## _4.3 Could We Use a Bunching Estimator to Bound the MVPF?_

Our empirical strategy amounts to estimating the reduction in the mass of households bunching at the old notch and the increase in the mass of households bunching at the new notch as a result of the reform using a difference-in-difference strategy. But one may wonder whether we could have used standard bunching techniques (developed in Kleven and Waseem (2013) and Saez (2010)) to back out the changes in these masses. Of course, even using standard bunching techniques, one must account for time variation, i.e., one must estimate how many people would have bunched at the old notch and at the new notch in the post-reform period had the reform not occurred. As an example of how one might use standard bunching techniques while also controlling for time variation, one could first use these techniques to back out the counterfactual reported income distributions that would result if there were no notches in both the pre- and post-reform period (e.g., in June 2014 and June 2016). To do so, one would use the observed reported income distributions in the pre- and post-reform periods and make the following two assumptions (which are standard in the bunching literature): (a) the reported income distribution would be smooth in absence of the notch, and (b) the reported income distribution sufficiently far away from the notch is unaffected by the notch. One could then assume that the relationship that exists between the counterfactual and the observed reported income distributions in the pre-reform period would have stayed the same in the post-reform period in absence of the reform. This assumption combined with the estimated counterfactual reported income distribution in the post-reform period would then allow one to calculate the reported income distribution in the post-reform period had the reform not happened. A similar strategy is employed in Carril (2022). However, this strategy is unlikely to be feasible in our setting. As seen in Figure 17 in Appendix C.3, the reported income distribution in Brazil is _extremely_ non-smooth. For instance, in June 2014, in addition to substantial bunching at the notch of R$70, there is extreme bunching at numbers equal to 0 mod 50, less severe bunching at numbers equal to 0 mod 10, and substantial bunching at R$60. R$60 was a previous BF eligibility threshold (the notch of R$60 was implemented in 2006 while the notch of R$70 was implemented in 2009; see Gazola Hellmann (2015)). While bunching estimators have been augmented to deal with “round-number” bunching and other “reference-point” bunching (e.g., Kleven and Waseem (2013) and Best and Kleven (2017)), we believe that the pervasiveness and the variability of round-number and reference-point bunching in our setting will make it too difficult to precisely identify the counterfactual distributions around our notches (in particular, around the original notch of R$70 which is equal to 0 mod 10). Thus, our empirical strategy highlights an alternative way to estimate changes in bunching when smoothness assumptions cannot be used to estimate counterfactual distributions.

# 5 Results

In this Section, we present results from our generalized difference-in-difference specification, Equation (15), and, in turn, calculate bounds on the MVPF. Based on these bounds, we then discuss the welfare implications of the reform. Next, we show robustness of our results to a variety of factors: we show robustness to Identification Assumption 1 by allowing the number of households between R$(56 _,_ 63] to also be impacted by the reform; we show robustness to a more general version of Identification Assumption 2; and, we show robustness of our results to household composition. Finally, we conduct a placebo exercise which partially tests the validity of our two identification assumptions.

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