Fuzzy difference in discontinuities

ABSTRACT

This paper discusses the conditions needed to estimate the local average treatment effect (LATE) of a policy using the ‘difference-in-discontinuities’ method while considering imperfect compliance. I show that simple and plausible assumptions about the participation and effects of the confounding existing policies allow the identification of the causal effect of a new policy. Identification is feasible even when information about the participation in other confounding policies is not available.

KEYWORDS:

• Fuzzy regression discontinuity
• difference in difference
• imperfect compliance
• public policy evaluation

JEL CLASSIFICATION:

• C21
• C26

Acknowledgments

I am grateful to Jan Stuhler, Luis Fernando Gamboa, Julián Roa and the anonymous referees whose comments improved this paper

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ As any FRD, the estimates identifies a LATE and not an ATE.

² Following the same notation: $y_t^{+}=li{m}_{z\to {\hat{z}}^{+}}E\left[y_{it}|z_{it}=z\right]$ and $y_t^{-}=li{m}_{z\to {\hat{z}}^{-}}E\left[y_{it}|z_{it}=z\right]$.

³ Borrowing from the difference in difference literature, parallel trends in this set up would imply that in the absence of $R$ time variation of potential outcomes just below and above the cut-off follow a parallel trend. If participation in or the effect of programme $W$ changes in time, by definition the time variation of the outcome variable will be different just below and just above the cut-off. Thus, in this set up the parallel trends assumption would not hold if assumptions A1 and/or A2 do not hold.