Comparative Statics for Difference-in-Differences

The stable unit treatment value assumption (SUTVA) in causal estimation rules out spillover effects, but spillover effects are the hallmark of many economic models. Testing model predictions with techniques that employ SUTVA are thus problematic. To address this issue, we first show that without the no interference component of SUTVA, the population difference-in-difference (DiD) identifies the difference in the average potential outcomes between the treated and untreated. We call this estimand the marginal average treatment effect among the treated with spillovers (MATTS). Then, in the context of a model whose equilibrium is characterized by a system of smooth equations, we provide comparative statics results which restrict the sign of MATTS. Specifically, we show that MATTS is positive for any nontrivial treatment group whenever treatment has a strictly positive direct effect if and only if the inverse of the negated Jacobian is a B0-matrix by columns. We then provide several conditions on the Jacobian such that its negated inverse is a B-matrix by columns. Additional related results are presented. These predictions can be tested directly within the DiD framework even when the SUTVA is violated. Consequently, the results in this paper render economic models rejectable with reduced form DiD methods.