Matching calipers and the precision of index estimation
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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.
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