TESTING IDENTIFYING ASSUMPTIONS IN FUZZY REGRESSION DISCONTINUITY DESIGN 1. INTRODUCTION

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1 TESTING IDENTIFYING ASSUMPTIONS IN FUZZY REGRESSION DISCONTINUITY DESIGN YOICHI ARAI a YU-CHIN HSU b TORU KITAGAWA c ISMAEL MOURIFIÉ d YUANYUAN WAN e GRIPS ACADEMIA SINICA UCL UNIVERSITY OF TORONTO ABSTRACT. We propose a test for the key identification and estimation conditions in Fuzzy Regression Discontinuity (FRD) design. We characterize the set of sharp testable implications of the FRD assumptions, for which the proposed test is uniformly valid under a class of distributions, is consistent against all fix alternatives and has non-trivial power against some local alternatives. Keywords: Fuzzy Regression Discontinuity Design, Heterogeneous Treatment Effect, Nonparametric Test, Inequality Restriction 1. INTRODUCTION In recent years Regression Discontinuity (RD) design, which was first introduced by Thistlethwaite and Campbell (1960), has become one of the most widely-used quasi-experimental method for identification and estimation of causal effect of policy interventions in applied researches. For example, Van der Klaauw (2002) and Angrist and Lavy (1999) estimate the effect of financial aid offers on students performance, Lee (2008) investigates the electoral advantage of incumbency in the United States House of Representatives, and DiNardo and Lee (2004) studies economic impacts of unionization on employers, among many others. See Imbens and Lemieux (2008) and Lee and Lemieux (2010) for more complete survey. Parallel to its wide applications in empirical researches, there have been developments on econometrics theory of identification and estimation in RD design. Hahn, Todd, and Van der Klaauw (2001, HTV) establishes conditions under which the (Fuzzy) RD design identifies average treatment effect for certain subpopulations in the presence of heterogeneity, Porter (2003) discusses estimation of RD Date: Wednesday 25 th January, *This paper is based on independent work of Arai and Kitagawa (2016) and Hsu, Mourifié, and Wan (2016). a. National Graduate Institute for Policy Studies (GRIPS), mailto:yarai@grips.ac.jp. b. Institute of Economics, Academia Sinica, ychsu@econ.sinica.edu.tw. c. Department of Economics, University College London, t.kitagawa@ucl.ac.uk. d. Department of Economics, University of Toronto, ismael.mourifie@utoronto.ca. e. Department of Economics, University of Toronto, yuanyuan.wan@utoronto.ca. 1

2 design, Lee (2008) and McCrary (2008) discuss testing the specification of RD design by examining the continuity of the density of running variable or other pre-determined variables, Frandsen, Frölich, and Melly (2012) studies identification and estimation of quantile treatment effect, Calonico, Cattaneo, and Titiunik (2014) provides robust confidence intervals for treatment effects, Dong (2014) considers cases in which the first stage regression can either poses a jump or kink at the cutoff point, and recently Gerard, Rokkanen, and Rothe (2016) considers partial identification in RD design when the running variable can be potentially manipulated. There have been different sets of identifying assumptions proposed in the literature. HTV is one of the earliest papers which formally provide assumptions to non-parametrically identify the heterogeneous treatment effect in the RD design. To identify the average treatment effect of the group of compliers (in the language of Imbens and Angrist, 1994), HTV imposes two main assumptions, namely (i) independence between potential variables and the running variable near the cutoff (local independence) and (ii) monotonic response of the treatment with respect to running variable near the cutoff (local monotonicity). These two assumptions play an analogous role as the independence and monotonicity assumptions in Imbens and Angrist (1994) for identification of local average treatment effect (LATE). Frandsen, Frölich, and Melly (2012, FFM) relaxes the local independence assumption by assuming that the conditional distribution (CDF) of potential outcomes given complying status and running variable is continuous in the latter near the cutoff (which FFM refers as local smoothness). Under local smoothness and local monotonicity, 1 FFM identifies the distribution of potential outcomes of compliers and then the local quantile treatment effect (LQTE). In this paper, we aim to test the sharp implications of the local monotonicity and a FFM-type local smoothness assumptions (see main text for formal definition). We focus on the FFM-type local smoothness because it is weaker than local independence and it is empirically more relevant in some applications as discussed in Dong (2016), and the sharp characterizations of local independence+local monotonicity are indeed the same as those under local smoothness+local monotonicity joint with other suitable conditions, as we will show later in Section 7.1. Our paper contributes to the RD design literature by providing "sharp testable implications" of RD assumptions, that is, it provides the most informative set of testable implications for detecting 1 The local monotonicity assumptions used in FFM and HTV have different forms but coincide in the limit. For now we refer both as local monotonicity. See more discussions in Section

3 observable violations of the Fuzzy Regression Discontinuity (FRD) design assumptions. To the best of our knowledge, it is the first result established in the RD design literature. Quite interestingly, analogous to the observation of Angrist, Imbens, and Rubin (1996) and Kitagawa (2015) make in the instrumental variable model framework, our sharp testable implication can also be interpreted as the non-negativity of the potential outcomes density functions for the compliers at the cut-off. Recently, Fiorini and Stevens (2014) discusses the importance of specification test in FRD design and suggests an idea of testing based on Kitagawa (2015) and Huber and Mellace (2015) and interpret it as a test of local monotonicity by assuming that local independence holds. Fiorini and Stevens (2014), however, does not discuss the sharpness of the testable implications and they test different assumptions from us. We propose nonparametric tests for the sharp implications of the FRD assumptions. We construct the test statistics using the local polynomial method (Fan and Gijbels, 1992) and propose to methods to compute the critical values: a multiplier bootstrap method and a pool bootstrap method. We incorporate both procedures with the generalized moment selection (GMS) method (see Andrews and Soares, 2010; Andrews and Shi, 2013) to improve the power of the test. The proposed tests are uniformly valid under a class of distributions, are consistent against all fix alternatives and has non-trivial power against some local alternatives. In the main theorems, we establish the results without additional covariates; in Section 7.3, we show that our testable implication and test can be adjusted straightforwardly to incorporate them. Our test focuses on different aspects to existing specification tests in FRD framework which often test the continuity of density function of running variable and/or other predetermined variables around the cutoff (see discussions in Lee, 2008). Rejecting the null hypothesis of continuity is interpreted as evidences of manipulation of the design. McCrary (2008) proposes a formal test for continuity of the density of the running variable at the cutoff point. It is worth noting that the continuity of the density of the running variable (and other predetermined variables) is neither sufficient nor necessary for identification of the LATE or LQTE. On the contrary, we test the necessary and sufficient conditions on the distribution of the observable variables for rationalizing the FRD design model under analysis. Therefore, our test is significantly different from the test suggested by McCrary (2008) and Lee (2008) and is a useful complementary to the literature. We will discuss this in more details in Section

4 Our paper also contributes to the growing literature of specification tests in causal inference. Recently, Kitagawa (2008, 2015), Huber and Mellace (2015), and Mourifié and Wan (2016) propose testing procedures to assess validity of key assumptions in the instrumental variable model discussed in Imbens and Angrist (1994). Even the parameter that one can identify in a FRD design can be interpret as LATE and some identifying assumptions play similar roles as their counterparts in instrumental variable models, in general FRD assumptions are invoked differently and the above mentioned tests cannot be directly applied to assess the validity of FRD assumptions. A formal test on FRD assumptions would be valuable by providing empirical researchers with a tool to assess the validity of FRD assumptions rather than just relying on economic intuition or on specification tests which are not very informative. The rest of the paper is organized as follows. In Section 2 we lay out the main identifying assumptions that we are interested in and derive their sharp characterizations in Section 3. We formally provide a test statistics and show how to obtain its critical values in Section 4. In Section 7, we discuss several extensions. All mathematical proofs are deferred to the Appendix. 2. ANALYTICAL FRAMEWORK AND IDENTIFICATION Consider the potential outcome model introduced in Rubin (1974). Let Y = Y 1 D + Y 0 (1 D), where Y is the observed outcome Y Y R, D {0, 1} is the observed treatment indicator, and (Y 1, Y 0 ) are potential outcomes. We define the potential treatment status when the running variable R R R be externally set to r as D(r), and hence the observed treatment status is D = D(R). The main research interest in such models is to identify the causal impact of the treatment D on the outcome Y. Let P be the joint distribution of (Y, D, R) and F R (r) denote the margin distribution function of R. We suppress covariates X for now and will show in Section 7.3 that it is straightforward to include covariates in the model. The main feature of the RD design is the existence of the running variable R which influences the probability of being treated in a discontinuous way when it is takes value around a threshold r 0. There are two main types of discontinuity designs considered in the literature: the sharp design and the fuzzy design. In the sharp RD design, the potential treatment D(r) is deterministic, while in the FRD design, D(r) is a Bernoulli random variable but the probability of taking value 1 varies discontinuously at r 0. This paper consider the FRD design. 4

5 Various sets of assumptions has been proposed in the literature in order to identify the LATE and the LQTE in the FRD design context, see HTV for the LATE and FFM for the LQTE, respectively. Identification of the LQTE is especially important for the analyses of heterogeneous treatment effects. As mentioned by Imbens and Rubin (1997) and Angrist and Pischke (2008), such object is useful for policy makers who want to take into account differences in the dispersion of the outcome of interest when contemplating the merits of one program or treatment versus another. In this paper we will focus mainly on the testability of the FFM-type of assumptions (local smoothness and monotonicity) at least for two reasons: (i) under FFM-type s assumptions, the distribution of the subpopulation of compliers can be identified, then the density (when it is well defined) therefore enable us to recover a wider class of parameter of interests in addition to the LATE and the LQTE, and (ii) the economic intuition behind the set of assumption required to identify only the LATE is similar to the one required by the FFM-type of assumptions to identify the LQTE. Furthermore, we will also discuss the testability of a variant of the FFM-type of assumptions (local independence instead of local smoothness) often invoked to identify the LQTE. We follow FFM and define D + = lim r r0 D(r) and D = lim r r0 D(r), if the limits exist. Based on this definition, we can define the compliance status as shown in the following table: We use TABLE 1. Subpopulations D D + Proportion A: Always-takers 1 1 π 11 D: Defiers 1 0 π 10 C: Compliers 0 1 π 01 N: Never-takers 0 0 π 00 π ij, i, j {0, 1} to denote the probability mass of corresponding groups. Indeed, there may exist a subpopulation for which D +, and D are not simultaneously well defined, however as in FFM, we maintain implicitly that such a subpopulation has a null mass, that is, π 11 + π 10 + π 01 + π 00 = 1. Throughout the paper, we implicitly assume that there are some observations close to the cutoff point r 0. Hereafter, let E P be the expectation under P and let P P denote the probability under P. 5

6 Assumption 1 (Discontinuity). The limits π + lim r r0 P P (D = 1 R = r) and π lim r r0 P P (D = 1 R = r) exist and π + = π. Assumption 1 represents the main feature of the FRD design in that the probability of receiving treatment differs on either side of the cutoff value. Let T denote the complying status and A, C, and N denote always-takers, compliers, and nevertakes, respectively. Assumption 2 (Local Smoothness). For d = 0, 1, t {A, C, N}, and B Y be a measurable set, the conditional probability P P (Y d B, T = t R = r) is continuous in r at r 0. Assumption 3 (Local Monotonicity-1). lim r r0 P P (D + D R = r) = 1. Assumption 2 essentially plays the same role as FFM Assumption I2. 2 The local smoothness condition is imposed on the conditional distribution rather than on the conditional mean (as in the HTV s spirit) and this stronger requirement allows to identify a wider class of parameter of interests, like the LQTE. Assumption 3 is the same as FFM Assumption I3 since we already assume that the subpopulation for which D + and D are not well-defined has mass zero. The direction of the Assumption 3 is without loss of generality and implies that π 10 = 0. Together with Assumption 1, it also implies that π + > π. Let 1{ } denote the indicator function. It can be shown that under Assumptions 1 to 3 (following FFM s Lemma 1 proof and hence omitted here), the compliers potential outcome distribution is identified: and F Y1 C,R=r 0 (y; P) = lim r r 0 E P [1{Y y}d R = r] lim r r0 E P [1{Y y}d R = r], (1) lim r r0 E P [D R = r] lim r r0 E P [D R = r] F Y0 C,R=r 0 (y; P) = lim r r 0 E P [1{Y y}(1 D) R = r] lim r r0 E P [1{Y y}(1 D)R = r]. lim r r0 E P [D R = r] lim r r0 E P [D R = r] (2) If we strengthen Assumption 2 to local independence, then it is possible to identify a wider class of parameters. Please see Section 7.1 for more discussions. 2 FFM-I2 assumes that (i) FYd D +,D,R (y d+, d, r; P) is continuous in r at r 0, for d +, d {0, 1} for all y Y and (ii) E P [D + R = r] and E P [D R = r] are both continuous in r at r 0, where F Yd D +,D,R (y d+, d, r; P) denote the conditional distribution of Y d conditioning on D + = d +,D = d and R = r under P for d +, d {0, 1}. 6

7 3. SHARP TESTABLE IMPLICATIONS In this section, we show that those assumptions together imply a set of restrictions on the data (Y, D, R) which will allow us to provide empirical content for the model. Let B Y be the collection of all Borel sets generated from Y and let B be an arbitrary element in B Y. The following proposition states the testable implication of Assumptions 1 to 3. Proposition 1. Let Assumptions 1 to 3 hold, then lim E P [1{Y B}D R = r] lim E P [1{Y B}D R = r] 0, (3) r r0 r r0 lim E P [1{Y B}(1 D) R = r] lim E P [1{Y B}(1 D) R = r] 0. (4) r r0 r r0 For the purpose of exposition, we collect all the proofs in appendix. Inequalities 3 and 4 represent the testable implications of Assumptions 1 to 3. While those inequalities form basis to assess the validity of the FRD assumptions, there are some important questions worth discussing. First, from an identification point of view, are inequalities 3 and 4 sharp testable implications of Assumptions 1 to 3? Could we identify a smaller set of inequalities that contain all the information deliver by the inequalities 3 and 4 regarding the testability of Assumptions 1 to 3. This is related to the issue of finding the low (lowest) cardinality class of sets that can characterize all the restrictions on the data imposed by the RD assumptions. 3 Second, from both inferential and empirical point of view, the set of inequalities 3 and 4 is difficult to analyze, for example the class of functions indexed by all Borel-set needs not be Donsker and it is certainly desirable to reduce the cardinality of the class of sets. Theorem 1 addresses these concerns while we will discuss the statistical property of our tests in details in Section 4. Theorem 1. Let {D(r), Y 1, Y 0 } a vector of unobserved potential treatments and outcomes, and (Y, D, R) the vector of observed variables where Y = Y 1 D + Y 0 (1 D) with the observed D defined as follows D D(R). Let r 0 the known cut-off point. 3 See Galichon and Henry (2006, 2011) and Chesher and Rosen (2015) for discussion of core determining classes. 7

8 (i) Under Assumptions 1 to 3, the following inequalities hold: η P,1 (y, y ) = lim r r0 E P [1{y Y y }D R = r] lim r r0 E P [1{y Y y }D R = r] 0 (5) η P,0 (y, y ) = lim r r0 E P [1{y Y y }(1 D) R = r] lim r r0 E P [1{y Y y }(1 D) R = r] 0 for all y, y Y, and inequalities hold strictly if Y [y, y ]. (ii) If inequalities 5 and 6 hold, there exists a joint distribution of ( D(r), Ỹ 1, Ỹ 0 ) such that Assumptions 1 to 3 hold, and the conditional distributions of (Ỹ, D) R = r and (Y, D) R = r satisfy (6) lim F Y,D R=r = lim FỸ, D R=r and lim F Y,D R=r = lim FỸ, D R=r, (7) r r0 r r0 r r0 r r0 that is, the observed distribution and the counter-factual distribution are the same as r approaches r 0 from each side, respectively. Remark: Theorem 1 part (i) shows a necessary condition that the distribution of observable variables should satisfy under the FRD assumptions. Part (ii) is more important. It shows that inequalities 5 and 6 are the most informative way to screen all the observables violations of the FRD assumptions. First, this means that we do not lose any information by reducing the collection of Borel sets B Y to the set of closed intervals. Second, inequalities 5 and 6 are necessary and sufficient conditions for the distribution of the observable variables (Y, D, R) to rationalize the FRD design entertained here, i.e., potential outcome model and Assumptions 1 to 3. To the best of our knowledge, this paper is the first in the literature that provides a set of inequalities that are the most informative to screen the violation of the FRD assumptions. Remark: Economic s interpretation of the "sharp" testable implications. Following Angrist, Imbens, and Rubin (1996) and Kitagawa (2008, 2015), we interpret the testable implication of those assumptions as the non-negativity of compliers potential outcome distributions. Consider a closed interval [y, y + h] for some h > 0, and follow the argument in the proof of Proposition 1, inequality 8

9 8 implies that lim E P [1{y Y y + h}d R = r] lim E P [1{y Y y + h}d R = r] r r0 r r0 = E P [1{y Y 1 y + h} C, R = r 0 ]P P (C R = r 0 ) 0. Dividing both side by h and let h 0 first (suppose we can switch the order of the limit and corresponding densities are well defined), we have lim P P (D = 1 R = r) f Y D,R (y 1, r; P) lim P P (D = 1 R = r) f Y D,R (y 1, r; P) r r0 r r0 = f Y1 C,R(y r 0 ; P)P P (C R = r 0 ) 0 where f Y1 C,R(y r; P) denote the potential outcome Y 1 probability density function for the subpopulation of compliers (when they exist). Therefore, our test can be interpreted as the non-negativity of the potential outcome density functions for the compliers at the cut-off. 4. TESTING Assume that we observe a random sample of (Y, D, R) of size n generated from the distribution P. Let F R denote the marginal distribution of P on R. Let n + and n denote number of observations such that R r 0 and R < r 0, respectively. Throughout the paper, we assume that n + /n converges to a constant bounded away from 0 and 1. For the purpose of inference, we define an equivalent class of inequalities of 5 and 6. Let G be the set of the indicator functions of a class of closed intervals C l Y, l L such that: G = {g l ( ) = 1{ C l } : l L}. We will propose later different choices of interval class L such that our test statistics has desirable asymptotic property and is easy to computed. As we show in Corollary 5 in appendix, the set of inequalities 5 and 6 are equivalent to the inequalities 8 and 9 below: ν P,1 (l) = lim r r0 E P [g l (Y)D R = r] lim r r0 E P [g l (Y)D R = r] 0, (8) ν P,0 (l) = lim r r0 E P [g l (Y)(1 D) R = r] lim r r0 E P [g l (Y)(1 D) R = r] 0. (9) 9

10 for all l L. We therefore define the hypothesizes H 0 and H 1 as H 0 : ν P,1 (l) 0 and ν P,0 (l) 0 for all l L, H 1 : H 0 does not hold. (10) 4.1. Test Statistics. We will construct our test statistics based on standardized nonparametric estimates of ν P,1 (l) and ν P,0 (l). To be specific, let m P,1,+ (l) = lim r r0 E P [g l (Y)D R = r], m P,1, (l) = lim r r0 E P [g l (Y)D R = r], m P,0,+ (l) = lim r r0 E P [g l (Y)(1 D) R = r] and m P,0, (l) = lim r r0 E P [g l (Y)(1 D) R = r], then we can estimate ν P,1 (l) and ν P,0 (l) respectively by ˆν 1 (l) = ˆm 1, (l) ˆm 1,+ (l), ˆν 0 (l) = ˆm 0,+ (l) ˆm 0, (l), where the right hand side term ˆm 1,+ (l), ˆm 1, (l), ˆm 0,+ (l) and ˆm 0, (l) are local linear estimators, which in turn can be constructed as the constant terms â 1,+ (l), â 1, (l), â 0,+ (l) and â 0, (l) in regressions of the form n ( Ri r 0 min 1(R i r 0 ) K â 1,+ (l),ˆb 1,+ (l) i=1 min n â 1, (l),ˆb 1, (l) i=1 min n â 0,+ (l),ˆb 0,+ (l) i=1 min n â 0, (l),ˆb 0, (l) i=1 h + ( Ri r 0 1(R i < r 0 ) K h ( Ri r 0 1(R i r 0 ) K h + ( Ri r 0 1(R i < r 0 ) K h ) [ g l (Y i )D i â 1,+(l) ˆb ] 2, 1,+(l)(R i r 0 ) ) [ g l (Y i )D i â 1, (l) ˆb ] 2, 1, (l)(r i r 0 ) ) [ g l (Y i )(1 D i ) â 0,+(l) ˆb ] 2, 0,+(l)(R i r 0 ) ) [ g l (Y i )(1 D i ) â 0, (l) ˆb ] 2, 0, (l)(r i r 0 ) where K( ) is a symmetric kernel function and (h +, h ) are the bandwidths for regression above and below the threshold, respectively. In particular, we let h + = c + h and h = c h, with (c +, c ) be positive constants and h 0. We follow the RD literature and suggest use the triangular kernel for boundary local linear estimators. As in Fan and Gijbels (1992), we write the local linear estimators as ˆm 1,+ (l) = ˆm 0,+ (l) = n w + n ni g l(y i )D i, ˆm 1, (l) = w ni g l(y i )D i, (11) i=1 i=1 n w + n ni g l(y i )(1 D i ), ˆm 0, (l) = w ni g l(y i )(1 D i ), (12) i=1 i=1 10

11 where the weights w + ni = 1(R i r 0 ) K( R i r 0 h + )[S + n,2 S+ n,1 (R i r 0 )] n i=1 1(R i r 0 ) K( R i r 0 h + )[S + n,2 S+ n,1 (R i r 0 )] w ni = 1(R i < r 0 ) K( R i r 0 h )[S n,2 S n,1 (R i r 0 )] i=1 n 1(R i < r 0 ) K( R i r 0 h )[S n,2 S n,1 (R i r 0 )]. and for j = 0, 1, 2,, S + n ( ) n,j = Ri r 0 1(R i r 0 ) K (R i r 0 ) j, i=1 h + S n ( ) n,j = Ri r 0 1(R i < r 0 ) K (R i r 0 ) j, i=1 h Finally, let ˆσ 1 (l) and ˆσ 0 (l) be consistent estimators of the asymptotic standard errors of ˆν 1 and ˆν 0, respectively, then we can define a Kolmogorov-Smirnov (KS) type test statistic as Ŝ n = nh sup d=0,1, l L where ξ d, d = 0, 1 are trimming constants chosen by researchers. ˆν d (l) max{ˆσ d (l), ξ d }, (13) 4.2. Critical Values. We propose three resampling procedures to obtain critical values. To estimate the (1 α)-th quantile of the asymptotic distribution of Ŝ n, we consider two resampling-based approaches: multiplier bootstrap and pooled bootstrap and recentered bootstrap. The first approach is based on Hsu (2016b). The second approach is along the idea of pooled bootstrap procedure considered in Abadie (2002) and Kitagawa (2015), which we tailor to the current context of fuzzy regression discontinuity design. In this section, we outline the steps of each procedure and will establish the asymptotic properties in next subsection. For both procedures, we need to specify bandwidths and kernel functions. We follow the RD literature and suggest use the triangular kernel. In simulations, we choose undersmoothed versions of conventional bandwidths options in RDD framework, such as the optimal bandwidths proposed by (Imbens and Kalyanaraman, 2011, IK), (Calonico, Cattaneo, and Titiunik, 2014, CCT), and (Arai and Ichimura, 2016, AI), respectively Multiplier Bootstrap. Let B be a large positive integer. 11

12 (1) Based on the original sample, compute ˆm 1,+ (l), ˆm 1, (l), ˆm 0,+ (l) and ˆm 0, (l) for each l L, as described in Equations (11) and (12). Calculate the test statistics using Equation (13). (2) For each l L, calculate the sample analog of the influence function ˆφ ν1,ni(l) = nh ( w + ni (g l(y i )D i ˆm 1,+ (l)) w ni (g l(y i )D i ˆm 1, (l)) ), ˆφ ν0,ni(l) = nh ( w ni (g l(y i )(1 D i ) ˆm 0, (l)) w + ni (g l(y i )(1 D i ) ˆm 0,+ (l)) ). (3) For each b = 1,, B, draw U b1, U b2, U bn as i.i.d. pseudo random variables with E[U bi ] = 0, E[Ubi 2 ] = 1 and E[U4 bi ] < that are independent of the sample path. (4) Calculated the simulated process (of l) as Φ u ν 1,n,b (l) and Φ u ν 0,n,b(l) be Φ u n ν 1,n,b (l) = U ib ˆφ ν1,ni(l), Φ u n ν 0,n,b (l) = U ib ˆφ ν0,ni(l). i=1 i=1 (5) Estimate the asymptotic variance by ˆσ 2 d (l) = n i=1 ˆφ 2 ν d,n,i (l) (6) Let B n be a sequences of non-negative numbers. For d = 0 and 1, define ψ nd (l) as ψ nd (l) = B n 1( nh ˆν d (l) < a n ), where a n is a sequence of non-negative numbers. (7) Let PB u denote the multiplier probability measure. For significance level α < 1/2, calculate the simulated critical value ĉ n,η (α) as ĉ n,η (α) = sup { q : P u B ( sup d=1,0, l L Φ u ν d,n,b (l) + ψ ) } nd(l) q 1 α + η + η, max{ˆσ d (l), ξ d } where η > 0 is an arbitrarily small positive number, e.g., That is, ĉ n,η (α) is the (1 { } Φ u B ν α + η)-th quantile of the simulated null distribution of sup d,n,b (l)+ψ nd(l) d=1,0, l L max{ˆσ d (l),ξ d } plus η. (8) Reject H 0 if Ŝ n > ĉ n,η (α). b=1 Remark. As most papers in the moment inequality literature, our paper uses the generalized moment selection method to construct the critical value (Step 5). The GMS method is introduced 12

13 by Andrews and Soares (2010) and Andrews and Shi (2013, 2010, 2016). It is similar to the recentering method in Hansen (2012) and Donald and Hsu (2016), as well as the contact set approach in Linton, Song, and Whang (2010). By doing this, one can construct a more powerful test than resorting to the least favorable configuration. Following Andrews and Shi (2013, 2016), we suggest a n = (0.3 ln(n)) 1/2 and B n = (0.4 ln(n)/ ln ln(n)) 1/2. Remark. The η constant is called an infinitesimal uniformity factor and is introduced by Andrews and Shi (2013) to avoid the problems that arise due to the presence of the infinite-dimensional nuisance parameter ν P,1 (l) and ν P,0 (l) Pooled Bootstrap. We introduce some additional notation. Let η n = n h (n h +n + h +. For l L and d {0, 1}, let b n (l, d) denote rescaled bias of local linear estimator ˆν d (l) It can be shown that b n (l, d) = E[ ˆν d (l)] ν d (l). b n (l, 1) p b(l, 1) κ [ ( ) 1 2 ηλ 2 r 2 P(Y A, D = 1 r ) ( )] 2 ηλ + r 2 P(Y A, D = 1 r +), (14) b n (l, 0) p b(l, 1) κ [ ( ) ηλ+ 2 2 r 2 P(Y A, D = 0 r +) ( )] 2 1 ηλ r 2 P(Y A, D = 0 r ), (15) where κ is the constant that depends only on the kernel function, κ κ2 2 κ 1κ 0 3 κ 2 κ 0 κ1 2, κ l u l k(u)du, and λ +, λ and η are probability limits of n + h 5 +, n + h 5 + and η n, respectively. Now we outline the steps of the pooled bootstrap procedure. (1) Based on the original sample, compute ˆm 1,+ (l), ˆm 1, (l), ˆm 0,+ (l) and ˆm 0, (l) for each l L, as described in Equations (11) and (12). Calculate the test statistics using Equation (13). 13

14 (2) (bias estimation) Estimate the bias terms shown in Equations (14) and (15) by plugging in estimator of the second derivatives obtained from, e.g., the local cubic regressions. (3) (construction of mixtures) Let Q n be the empirical distribution constructed by {(Y i, D i, R i ) : R i < c, i = 1,..., n } and ˇQ n be the empirical distribution constructed by {(Y i, D i, 2c R i ) : R i < c, i = 1,..., n }. Similarly, let P n+ be the empirical distribution constructed by {(Y j, D j, R j ) : R j c, j = 1,..., n + } and ˇP n+ be the empirical distribution constructed by {(Y j, D j, 2c R j ) : R j c, j = 1,..., n + }. Note that ˇQ n and ˇP n+ flip the original values of R symmetrically around the cut-off value. Define mixtures of the empirical distributions H (1 η n )Q n + η n ˇP n+, H + (1 η n ) ˇQ n + η n P n+. (4) (bootstrap) Draw n iid observations of (Y, D, R) from H and n + iid observations from H +. Using thus-generated bootstrap sample, compute ˆν d (l) and ˆσ d (l), which are denoted by ˆν d (l) and ˆσ d (l). We then form the bootstrap statistic { [ sup l L [ξ 1 ˆσ 1 Ŝ boot = max (l)] 1 ˆν 1 (l) + ˆb(l, ]} 1) { [ sup l L [ξ 0 ˆσ 0 (l)] 1 ˆν 0 (l) + ˆb(l, ]} 0), where ˆb(l, d) are the estimates of the bias constructed in Step 2. (5) Repeat Step 4 many times and use ĉ 1 α the empirical (1 α)-th quantile of Ŝ boot as a critical value of the test. Reject the null hypothesis if Ŝ n > ĉ 1 α. Remark. If the bias is controlled by under-smoothing, then λ = λ + = 0, implying that b n (l, d) is asymptotically negligible Asymptotics of the Proposed Tests Multiplier Bootstrap. In this subsection we derive the asymptotic properties of the multiplier bootstrapping procedure. We let the observed outcome Y Y [0, 1]. 4 We establish the results for 4 Assuming that the suppose of Y is a subset of [0,1] is without loss of generality. If not, we can define Ỹ = Φ(Y) where Φ( ) is the CDF of standard normal. 14

15 the following choice of set L: G = {g l ( ) = 1( C l ) : l (y, c) L}, where C l = [y, y + c] Y and L = { (y, c) : c 1 = q, and q y {0, 1, 2,, (q 1)} for q = 1, 2, }. (16) Let h 2 (, ) be a covariance kernel on L L. Let H 2 be the collection of all possible covariance kernel functions on L L. For any pair of h (1) 2 and h (2) 2, we define the distance between them as d(h (1) 2, h(2) 2 ) = sup h (1) 2 (l 1, l 2 ) h (2) 2 (l 1, l 2 ). (17) l 1,l 2 L Let σ 2 P,1,+ (l 1, l 2 ) = lim r r0 Cov P,1 (g l1 (Y)D, g l2 (Y)D R = r) be the conditional covariance of g l1 (Y)D and g l2 (Y)D when r approach to r 0 from above. Let σ 2 P,0,+ (l 1, l 2 ) = lim r r0 Cov P,0 (g l1 (Y)(1 D), g l2 (Y)(1 D) R = r) be the conditional covariance of g l1 (Y)(1 D) and g l2 (Y)(1 D) when r approach to r 0 from above. Define σ 2 P,1, (l 1, l 2 ) and σ 2 P,0, (l 1, l 2 ), similarly. For j = 0, 1, 2,..., let ϑ j = 0 uj K(u)du. For d = 0, 1, Define 0 h 2,P,d (l 1, l 2 ) = (ϑ 2 uϑ 1 ) 2 K 2 (u)du σp,d,+ 2 (l 1, l 2 ) + σp,d, 2 (l 1, l 2 ) (ϑ 2 ϑ 0 ϑ1 2)2 f r (r 0 ) which is the covariance kernel of Φ u ν 1,n(l) under distribution P. Also, f r (r 0 ) is the density function of R evaluated at R = r 0 that is assumed to be not depending on P. Let P be the collection of probability distributions. Let f R (r) denote the probability density function of F R under P P. We consider the case in which f R (r) is the same for all P P. For d = 0, 1, let f Y D,R (y d, r; P) denote the conditional pdf of Y conditional on D = d and R = r that is equivalent to f Yd r(y r; P) under P. Let ρ P,+ (r) = E P [D(r) R = r] for r r 0 and ρ P, (r) = E P [D(r) R = r] for r < r 0. Let f and f denote the first and second derivatives of function f. Let for any δ > 0, N δ (r 0 ) = {r : r r 0 δ} denote a neighborhood of r around r 0. Let N + δ (r 0) = {r : 0 = r r 0 δ} and N δ (r 0) = {r : 0 < r 0 r δ}. Assumption 4. Assume that for all P P, (1) P r is the same and f R (r) is twice continuously differentiable in r on N δ (r 0 ), (2) f r (r) is bounded away from zero on N δ (r 0 ), 15

16 (3) ρ P,+ (r) is twice continuously differentiable in r on N + δ (r 0) and ρ P, (r) is twice continuously differentiable in r on N δ (r 0). (4) for d = 0, 1 and for each y Y, f y d,r (y, r; P) is twice continuously differentiable in r on N δ (r 0 ), (5) dρ P,+ (r)/dr M and d 2 ρ P,+ (r)/drdr M on N + δ (r 0), and dρ P, (r)/dr M and d 2 ρ P, (r)/drdr M on r on N δ (r 0), (6) for d = 0, 1, f Y D,R (y d, r; P)/ r M and 2 f Y D,R (y d, r; P)/ r r M on r N δ (r 0 ) for each y Y, (7) for d = 0, 1, f Y D,R (y d, r; P) is continuous on y Y for all r N δ (r 0 ), (8) M and δ are positive constants not dependent on P. Assumption 4 (ii)-(vi) are standard in nonparametric estimation. Assumption 4(v) and (vi) are needed to show that the bias terms of the ˆν 1 (l) and ˆν 0 (l) are asymptotically negligible uniformly over l L. Assumption 4(vii) is assumed for notational simplicity. It requires that the conditional distribution of Y conditioning on R = r and D = d is continuous and this basically rules out the cases in which Y is a discrete variable or a mixture of discrete and continuous variables. Extensions of our results to these cases are straightforward and we omit them for brevity. Assumption 5. Assume that (1) The K( ) is a non-negative symmetric bounded kernel with a compact support in R (say [ 1, 1]). (2) K(u)du = 1, (3) h 0, nh and nh 5 0 as n. Assumption 5 is standard for nonparametric estimation. Note that nh 5 0 as n implies undersmoothing so that the bias terms converge to zero even after we multiply it with nh. This condition is standard if one wants to obtain the asymptotic normality of the estimators. Assumption 6. Let {U i : 1 i n} be a sequence of i.i.d. random variables E[U] = 0, E[U 2 ] = 1, and E[ U 4 ] < M 1 for some M 1 > 0, and {U i : 1 i n} is independent of the sample path {(Y i, D i, R i ) : 1 i n}. 16

17 Assumption 6 is standard for the multiplier bootstrap as in Hsu (2016b) and E[ U 4 ] < M 1 is needed for the multiplier bootstrap for nonparametric method. Assumption 7. Assume that: (i) Let a n be a sequence of non-negative numbers satisfying lim n a n = and lim n a n / nh = 0. (ii) Let B n be a sequence of non-negative numbers satisfying that B n is non-decreasing, lim n B n = and lim n B n /a n = 0. The following theorem summarizes the uniform size of our test. Let λ( ) denotes the Lebesgue measure. Define YP,d o {y Y : f Y d C,R(y, r o ; P)P P (C R = r 0 ) = 0} and L o P,d {l L : ν P,d (l) = 0}. It is straightforward to see that when λ(yp,d o ) > 0, then Lo P,d is not an empty set. Assumption 8. Let P 0 be the subset of P that satisfies Assumption 4 such that the null hypothesis in Equation (10) holds under P if P P 0. Assumption 9. Assume that there exists P c P 0 such that (1) max d=0,1 λ(yp o c,d ) > 0 under P c. (2) For d = 0, 1, h 2,Pc,d H 2,cpt. (3) Either h 2,Pc,1 restricted to L o P c,1 Lo P c,1 is not a zero function or h 2,P c,0 restricted to L o P c,0 L o P c,0 is not a zero function. Theorem 2. Suppose Assumption 4-Assumption 8 hold. We reject H 0 when Ŝ n > ĉ η. Then, for every compact subset H 2,cpt of H 2 (a) lim sup n sup {P P0 : h 2,P,1,h 2,P,0 H 2,cpt } P(Ŝ n > ĉ n,η (α)) α. (b) if Assumption 9 also holds, then lim lim sup η 0 n sup P(Ŝ n > ĉ n,η (α)) = α. {P P 0 :h 2,1,P,h 2,0,P H 2,cpt } Theorem 2(a) shows that our test have correct uniform asymptotic size over a compact sets of covariance kernels which is similar to Theorem 2(a) of Andrews and Shi (2013). Theorem 2(b) shows that our test is at most infinitesimally conservative asymptotically when there exists at least one P c such that max d=0,1 λ(yp o c,d ) > 0 under P c and one of h 2,Pc,d restricted to L o d Lo d is not a 17

18 zero function. Theorem 2 is similar to Theorem 2 of Andrews and Shi (2013) and Theorem 5.1 of Hsu (2016a) except that we consider a testing problem in regression discontinuity design. We present the power of our test against fixed alternatives. Let P 1 P such that one of f Yd C,R(y r o ; P 1 )P P1 (C R = r 0 ) > 0 for some y Y under P 1. Then by the continuity of f Yd C,R(y r o ; P 1 ) in y, there exists a neighborhood N (y ) around y such that f Yd C,R(y r o ; P 1 )P P (C R = r 0 ) > 0 for all y N (y ) and λ(n (y )) > 0. This implies that there exists l L such that ν P1,d(l ) > 0. The following theorem shows the consistency of our test. Theorem 3. Suppose Assumption 4-Assumption 7 hold and α < 1/2. Let P 1 P such that at least one of f Yd C,R(y r o ; P 1 )P P1 (C R = r 0 ) > 0 for some y Y under P 1. Then, lim n P(Ŝ n > ĉ n,η (α)) = 1. Theorem 3 follows from the fact that the test statistic Ŝ n diverges to positive infinity under P 1 and the critical value ĉ n,η (α) is bounded in probability. We show that our test is unbiased against some nh 1/2 local alternatives. Define A\B {y : y A but y B} for any two sets A and B. We consider a sequence of P n P\P 0 such that f Y1 C,R(y r o ; P n ) = f Y1 C,R(y r o ; P o ) + δ 1(y) nh, f Y1 C,R(y r o ; P n ) = f Y0 C,R(y r o ; P o ) + δ 0(y) nh, (18) for some P o P 0. We impose the following conditions on the local alternatives we consider. Assumption 10. Let {P n P\P 0 : n 1} and P o P 0 such that: (1) Equation (18) holds under P n, (2) max d=0,1 λ(yp o o,d ) > 0, (3) for d = 0, 1, δ d (y) 0 if y YP o o,d, (4) max d=0,1 λ(δ d Y P o o,d ) > 0 where δ d {y : δ d(y) > 0}. (5) for d = 0, 1, lim n d(h 2,Pn,d, h2 ) = 0 for some h 2,d H 2. Assumption 10(i) requires that the local alternatives converge to a null hypothesis at the rate (nh) 1/2. Assumption 10(ii) requires that the null hypothesis is not an interior one so that we can have nontrivial local power. Assumption 10(iii) ensures that our test is unbiased. Assumption 10(iv) 18

19 specifies the asymptotic behavior of the covariance kernels and this is similar to LA1(c) of Andrews and Shi (2013). The following theorem shows that the asymptotic local power of our test is greater than or equal to α when η tends to zero, i.e., our test is unbiased against those local alternatives that satisfy Assumption 10. Theorem 4. Suppose Assumption 4-Assumption 7 hold and α < 1/2. Under {P n : n 1} that satisfies Assumption 10, lim η 0 lim n P(Ŝ n > ĉ n,η (α)) α Pooled Bootstrap. 5. SIMULATION In this section, we provide intensive Monte Carlo experiments to investigate the finite sample performance of the proposed procedures. We consider 10 DGPs, including 4 DGPs for examining size property and six for power property. We collect the details of the design and full sets of simulation results in Appendix 3 for the purpose of exposition. For each DGP, we generate random sample of four sizes: 1000, 2000, 4000 and These sample sizes are comparable to, if not smaller than, the sample sizes of many empirical applications of RD designs. We consider three choices of bandwidths: AI, IK and CCT. For each bandwidth, we conduct undersmoothing by multiplying each bandwidths by a factor of n c, with c = 4.5. We conduct simulations for other choices of c [3, 5] and the results are qualitatively robust. We consider two classes of intervals. One is the interval class defined in Equation (16), for which we choose q = 1, 2,, L and choose L = 10. We also replicate all the simulations with L = 15 as a robustness check. The second class of intervals are formed by observed Y-values. For the observations with D=1, we first sort the Y-values. Suppose there are in total L 1 number of distinct observed values of Y. We then create the equal-distant integer subsequence of (1, 2,..., L 1 ) with length L 1 /10 (rounded up if fractional), and construct the grids by picking up the corresponding ordered statistics of Y. To put it in another way, we construct grids {g 1, g 2,, g L1/10 } such that there are 10 distinguished Y-values in each interval [g j, g j+1 ). Based on these grids, we construct intervals associate with D = 1 as the intervals formed by every possible combination of those grid points. The same operation applies to the observations with D = 0. 19

20 To briefly summarize, we consider ten DGPs. For each DGP, we consider four sample sizes, three nominal levels (1%, 5%, 10%), three bandwidth choices (AI, IK, CCT), two interval classes, and two bootstrap approaches. We conduct 1000 replications for each simulation and choose bootstrap sample size B = 300. Overall, the simulations results demonstrate very good finite sample performances. Please see Appendix 3 for details. To be written APPLICATION 7. EXTENSIONS AND DISCUSSIONS 7.1. Local smoothness and local independence. In this section, we discuss the identification and specification testing under a set of alternative assumptions that are often invoked in the literature. Let ɛ > 0 be a small constant, Assumption 11 (Local Independence). (Y d, D(r)) is jointly independent of R in a small neighborhood (r 0 ɛ, r 0 + ɛ). Assumption 12 (Local Monotonicity-2). D(r 0 + e) D(r 0 e) for any e (0, ɛ). Assumption 11 is often referred as local independence assumption, which assumes that the potential outcomes and treatment variables are independent of the running variable near the cut-off point. Note that local independence implies local smoothness (Assumption 2) since if the former is true, the conditional distribution will be flat and hence differentiable in r in the neighhood of r 0. Recently, Dong (2016) gives an interesting discussion regarding the local smoothness vs the local independence and provides some empirical evidences for which the local smoothness would be a more interesting assumption. Assumption 12, which we refer as local monotonicity-2, is an alternative monotonicity assumption from Assumption 3 in this paper. Local monotonicity-2 imposes restrictions on D(r) in the neighbourhood of r 0, rather than just at the limit. It is interesting to note that with Assumption 11, one can identify some parameters directly. For example, a researcher would be interested in knowing the impact of the treatment or a program on the dispersion measure like the variance because even when a treatment has a zero (local) average 20

21 treatment effect it would improve social welfare by lowering the dispersion of the potential outcome distribution. The following corollary provides such results. Corollary 1. Suppose Assumptions 1, 11 and 12 are satisfied. Let G( ) : Y R be a measurable function, then and E P [G(Y 1 ) C, R = r 0 ] = lim r r 0 E P [G(Y)D R = r] lim r r0 E P [G(Y)D R = r], lim r r0 E P [D R = r] lim r r0 E P [D R = r] E P [G(Y 0 ) C, R = r 0 ] = lim r r 0 E P [G(Y)(1 D) R = r] lim r r0 E P [G(Y)(1 D) R = r]. lim r r0 E P [D R = r] lim r r0 E P [D R = r] Note that for G(Y) = Y, we recover the LATE estimand derived in HTV, and for G(Y) = 1{Y y} we obtain the identification of the CDF of the potential outcomes for the compliers derived in FFM. An interesting direct application of this general identification result is the identification of the variance of the potential outcome distribution V P (Y 1 C). V P (Y 1 C) = lim r r 0 E P [Y 2 D R = r] lim r r0 E P [Y 2 D R = r] lim r r0 E P [D R = r] lim r r0 E P [D R = r] ( ) limr r0 E P [YD R = r] lim r r0 E P [YD R = r] 2. lim r r0 E P [D R = r] lim r r0 E P [D R = r] It is worth noting that V P (Y 1 C) can also been identified from the result in Equations (1) and (2) since it identifies the potential outcome distribution of the compliers F Y1 C(.) and calculate V P (Y 1 C) subsequently. However, this procedure involves multiples steps and make the inference procedure more involved, this would justify why empirical researchers do not reports such important parameter of interest in their study. This present identification result proposes an easy way to identify and estimate the V P (Y 1 C). Notice that because the variance is always positive, this identification result implies also an additional restriction on the data (Y, D, R) which is: lim r r0 E P [Y 2 D R = r] lim r r0 E P [Y 2 D R = r] lim r r0 E P [D R = r] lim r r0 E P [D R = r] ( ) limr r0 E P [YD R = r] lim r r0 E P [YD R = r] 2. lim r r0 E P [D R = r] lim r r0 E P [D R = r] More generally the local independence and local monotonicity-2, i.e., Assumptions 1, 11 and 12 deliver the following set of testable restrictions: Let G + a measurable function G + : Y R +. 21

22 Under Assumptions 1, 11 and 12, for any e (0, ɛ) we have: E P [G + (Y)D R = r 0 + e] = E P [G + (Y 1 )D(r 0 + e) R = r 0 + e] = E P [G + (Y 1 )D(r 0 + e)] E P [G + (Y 1 )D(r 0 e)] = E P [G + (Y 1 )D(r 0 e) R = r 0 e] = E P [G + (Y)D R = r 0 e], where the first and fourth equalities are by definition of potential variables, the second and third equalities are by local independence assumption, and the inequality is due to monotonicity and the non-negativity of G +. Similarly, we can derive the second testable implication involving Y 0. Both can be summarized as follows: For any e (0, ɛ) we have: E P [G + (Y)D R = r 0 e] E P [G + (Y)D R = r 0 + e] 0, (19) E P [G + (Y)(1 D) R = r 0 + e] E P [G + (Y)(1 D) R = r 0 e] 0. (20) Remark 3: The local independence appears to impose more restrictions on the observables. Indeed, by letting ɛ converge to 0 and restrict the class of positive measurable function G + to G we obtain inequalities (8) and (9). This is not surprising since the local independence is a stronger requirement. As before, the class of positive measurable function G + can also be reduced to G without losing information on the testability of Assumptions 1, 11 and 12. Remark 4: If the ɛ appears in Assumptions 11 and 12 were unknown to researchers, the sharp testable implications of local independence and local monotonicity-2 is actually the same as described in inequalities (8) and (9) since we can only look at the limit without knowledge of ɛ. However, if ɛ were known, then Assumptions 11 and 12 still imposes more restrictions on the data. It is worth stressing that while we may think that ɛ is unknown in general, in practice, applied researchers usually move away from the cut-off point to collect more observations in order to increase the significance of their estimates. This deviation is sometimes captured by the type of bandwidth they use in their non-parametric estimation. This result shows that researcher should be careful about the bandwidth they use since a wider one imposes more restriction on the data which make more likely the RD assumptions to be violated. The following corollaries summarize our results. 22

23 Corollary 2. Let D + and D be well defined. If ɛ is unknown, then the statement of Theorem 1 holds with Assumption 2 and Assumption 3 being replaced by Assumption 11 and Assumption 12, respectively. Corollary 3. Let D + and D be well defined and assume that ɛ is known. Let {D(r), Y 1, Y 0 } a vector of unobserved potential treatments and outcomes, and (Y, D, R) the vector of observed variables where Y = Y 1 D + Y 0 (1 D) with the observed D defined as follows D D(R). Let r 0 the observed cut-off point. (i) Under Assumptions 1, 11 and 12, then η 1 (l, e)e P [1{Y C l }D R = r + e] E P [1{Y C l }D R = r e] 0 (21) η 0 (l, e) = E P [1{Y C l }(1 D) R = r + e] E P [1{Y C l }(1 D) R = r e] 0 (22) for all l L and e (0, ɛ), and the inequalities hold strictly for C l = Y and sufficiently small e. (ii) If inequalities 21 and 22 hold, there exists a joint distribution of ( D(r), Ỹ 1, Ỹ 0 ) such that Assumptions 1, 11 and 12 hold for r (r 0 ɛ, r 0 + ɛ), and the conditional distributions of (Ỹ, D) R = r and (Y, D) R = r satisfy F Y,D R=r = FỸ, D R=r r (r 0 ɛ, r 0 + ɛ). (23) 7.2. Discussion on other specification tests. McCrary (2008) density test is the of the most used tests by empirical researchers to provide evidence that either supports or discounts the validity of the regression discontinuity design. McCrary (2008) suggests to examine the density of observations of the running variable, especially at the cut-off point r 0. If there is a discontinuity in the density at r 0, this may suggest that agents were able to perfectly manipulate their treatment status, in other terms being just above or below the cut-off would be no longer random. In such a case, it would "intuitively" provide some suggestive evidence of the violation of the local independence Assumption 11. Although this test has a very intuitive interpretation, the theoretical justification of this test was not very clear, especially in presence of Fuzzy design. Dong (2016) recently suggested a theoretical justification of this test that we will investigate below. Dong (2016) proposed the following assumption as an alternative assumption of Assumption 11 or Assumption 2 for identification of 23

24 parameters of interest. Denote f R Y1,Y 0,D +,D (.) the conditional density of the running variable R conditional to the joint vector of potential outcomes and the compliance groups. Assumption 13 (Strong Smoothness). f R Y1,Y 0,D +,D (.) is continuous in the neighborhood of r 0, and f R (.) is continuous and strictly positive in the neighborhood of r 0. First, notice that Assumption 13 implies Assumption 2 (i), then stronger that what required for identification. 5 Second, because the conditioning variable involves the unobserved potential outcomes, Assumption 13 is not directly testable but has as a testable implication the continuity of the unconditional density of the running variable at r 0 which is the test proposed by McCrary (2008). In addition of that, McCrary (2008) s test is not related to the monotonicity assumption which is required for identification in presence of heterogenous treatment effect as mainly in HTV and FFM. Therefore, unlike our proposed test, McCrary (2008) test is neither a sufficient nor necessarily for assessing the validity of the FRD assumptions, as illustrated on Figure 1. We assume that all data generated processes (DGPs) represented on this figure satisfy Assumption 1. The intersection of the "black" and the "red" ellipses represents all DGPs that respect the FRD design s assumptions. The "blue" ellipse depicted DGPs that respect the McCrary s null hypothesis. We clearly see that McCrary s test is not, in general, very informative about the validity of the FRD assumptions, especially when the treatment effect is heterogenous as entertained here. However, our proposed null hypothesis depicted in "green" covers all DGPs that respect the FRD assumptions. This shows the necessity of our test. In this figure, the sharpness (or sufficiency of our inequalities, in Pearl (1994) s language) means that there is no testable inequalities for which their set representation cover the FRD s assumption and is included in the "green" ellipse Incorporating Covariates. We briefly illustrate that it is straightforward to incorporate additional covariates X X R d x. Again, we implicitly assume that there are observations near the cutoff point conditioning on each realization of x. We make the following assumptions. Assumption 14. The limits π + (x) lim r r0 P P (D = 1 R = r, X = x) and π (x) lim r r0 P P (D = 1 R = r, X = x) exist and π + (x) = π (x) for all x X. 5 Refer to proof of Dong (2016) s Lemma. 24