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In Shigeoka's context, these conditions are met because the heaping is in the day of birth and treatment coincides with the beginning of the month after age 70.
They note that their data exhibits heaping at ages in round decades and highlight that this heaping is non-random as "women at age 60 generally look different than would be predicted by the trend prior to age 60 and the trend after 60." In line with the solutions we describe above, they exclude women at age 60 from estimation and explain that this alters the population to which the results are applicable.
In particular, the fact that self-employed taxpayers bunch at tax kink points but others do not indicates non-random heaping. Taking a closer look at income data based on the PSID shows even more systematic heaping.
In this paper, we have demonstrated that the RD design's smoothness assumption is inappropriate when there is non-random heaping. In particular, we have shown that RD-estimated effects are afflicted by composition bias when attributes related to the outcomes of interest predict heaping in the running variable.
While the importance of showing disaggregated mean plots is well established as a way to visually confirm that estimates are not driven by misspecification (Cook and Campbell 1979), our examples demonstrate that researchers should highlight data at reporting heaps in such plots in order to visually inspect whether there is nonrandom heaping. As a more-formal diagnostic to be used when the problem is not obvious, we suggest that researchers estimate the extent to which characteristics at heap points "jump" off of the trend predicted by non-heaped data.
We consider several different approaches to addressing the bias that non-random heaping introduces into standard RD estimates.
These lines show that the non-random heaping captured in DGP-2 leads to an estimate that is negatively biased.
However, as we show below, the non-random nature of the heaping will cause the standard RD estimated effects to go awry.
In particular, this behavior will produce more of a shift in the distribution at the treatment threshold, whereas heaping produces blips in the distribution that may or may not coincide with the treatment threshold.
As we here consider a proxy variable indicating heaped types, then, we surmise that such a proxy may be arrived at in practice by institutional knowledge, or common practice (as might be the case in rounding leading to heaping, for example).
(11.) The relationship highlighted in Panel A of Figure 3--that the sign of the bias depends on the location of the heap relative to the cutoff--also reveals a potential special case in which the heaping is such that equal and opposing biases of the estimates of the conditional expectation function on each side of the threshold results in an unbiased (though imprecise) estimate of the true treatment effect.