Scrutinizing the Monotonicity Assumption in IV and fuzzy RD designs*

Abstract

Whenever treatment effects are heterogeneous, and there is sorting into treatment based on the gain, monotonicity is a condition that both instrumental variable (IV) and fuzzy regression discontinuity (RD) designs must satisfy for their estimate to be interpretable as a local average treatment effect. However, applied economic work often omits a discussion of this important assumption. A possible explanation for this missing step is the lack of a clear framework to think about monotonicity in practice. In this paper, we use an extended Roy model to provide insights into the interpretation of IV and fuzzy RD estimates under various degrees of treatment effect heterogeneity, sorting on gain and violation of monotonicity. We then extend our analysis to two applied settings to illustrate how monotonicity can be investigated using a mix of economic insights, data patterns and formal tests. For both settings, we use a Roy model to interpret the estimate even in the absence of monotonicity. We conclude with a set of recommendations for the applied researcher.

Patient Outcomes After Hospital Discharge to Home With Home Health Care vs to a Skilled Nursing Facility

“…We used differential distance to create a dichotomous measure that equals 1 if the beneficiary lives closer to a home health agency than an SNF and zero if the beneficiary lives equidistant between a home health agency and SNF or closer to an SNF than to a home health agency (eTable 2 in the Supplement). We dichotomized differential distance because the relationship between choice of home health care vs SNF and distance is not linear and the dichotomous version was more likely to meet the monotonicity assumption of the instrumental variable model …”

“…We dichotomized differential distance because the relationship between choice of home health care vs SNF and distance is not linear and the dichotomous version was more likely to meet the monotonicity assumption of the instrumental variable model. 17 We first tested whether the instrument was correlated with the treatment of interest, in this case treatment by a home health agency (rather than an SNF). We found that living closer to a home health agency was associated with discharge to a home health agency (F = 263.4; eTable 3 in the Supplement).…”

Sibling health, schooling, and longer-term developmental outcomes1

“…A second issue is that of monotonicity (Fiorini, Stevens, Taylor & Edwards 2013). This occurs because the children who are born before the cut-off (and thus eligible to enter school when relatively young) are held back and end up going to school at an even later age than children born after the cut-off.…”

The Intergenerational Effects of Recessions

“…This is similar to the no‐defier assumption in the context of binary instruments. In models where the treatment takes more than two values, the monotonicity assumption generates the testable implication of stochastic dominance of treatment outcomes under different instrument values (Angrist and Imbens 1995; Fiorini and Stevens 2021). Appendix Figures A1 and A2 show that the cumulative distribution function (CDF) of the treatment corresponding to high values of the instrument (higher quartiles) lies below the CDF of the treatment corresponding to lower values of the instrument (lower quartiles).…”