Generalized Nonlinear Difference-in-Difference-in-Differences

Abstract

Difference-in-difference-in-differences (DiDiD) allow for the correction of unmeasured con-founding and function as a robustness check for difference-in-differences (DiD) techniques. However, this technique is not scale invariant and requires that the outcome variable be measured on units for which the treatment could have had no effect in either the pretreatment or post-treatment periods. Athey and Imbens (2006) provides a scale invariant, nonlinear DiD approach known as Changes-in-Changes (CiC). Sofer et al. (2016) extends CiC by showing that pre-treatment outcome measures are a special case of placebo (negative) outcomes and proposes a generalization of CiC called Negative Outcome Control (NOC). We develop a generalized nonlinear DiDiD approach we call NOCNOC that can be used either in the traditional DiDiD setting or when a placebo outcome is available in the pre and post-treatment data. We show that NOCNOC can correct for bias in Di-DiD, CiC, and NOC. We apply this method to a study of whether exposure to candidate debates affected Nepalese citizens' sense of political efficacy.