us discuss the statistical and economic interpretations of Identification Assumption 2. From a statistical perspective, Identification Assumption 2 is simply a generalization of the standard difference-in-difference assumption (this generalization is employed in, for example, Wolfers (2006) and is discussed more generally in Mora and Reggio (2013)).<sup>37</sup> Setting _K_ = 0 results in a standard difference-in-difference estimator in which the control groups and treatment groups are assumed to have “parallel trends” pre-reform that would have persisted in the post-reform period had the reform not occurred. The standard difference-in-difference strategy therefore assumes that the difference between the treatment and control groups follow a 0<sup>_th_</sup> order polynomial in absence of the reform (i.e., that they differ by a constant). Larger values of _K_ require progressively weaker parallel assumptions. For instance, setting _K_ = 1 requires the “parallel growths” assumption which asserts that the growth rates in the treatment and control groups over time are the same (i.e., that second and higher differences between treatment and control groups are constant over time).<sup>38</sup> Setting _K_ = 2 leads to the “parallel accelerations” assumption which asserts that the acceleration rates in the treatment and control groups over time are the same (i.e., third and higher differences between treatment and control groups are constant over time). Setting _K >_ 2 results in what Mora and Reggio (2013) refer to as the “parallel- _K_ ” assumption which asserts that the _K_ + 1 and higher differences between the treatment and control groups are constant over time.

But what is the economic meaning of Identification Assumption 2? Fundamentally, there are complex structural relationships between the number of people reporting in each income bin; these relationships are determined by how the true income distribution is evolving over time, how the true income distribution is mapped to the reported income distribution, as well as growth of the BF registry over time. By making Identification Assumption 2, we are essentially assuming that the differences over time between the number of people reporting in each income bin (which are determined by unknown structural relationships) are well approximated by polynomials which would have persisted in absence of the reform. Identification Assumption 2 is not fully testable in the same way that the standard differences-in-differences assumption (that parallel pre-trends persist post treatment) is not fully testable. However, as in the case of the standard “parallel trends” assumption, we can gauge whether Identification Assumption 2 seems sensible by looking at prereform data. Identification Assumption 2 implies that the difference between the log number of households in any two 7-increment bins below R$77 is governed by a stable polynomial in absence of the reform. Appendix C.4 shows how the differences log � _N_ (70 _,_ 77]� _−_ log � _N_ ( _x−_ 7 _,x_ ]� and log � _N_ (63 _,_ 70]� _−_ log � _N_ ( _x−_ 7 _,x_ ]� evolved in the pre-reform period for _x ∈{_ 7 _,_ 14 _, ...,_ 63 _}_ . These differences follow stable, low order polynomial relationships in the pre-period. While this of course does not imply that these relationships would have persisted into the post-reform period (just as parallel pre-trends do not imply that the trends would continue on a parallel path post treatment), it is at least suggestive that stable relationships exists between the various income bins.

However, because we have several control bins, we can _partially_ test the validity of our two identification assumptions. In particular, under Identification Assumption 1, our control bins should be unaffected by the reform. Thus, in the post-reform period, we can test whether the number of households locating in each of our control bins can be accurately predicted by our other control bins using Identification Assumption 2. Thus, at the end of Section 5, we run a number of placebo tests, suggesting that our two identification assumptions are reasonable.

> 37. Equation (2) in Wolfers (2006) uses an analogous generalized difference-in-difference accounting for differential quadratic pre-trends. Our Equation (15) is a combination of Equation (17) and Equation (20) from Mora and Reggio (2013). 38. A number of papers discuss the extended version of differences-in-differences under the “parallel growths” assumption (e.g., Mora and Reggio (2013), Rambachan and Roth (2020), and Bilinski and Hatfield (2019)).

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