Violations of Monotonicity
The bias due to violations of monotonicity is composed of two factors. The first factor, A = P(i is a defier)/(P(i is a complier) – P(i is a defier)), is related to the proportion of defiers and is equal to zero under the monotonicity assump- tion. The smaller the proportion of defiers, the smaller the bias will be from violations of the monotonicity assump- tion. However, because the denominator of this factor is the average causal effect of Z on D, the bias can be large even if there are few defiers, as long as the average causal effect of Z on D is small. Note again that the stronger the instrument, the less sensitive the IV estimand is to viola- tions of the monotonicity assumption. The second factor isthe difference in average causal effects of D on Y for the compliers and defiers. If the average causal effects of D on Y are identical for defiers and compliers, violations of the monotonicity assumption generate no bias. The less varia- tion there is in the causal effect of D on Y, the smaller the bias from violations of the monotonicity assumption.For more information on Violations of Monotonicity check on: https://en.wikipedia.org/wiki/Monotonicity_criterion
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