Some Notes on Using Regression on Paired Data from Statistical Matching

This paper examines the use of OLS regression on paired data obtained through statistical matching to estimate treatment effects in time-series contexts where the baseline outcome is unobservable during the treatment period. The study formalizes a model in which treatment produces both a fixed shift and a proportional change. Because the untreated baseline must be estimated using matching procedures, the resulting regression framework violates the standard exogeneity assumption required for unbiased ordinary least squares (OLS) estimation. The paper tries to demonstrate analytically that bias. The analysis further shows that restricting estimation to selected sub-samples can reduce bias, although at the cost of increased variance and reduced efficiency. In the special case of proportional treatment effects, the no-constant least squares estimator is shown to underestimate the treatment parameter, while a method-of-moments estimator provides lower distortion under certain assumptions. The findings highlight important limitations of regression-based causal inference using statistically matched time-series data and suggest avenues for improving estimator performance through sample selection and alternative estimation strategies.