Treatment allocation for linear models

Methods of systematically balanced treatment allocation for economic experiments, as an alternative to random allocation, are gaining increasing attention in recent years. This paper analyzes the benefits and the limits of a systematic allocation of treatments within a linear model framework. Linear models do not necessarily require the treatment allocation to be random. Since the variance of the treatment estimator within linear models does not depend on the realization of the dependent variable, whenever the covariate information is available prior to allocating treatments it is possible to allocate treatments in a way that minimizes the variance of the treatment estimator. I show that in each experiment satisfying the linear model assumptions, there exists at least one deterministic optimal design, i.e., a deterministic way of allocating treatments that minimizes the variance of the treatment estimator over all alternative ways of allocating treatments. In finite samples, optimal design reduces the variance of the treatment estimator and increases statistical power compared to random allocation. For a given linear model with m covariates, optimal design reduces the required sample size of the experiment to achieve a predefined power by approximately m. However, asymptotically, as the sample size goes to infinity, neither optimal design nor any alternative design yields any benefit over random allocation.