Finite Sample Performance of Inference Methods for Propensity Score Matching and Weighting Estimators (2018, unknown): “Then, as shown in Abadie and Imbens (2012), we can write the matching estimator as √ n(θ̂match − θ ) = T1n + T2n + op(1), where T1n = √ n n1 n∑ i=1 Di(μ( p̂(Xi), 1) − μ( p̂(Xi), 0) − θ ), T2n = √ n n1 n∑ i=1 (Di − (1 − Di)Ki) (Yi − μ( p̂(Xi),Di)) .”
Shining a spotlight on small rural businesses: How does their performance compare with urban? (2019, methodology): “If balancing tests are passed, the average treatment effect is calculated as the mean difference in the outcome across these two groups (Abadie and Imbens, 2012).”
Econometric Methods for Program Evaluation (2018, background): “Abadie and Imbens (2006, 2011, 2012) derive large sample results for estimators that match directly on the covariates, X. Abadie and Imbens (2016) provide large sample results for the case where matching is done on an estimated propensity score.”
Propensity score matching and subclassification in observational studies with multi‐level treatments (2015, unknown): “These re-sults motivate us to compare the seven estimators in settings ) Abadie and Imbens (2006) variance estimator for COV matching and PSSM; (3) Abadie and Imbens (2012) variance estimator for PPSM and GPSM. with a large number of treatment levels, and where some of the treatment levels have low…”
When Should We Use Unit Fixed Effects Regression Models for Causal Inference with Longitudinal Data? (2019, methodology): “This equivalence allows us to construct simple model-based standard errors instead of more complex and computationally intensive standard errors proposed in the literature (e.g., Abadie and Imbens 2006, 2012; Otsu and Rai 2017).”