Research

Works in Progress

• Comparative Statics for Difference-in-Differences

             Informal summary in Q&A style:

Q: What can we identify in the canonical DiD framework when we allow for spillovers? 

A: The average impact of treatment on the treated net of any spillovers to the untreated.

Q: But how do you identify spillover effects? 

A: We don't. In this paper we are looking at total effects of treatment, inclusive of spillovers.

Q: Why do we care about this? 

A: Many reasons. Here are two: (1) In practice, policies and interventions are implemented when spillovers are present. From a policy perspective it is interesting to know when the impact of treatment is greater among the treated than the untreated, on average. (2) DiD is  a simple and intuitive technique. It would be great if we can use it, even in the presence spillovers, to test economic theory by way of comparative statics analysis. 

Q: On that last point, wouldn't the economic theories have to make unambiguous predictions about how a treatment, or shock, affects average outcomes among units that are treated compared to units that are untreated? 

A: Yes.

Q: Isn't that complicated?

A: Yes, but one of the main contributions of the paper is to show that it is feasible. In fact, we give a variety of conditions under which the predictions hold no matter who is treated and whether or not they received the full intended treatment. Surprisingly, the conditions are not much stronger than existing conditions in the literature under which more narrow predictions hold. In fact, in some cases the conditions are weaker!

Q: How does that work?

A: B-matrices. This is a really cool class of P-matrices, and it turns out that this is exactly the kind of matrix we need to make DiD predictions. 

Abstract. 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 B-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.

• The Role of Learning in Search with Statistical Discrimination