Correcting for Endogeneity in Models with Bunching

Abstract

In this online appendix, we provide additional details and extensions, including the following sections: A, which contains extensions of the approach into nonlinear and nonparametric models; B, a supplementary identification section that provides formal details of the claims made in the remarks from Section 2; C, a supplementary estimation section which contains technical assumptions and further discussion, as well as some additional results; D, which details the clustering discretization strategy; E, which discusses the implementation of the tests of the identification assumptions in our application and provides some additional technical details; F, an empirical Monte Carlo study; and G, where all the proofs are gathered. A Nonlinear/Nonparametric Model SpecificationsThis section presents some generalizations of the model in Section 2. Note that equation (3) can be relaxed both directly, in its linear structure, and in how flexibly one defines ⌘ in equation (2). We discuss several possible generalizations here, although these examples are by no means exhaustive. A.1 Linear Correlated Random CoefficientsWe relax linearity by allowing the treatment effect to be a known parametric function of X and Z. For example, suppose that = ↵ 0 X + Z 0 ↵ in equation ( 3). This is equivalent to Garen (1984) 's model (see, e.g., Chay and Greenstone 2005), except that we do not exclude Z from the structural equation, and we do not require that Z and ⌘ are independent. Then,is identified, then so are ↵ 0 and ↵, and thus we can identify the treatment effects. Our estimation results also cover this model (see Remark C.3 in Appendix C.1).Other models where = g 1 (X, Z; ↵ 1 ) for some function g 1 known up to the finite parameter vector ↵ 1 , may be identified analogously. If E[Y |X, Z] is linear in parameters, estimation is covered by the results in Section 4. Otherwise, this is a special case of the next model. A.2 Nonlinear Correlated Random EffectsDefine ⌘ as X ⇤ = h 1 (Z;  1 ) + h 2 (Z;  2 )⌘,