  This paper studies inference on the average treatment effect in experiments
in which treatment status is determined according to "matched pairs" and it is
additionally desired to adjust for observed, baseline covariates to gain
further precision. By a "matched pairs" design, we mean that units are sampled
i.i.d. from the population of interest, paired according to observed, baseline
covariates and finally, within each pair, one unit is selected at random for
treatment. Importantly, we presume that not all observed, baseline covariates
are used in determining treatment assignment. We study a broad class of
estimators based on a "doubly robust" moment condition that permits us to study
estimators with both finite-dimensional and high-dimensional forms of covariate
adjustment. We find that estimators with finite-dimensional, linear adjustments
need not lead to improvements in precision relative to the unadjusted
difference-in-means estimator. This phenomenon persists even if the adjustments
are interacted with treatment; in fact, doing so leads to no changes in
precision. However, gains in precision can be ensured by including fixed
effects for each of the pairs. Indeed, we show that this adjustment is the
"optimal" finite-dimensional, linear adjustment. We additionally study two
estimators with high-dimensional forms of covariate adjustment based on the
LASSO. For each such estimator, we show that it leads to improvements in
precision relative to the unadjusted difference-in-means estimator and also
provide conditions under which it leads to the "optimal" nonparametric,
covariate adjustment. A simulation study confirms the practical relevance of
our theoretical analysis, and the methods are employed to reanalyze data from
an experiment using a "matched pairs" design to study the effect of
macroinsurance on microenterprise.
