to : Tolerating de ance ?

This paper gathers the supplementary material to de Chaisemartin (2016). I show that one can estimate quantile treatment e ects in the comvivors subpopulation, that one can test the CD condition, and that my results extend to multivariate treatment and instrument, to fuzzy regression discontinuity designs, and to the bounds in Lee (2009) for survey nonresponse. ∗Department of Economics, University of Warwick, [email protected] 1 Testability and quantile treatment e ects. In this section, I show that random instrument, exclusion restriction, and CD have a testable implication. Then, I discuss how the testable implications of random instrument, exclusion restriction, and ND studied in Huber & Mellace (2012), Kitagawa (2013), and Mouri e & Wan (2014) relate to the CD condition. For any random variable R, let S(R) denote the support of R. Let Udz be a random variable uniformly distributed on [0, 1] denoting the rank of an observation in the distribution of Y |D = d, Z = z. If the distribution of Y |D = d, Z = z is continuous, Udz = FY |D=d,Z=z(Y ). If the distribution of Y |D = d, Z = z is discrete, one can randomly allocate a rank to tied observations by setting Udz = FY |D=d,Z=z(Y −) + V (FY |D=d,Z=z(Y )− FY |D=d,Z=z(Y −)), where V is uniformly distributed on [0, 1] and independent of (Y,D,Z), and Y − = sup{y ∈ S(Y |D = d, Z = z) : y < Y }. For every d ∈ {0, 1}, let pd = FS P (D=d|Z=d) . Notice that both p0 and p1 are included between 0 and 1. Finally, let L = E(Y |D = 1, Z = 1, U11 ≤ p1)− E(Y |D = 0, Z = 0, U00 ≥ 1− p0) L = E(Y |D = 1, Z = 1, U11 ≥ 1− p1)− E(Y |D = 0, Z = 0, U00 ≤ p0). Theorem S1 If Assumptions 1, 2, and 5 are satis ed, L ≤ W ≤ L. (29) The intuition of this result goes as follows. Under random instrument, exclusion restriction, and CD, E(Y1 − Y0|CV ) is point identi ed: following Theorem 2.1 in de Chaisemartin (2016), it is equal to W . It is also partially identi ed. CV is included in {D = 0, Z = 0}, and it accounts for p0% of this population. E(Y0|CV ) can therefore not be larger than the mean of Y0 of the p0% of this population with largest Y0. It can also not be smaller than the mean of Y0 of the p0% with lowest Y0 (see Horowitz & Manski, 1995 and Lee (2009)). Combining this with a similar reasoning for E(Y1|CV ) yields worst case bounds for E(Y1 − Y0|CV ), L and L. The estimand of E(Y1 − Y0|CV ) should lie within its bounds, hence the testable implication. To implement this test, one can use results from Andrews & Soares (2010). Let θ satisfy the two following moment inequality conditions: 0 ≤ E ( Y Z P (Z = 1) − Y (1− Z) P (Z = 0) − FS ( θ + Y DZ1{U11 ≤ p1} P (D = 1, Z = 1, U11 ≤ p1) − Y (1−D)(1− Z)1{U00 ≥ 1− p0} P (D = 0, Z = 0, U00 ≥ 1− p0) )) 0 ≤ E ( FS ( θ + Y DZ1{U11 ≥ 1− p1} P (D = 1, Z = 1, U11 ≥ 1− p1) − Y (1−D)(1− Z)1{U00 ≤ p0} P (D = 0, Z = 0, U00 ≤ p0) ) − ( Y Z P (Z = 1) − Y (1− Z) P (Z = 0) )) . This de nes a moment inequality model with preliminary estimated parameters. This model satis es all the technical conditions required in Andrews & Soares (2010) if E(|Y |) < +∞ Y − should be understood as −∞ if Y = inf{S(Y |D = d, Z = z)}.