Tolerating defiance? Local average treatment effects without monotonicity
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de Chaisemartin, Clément Article Tolerating defiance? Local average treatment effects without monotonicity Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: de Chaisemartin, Clément (2017) : Tolerating defiance? Local average treatment effects without monotonicity, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 8, Iss. 2, pp. 367-396, https://doi.org/10.3982/QE601 This Version is available at: https://hdl.handle.net/10419/195543 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by-nc/4.0/
Quantitative Economics 8 (2017), 367–396 1759-7331/20170367 Tolerating defiance? Local average treatment effects without monotonicity Clément de Chaisemartin Department of Economics, University of California at Santa Barbara Instrumental variables (IVs) are commonly used to estimate the effects of some treatments. A valid IV should be as good as randomly assigned, it should not have a direct effect on the outcome, and it should not induce any unit to forgo treatment. This last condition, the so-called monotonicity condition, is often implausible. This paper starts by showing that actually, IVs are still valid under a weaker condition than monotonicity. It then derives conditions that are sufficient for this weaker condition to hold and whose plausibility can easily be assessed in applications. It finally reviews several applications where this weaker condition is applicable while monotonicity is not. Overall, this paper extends the applicability of the IV estimation method. Keywords. Monotonicity, defiers, instrumental variable, average treatment effect, partial identification. JEL classification. C21, C26. 1. Introduction Applied economists study difficult causal questions, such as the effect of juvenile incarceration on educational attainment or the effect of family size on mothers’ labor supply. For that purpose, they often use instruments that affect entry into the treatment being studied and then estimate a two stage least squares regression (2SLS). As is well known, a valid instrument should be as good as randomly assigned and should not have a direct effect on the outcome. But even with an instrument satisfying these two conditions, the resulting 2SLS estimate might not capture any causal effect. People’s treatment participation can be positively affected, unaffected, or negatively affected by the instrument. Those in the first group are called compliers, those in the second are called noncompliers, and those in the third are called defiers. Noncompliers reduce the instrument’s statistical power as well as the external validity of the effect it estimates. But they do not threaten its internal validity. Indeed, Imbens and Angrist (1994) show that if the population only contains compliers and noncompliers, 2SLS estimates Clément de Chaisemartin: [email protected] I am very grateful to Josh Angrist, Sascha Becker, Stéphane Bonhomme, Federico Bugni, Laurent Davezies, Xavier D’Haultfœuille, Sara Geneletti, Walker Hanlon, Toru Kitagawa, Andrew Oswald, Azeem Shaikh, Roland Rathelot, Ed Vytlacil, Fabian Waldinger, Chris Woodruff, the co-editor, three anonymous referees, and participants at various conferences and seminars for their helpful comments. Copyright ©2017 The Author. Quantitative Economics. The Econometric Society. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://www.qeconomics.org. DOI: 10.3982/QE601
368 Clément de Chaisemartin Quantitative Economics 8 (2017) the average effect of the treatment among compliers, the so-called local average treatment effect (LATE). Defiers are a much more serious concern. If there are defiers in the population, we only know that 2SLS estimates a weighted difference between the effect of the treatment among compliers and defiers (see Angrist, Imbens, and Rubin (1996)). This difference could be a very misleading measure of the treatment effect: it could be negative, even when the effect of the treatment is positive in both groups. Defiers could be present in a large number of applications, and I will now give four examples which illustrate this situation. First, a number of papers have used randomly assigned judges with different sentencing rates as an instrument for incarceration (see Aizer and Doyle (2015) and Kling (2006)) or receipt of disability insurance (see Maestas, Mullen, and Strand (2013),French and Song (2014),andDahl, Kostøl, and Mogstad (2014)). Imbens and Angrist (1994) argue that the “no-defiers” condition is likely to be violated in these types of studies. In this context, ruling out the presence of defiers would require that a judge with a high average of strictness always hands down a more severe sentence than that of a judge who is on average more lenient. Assume judge A only takes into account the severity of the offense in her decisions, while judge B is more lenient toward poor defendants and more severe with well-off defendants. If the pool of defendants bears more poor than rich individuals, B will be on average more lenient than A, but she will be more severe with rich defendants. Second, defiers could be present in studies relying upon sibling-sex composition as an instrument for family size, because some parents are sex-biased. In the United States, parents are more likely to have a third child when their first two children are of the same sex. Angrist and Evans (1998) use this as an instrument to measure the effect of family size on mothers’ labor supply. However, some parents are biased toward one or the other sex. Dahl and Moretti (2008) show that in the United States, fathers have a preference for boys. Because of sex bias, some parents might want two sons, while others might want two daughters; such parents would be defiers. Third, defiers could be present in randomized controlled trials relying on an encouragement design. Duflo and Saez (2003) measure the effect of attending an information meeting on the take-up of a retirement plan. To encourage the treatment group to attend, subjects were given a financial incentive upon attendance. Deci (1971) and Frey and Jegen (2001) provide evidence showing that financial incentives sometimes backfire because they crowd-out intrinsic motivation. Sometimes, the crowding-out effect even seems to dominate: Gneezy and Rustichini (2000) find that fining parents who pick up their children late at daycare centers actually increased the number of late-coming parents. Accordingly, paying subjects to get treated in encouragement designs could lead some of them to forgo treatment. In this paper, I show that 2SLS still estimates a LATE if the no-defiers condition is replaced by a weaker “compliers–defiers” condition. If a subgroup of compliers accounts for the same percentage of the population as defiers and has the same LATE, 2SLS estimates the LATE of the remaining part of compliers. Compliers–defiers is the weakest condition on compliance types under which 2SLS estimates a LATE: if it is violated, 2SLS does not estimate a causal effect.
Quantitative Economics 8 (2017) Tolerating defiance? 369 The compliers–defiers (CD) condition is somewhat abstract, so I derive more interpretable sufficient conditions. I start by showing that CD holds if in each stratum of the population with the same value of their treatment effect there are more compliers than defiers. If that is the case, within each stratum one can form a subgroup of compliers with as many units as defiers. Pooling these subgroups across strata yields a subgroup of compliers accounting for the same percentage of the population as defiers and with the same LATE. I further show that with binary outcomes, CD holds if defiers’ LATE and the 2SLS coefficient are both of the same sign, or if defiers’ and compliers’ LATE are both of the same sign and the ratio of these two LATEs is lower than the ratio of the shares of compliers and defiers in the population, or if the difference between compliers’ and defiers’ LATEs is not larger than some upper bound that can be estimated from the data. These results have practical applicability. Maestas, Mullen, and Strand (2013) study the effect of disability insurance on labor market participation. Their 2SLS coefficient is negative. In standard labor supply models, disability insurance can only reduce labor market participation because it increases nonlabor income. It is therefore plausible that defiers’ LATE is negative and has the same sign as their 2SLS coefficient, thus implying that CD should hold in this study. Therefore, even though their coefficient might not estimate the LATE of compliers, it follows from my results that it still estimates the LATE of a subgroup of compliers. Later in the paper, I argue that this restriction on the sign of defiers’ LATE is also plausible in French and Song (2014),Aizer and Doyle (2015),and Duflo and Saez (2003).Angrist and Evans (1998) study the effect of having a third child on mothers’ labor market participation. I estimate the upper bound mentioned in the previous paragraph in their data, and find that it is large. On the other hand, there is no reason to suspect that defiers and compliers have utterly different LATEs: selection into one or the other population is driven by parents’ preferences for one or the other sex, not by gains from treatment. Therefore, CD should also hold in this application. Overall, the 2SLS method is applicable in studies in which defiers could be present, provided one can reasonably assume that defiers’ LATE has the same sign as the 2SLS coefficient or that compliers’ and defiers’ LATE do not differ too much. As I explain in more details later, my CD condition is also more likely to hold when the instrument has a large first stage. 2SLS is not the only statistical method requiring that there be no defiers. An important example is bounds for the average treatment effect (ATE) derived under the assumption that treatment effects have the same sign for all units in the population (see Bhattacharya, Shaikh, and Vytlacil (2008),Chesher (2010),Chiburis (2010),Shaikh and Vytlacil (2011),andChen, Flores, and Flores-Lagunes (2012)).1All of these bounds rely on the assumption that there are no defiers in the population. Actually, I show that these bounds are still valid under my CD condition. Other papers have studied relaxations of the no-defiers condition. Klein (2010) considers a model in which a disturbance uncorrelated with treatment effects leads some subjects to defy. By contrast, under my CD condition the factors leading some subjects 1Actually, Chen, Flores, and Flores-Lagunes (2012) only require that the LATEs of compliers, never-takers, always-takers, and defiers all have the same sign.
370 Clément de Chaisemartin Quantitative Economics 8 (2017) to defy can be correlated with treatment effects. Small and Tan (2007) show that if in each stratum of the population with the same value of their two potential outcomes there are more compliers than defiers, a condition they refer to as stochastic monotonicity,then 2SLS estimates a weighted average treatment effect. Nevertheless, some of their weights are greater than 1, so their parameter does not capture the effect of the treatment for a well defined subgroup, making it hard to interpret. Moreover, stochastic monotonicity is a stronger condition than CD. DiNardo and Lee (2011) derive a result similar to Small and Tan (2007).Huber and Mellace (2012) consider a local monotonicity assumption that requires that there be only compliers or defiers conditional on each value of the outcome. The CD condition allows for both compliers and defiers conditional on the outcome. Finally, Fiorini, Stevens, Taylor, and Edwards (2013) provide practitioners with recommendations as to how they should investigate the plausibility of the no-defiers condition in their applications. The remainder of the paper is organized as follows. Section 2concerns identification, Section 3concerns inference, Section 4concerns results of a simulation study, Section 5 concerns empirical applications, and Section 6concludes. Most proofs are deferred to the Appendix. For the sake of brevity, I consider some extensions in the Supplemental Material, available in a supplementary file on the journal website, http://qeconomics. org/supp/601/supplement.pdf, where I show that one can estimate quantile treatment effects among a subpopulation of compliers even if there are defiers, that one can test the CD condition, and that my results extend to multivariate treatment and instrument. 2. Identification 2.1 Identification of a LATE with defiers In this section, I show that with a binary instrument at hand, one can identify the LATE of a binary treatment on some outcome under a weaker assumption than no-defiers. The results presented in this section extend to more general settings with multivariate instrument and treatment. These extensions are deferred to the Supplemental Material. Imbens and Angrist (1994) study the causal interpretation of the coefficients of a 2SLS regression with binary instrument and treatment. Let Zbe a binary instrument. Let Dz∈{0;1}denote a subject’s potential treatment when Z=z.LetYdz denote her potential outcomes as functions of the treatment and of the instrument. Only Z,D≡DZ, and Y≡YDZ are observed. Following Angrist, Imbens, and Rubin (1996),letnevertakers (NT) be subjects such that D0=0and D1=0, let always-takers (AT) be such that D0=1and D1=1, let compliers (C)besuchthatD0=0and D1=1, and let defiers (F)2be such that D0=1and D1=0.LetFS=P(D =1|Z=1)−P(D =1|Z=0)denote the probability limit of the coefficient of the first stage (FS) regression of Don Z.Let RF =E(Y|Z=1)−E(Y|Z=0)denote the probability limit of the coefficient of the reduced form (RF) regression of Yon Z. Finally, let W=RF FS denote the probability limit of the coefficient of the second stage regression of Yon D. 2In most of the treatment effect literature, treatment is denoted by D. To avoid confusion, defiers are denoted by the letter Fthroughout the paper.
Quantitative Economics 8 (2017) Tolerating defiance? 371 Angrist, Imbens, and Rubin (1996) make a number of assumptions. First, they assume that FS =0. I will further assume throughout the paper that FS >0.Thisisamere normalization: if it appears from the data that FS <0, one can switch the words “defiers” and “compliers” in what follows. Under Assumption 1(see below), this normalization implies that more subjects are compliers than defiers: P(C)>P(F). Second, they assume that the instrument is independent of potential treatments and outcomes. Assumption 1 (Instrument Independence). We have (Y00Y01Y10Y11D0D1)⊥⊥Z Third, they assume that the instrument has no direct effect on the outcome. Assumption 2 (Exclusion Restriction). For all d∈{01}, Yd0=Yd1=Yd Last, they assume that there are no defiers in the population or that defiers and compliers have the same average treatment effect. Assumption 3 (No-Defiers (ND)). We have P(F)=0 Assumption 4 (Equal LATEs for Defiers and Compliers (ELATEs)). We have E(Y1−Y0|C) =E(Y1−Y0|F) The following proposition summarizes the three main results in Imbens and Angrist (1994) and Angrist, Imbens, and Rubin (1996). LATE Theorems (Imbens and Angrist (1994) and Angrist, Imbens, and Rubin (1996)). (i) Suppose Assumptions 1and 2hold.Then FS =P(C)−P(F) (1) W=P(C)E(Y1−Y0|C)−P(F)E(Y1−Y0|F) P(C)−P(F) (2) (ii) Suppose Assumptions 1,2,and 3hold.Then FS =P(C) (3) W=E(Y1−Y0|C) (4) (iii) Suppose Assumptions 1,2,and 4hold.Then W=E(Y1−Y0|C) (5)
372 Clément de Chaisemartin Quantitative Economics 8 (2017) Under random instrument and exclusion restriction alone, Wcannot receive a causal interpretation, as it is equal to a weighted difference of the LATEs of compliers and defiers. If there are no defiers, (1)and(2), respectively, simplify into (3)and(4). Then Wis equal to the LATE of compliers, while FS is equal to the percentage of the population compliers account for. Finally, when ND does not sound credible, Wcan still capture the LATE of compliers provided one is ready to assume that defiers and compliers have the same LATE, as shown in (5). In this paper, I substitute the following condition to Assumption 3or 4. Assumption 5 (Compliers–Defiers (CD)). There is a subpopulation of compliers CFthat satisfies P(CF)=P(F) (6) E(Y1−Y0|CF)=E(Y1−Y0|F) (7) CD is satisfied if a subgroup of compliers accounts for the same percentage of the population as defiers and has the same LATE. I call this subgroup compliers–defiers.CD is weaker than Assumptions 3and 4. If there are no defiers, one can find a zero probability subset of compliers with the same LATE as defiers. Similarly, if compliers and defiers have the same LATE, one can randomly choose P(F) P(C)%of compliers and call them compliers-defiers: this will yield a subgroup accounting for the same percentage of the population and with the same LATE as defiers. I can now state the main result of this paper. Theorem 2.1. Suppose Assumptions 1and 2hold.If a subpopulation of compliers CF satisfies (6)and (7), then CV=C\CFsatisfies P(CV)=FS(8) E(Y1−Y0|CV)=W (9) Conversely,if a subpopulation of compliers CVsatisfies (8)and (9), then CF=C\CV satisfies (6)and (7). Proof.(⇒)Wehave FS =P(C)−P(F)=P(CV)+P(CF)−P(F)=P(CV) The first equality follows from (1); the last follows from (6). This proves that CVsatisfies (8). Then E(Y1−Y0|C) =P(CV|C)E(Y1−Y0|CV)+P(CF|C)E(Y1−Y0|CF) =P(C)−P(F) P(C) E(Y1−Y0|CV)+P(F) P(C)E(Y1−Y0|F)
Quantitative Economics 8 (2017) Tolerating defiance? 373 where the last equality follows from (6)and(7). Plugging this into (2) yields W=E(Y1−Y0|CV) This proves that CVsatisfies (9). (⇐)Wehave P(CF)=P(C)−P(CV)=P(C)−FS =P(C)−P(C)−P(F)=P(F) The second step follows from (8); the third step follows from (1). This proves that CF satisfies (6). Then E(Y1−Y0|C) =P(CV|C)E(Y1−Y0|CV)+P(CF|C)E(Y1−Y0|CF) =FS P(C)W+P(F) P(C)E(Y1−Y0|CF) where the last equality follows from (8), (9), and (6). Plugging this equation into (2) yields E(Y1−Y0|F)=E(Y1−Y0|CF) This proves that CFsatisfies (7). This result is derived from (1)and(2), after using the law of iterated expectations and invoking Assumption 5. The intuition underlying it is as follows. Under CD, compliers– defiers and defiers cancel one another out, and the 2SLS coefficient is equal to the effect of the treatment for the remaining part of compliers. I hereafter refer to the CVsubpopulation as surviving-compliers, as they are compliers who “out-survive” defiers. The LATE in Theorem 2.1 is harder to grasp than the LATE identified under the nodefiers assumption. It does not apply to all compliers, but only to a subset of them, the surviving-compliers subpopulation. Note that under the no-defiers assumption, compliers account for the same percentage of the population as surviving-compliers under the CD assumption. Therefore, the LATE in Theorem 2.1 does not apply to a smaller population than the LATE identified under the no-defiers assumption. Moreover, as I show inthenextsubsection,onecanestimatethemeanofanycovariate(age,sex,...)among surviving-compliers under a mild strengthening of the CD assumption. Thus, the analyst can assess whether surviving-compliers strongly differ from the entire population. Still, surviving-compliers differ from compliers in that they are not fully characterized by their potential treatments. Knowing D0and D1is not sufficient to distinguish survivingcompliers (comvivors) from compliers–defiers (comfiers). Actually, in most instances even knowing Y1−Y0is not sufficient to tell the two populations apart. If a comvivor and a comfier have the same value of Y1−Y0, switching the comvivor to the comfier population, and the comfier to the comvivor population will not change the LATE and the size of the new comvivor and comfier populations. Thus, as soon as the supports of Y1−Y0in the two populations overlap, they are not uniquely defined.
374 Clément de Chaisemartin Quantitative Economics 8 (2017) This raises the question of whether this LATE is an interesting parameter. Some authors consider that treatment effect parameters are worth considering if they can inform treatment choice (see Manski (2005)). From that perspective, LATEs are not necessarily interesting: to decide whether she should give some treatment to her population, a utilitarian social planner needs to know the average treatment effect (ATE), not the LATE (see, e.g., Heckman and Urzúa (2010)). However, other authors have argued that researchers should still report an estimate of the LATE of compliers, along with the bounds on the ATE (see Imbens (2010)). Their arguments can be summarized as follows: reporting only the bounds might leave out relevant information; the LATE of compliers can give researchers an idea of the magnitude of the treatment effect; under some assumptions this LATE can be extrapolated to other populations (see Angrist and Fernandez-Val (2013)). In a world with defiers, these arguments no longer apply. In such a world, the LATE of compliers is not even identified. Only the LATE of surviving-compliers can be identified. Accordingly, it is this parameter that should be reported along with bounds on the ATE.3 A great appeal of the ND condition is that it is simple to interpret. On the contrary, CD is an abstract condition. I try to clarify its meaning by deriving more interpretable conditions under which it is satisfied. A sufficient condition for CD to hold I start by considering a condition that is sufficient for CD to hold irrespective of the nature of the outcome. Let R(P(F)) =1+FS P(F). Notice that (1)impliesthatR(P(F)) =P(C) P(F). Therefore, R(P(F)) is merely the ratio of the shares of compliers and defiers in the population. Assumption 6 (More Compliers Than Defiers (MC)). For every δin the support of Y1−Y0, fY1−Y0|F(δ) fY1−Y0|C(δ) ≤RP(F)(10) I call this condition the more compliers than defiers condition. Indeed, as R(P(F)) = P(C) P(F),(10) is equivalent to P(F|Y1−Y0)≤P(C|Y1−Y0) (11) Equation (11) requires that each subgroup of the population with the same value of Y1−Y0comprise more compliers than defiers. This condition is weaker but closely related to the stochastic monotonicity assumption in Small and Tan (2007). For instance, their condition is satisfied if P(F|Y0Y1)≤P(C|Y0Y1), that is, if in each stratum of the population with the same value of their two potential outcomes there are more compliers than defiers. As shown in Angrist, Imbens, and Rubin (1996), 2SLS estimates a LATE if there are no defiers or if defiers and compliers have the same distribution of Y1−Y0.Theseassumptions are “polar cases” of MC. MC holds when defiers and compliers have the same 3The extrapolation strategy proposed in Angrist and Fernandez-Val (2013) under the no-defiers assumption can also be used under the compliers–defiers assumption introduced in this paper.
Quantitative Economics 8 (2017) Tolerating defiance? 381 loss of generality, one can assume that V0and V1have the same marginal distributions, and that v1≤v0to account for the fact that P(D1=1)≥P(D0=1). Compliers satisfy {V0<v 0V1≥v1}. Defiers satisfy {V0≥v0V1<v 1}: the instrument substantially diminishes their taste for treatment, which induces them not to get treated when they receive it. Vytlacil (2002) shows that ND is equivalent to imposing V0=V1.Iwillnotmake this assumption here to allow for defiers. On the other hand, I will assume that (V0V1 Y1−Y0)is jointly normal: ⎛ ⎜ ⎝ V0 V1 Y1−Y0 ⎞ ⎟ ⎠→N⎛ ⎜ ⎝⎛ ⎜ ⎝ 0 0 μ⎞ ⎟ ⎠⎛ ⎜ ⎝ 1ρV0V1σΔρV0Δ ρV0V11σΔρV1Δ σΔρV0Δ σΔρV1Δ σ2 Δ ⎞ ⎟ ⎠⎞ ⎟ ⎠ Let Σdenote the variance of this vector, and let V0and V1be normalized to have mean 0and variance 1. I further assume that σ2 Y0=1and σ2 Y0=σ2 Y1. The first assumption is a mere normalization, which corresponds to the common practice of standardizing the outcome by its standard deviation in empirical work. The second assumption is a homoscedasticity condition. Together, they imply that σ2 Δ≤4. The data also impose a number of restrictions on the parameters of this model, revealing v0and v1: vz=−1(P(D =0|Z=z)),where(·)denotes the cumulative distribution function (c.d.f.) of a standard normal variable. It also imposes that ρV1Δ be written as a function of μ,σΔ,andρV0Δ, ρV1Δ =RF −μFS σΔφ(v1)+ρV0Δ φ(v0) φ(v1) where φ(·)is the probability density function (p.d.f.) of a standard normal. Combining the last equation with 0≤σΔ≤√4,−1≤ρV0Δ ≤1,and−1≤ρV1Δ ≤1,onecanshow that the data also bound μ: μ=RF −2φ(v0)+φ(v1) FS ≤μ≤μ=RF +2φ(v0)+φ(v1) FS Overall, the parameters of the model are partially identified, and the identified set is defined by the constraints θ=μσ2 ΔρV0V1ρV0Δ∈Θ=[μμ]×[04]×[−11]×[−11] ρV1Δ(θ) =RF −μFS σΔφ(v1)+ρV0Δ φ(v0) φ(v1)∈[−11] Σis positive definite Finally, note that if ρV0Δ =ρV1Δ, CD is satisfied. Indeed, we then have (V0V1)|Y1−Y0∼ (V1V0)|Y1−Y0,soCF={V1≥v0V0<v 1}satisfies (6)and(7): P(CF)=P(V1≥v0V0<v 1)=P(V0≥v0V1<v 1)=P(F) E(Y1−Y0|CF)=E(Y1−Y0|V1≥v0V0<v 1)(18) =E(Y1−Y0|V0≥v0V1<v 1)=E(Y1−Y0|F)
382 Clément de Chaisemartin Quantitative Economics 8 (2017) In my simulations, I consider a first numerical example in which P(D =1|Z=1)=04, P(D =1|Z=0)=01,andW=02. This could, for instance, correspond to a randomized experiment with a first stage of 30% and with a 2SLS coefficient equal to 20% of the standard deviation of the outcome. I also consider a second numerical example in which P(D=1|Z=1)=02,P(D=1|Z=0)=01,andW=02. This could, for instance, correspond to a randomized experiment with a weaker first stage of 10% and the same 2SLS coefficient. For each numerical example, I draw a sample of 4,000 vectors of parameters representative of the population of parameters compatible with the data. To do so, Idrawvaluesforθfrom the uniform distribution on Θ, and keep only those such that ρV1Δ(θ) ∈[−11]and Σis positive definite. For each vector of parameters, I draw 100,000 realizations from the corresponding distribution of (V0V1Y1−Y0). This also gives me 100,000 realizations of (D0D1Y1−Y0). For each of these 4,000 empirical distributions of (D0D1Y1−Y0), I assess whether it satisfies the CD assumption using an algorithm presented in the Appendix. The main results from this exercise are as follows. First, the larger the instrument’s first stage, the more CD is likely to hold. While in the first numerical example CD is satisfied for 67% of the 4,000 data generating points (DGPs) considered, in the second example it is only satisfied for 43% of them. Second, CD is more likely to hold when the LATE of defiers has the same sign as the 2SLS coefficient. Most DGPs for which E(Y1−Y0|F) ≥0satisfy CD. However, some DGPs for which E(Y1−Y0|F) is very large violate it. For instance, across the 4,000 DGPs in the first numerical example, the DGP with the lowest positive value of E(Y1−Y0|F) for which CD is violated has E(Y1−Y0|F)=086σY0, a very large treatment effect. Third, the difference between ρV1Δ and ρV0Δ seems to be the main determinant of whether CD is satisfied or not in this model. A regression of a dummy for whether CD is satisfied on |ρV1Δ −ρV0Δ|has an R2 of 066. Adding (μσ2 ΔρV0V1ρV0ΔρV1Δ)to this regression hardly adds any explanatory power. These results might help applied researchers to assess whether CD is likely to hold when their outcome of interest is continuous. When their 2SLS coefficient is, say, positive, they can assess whether defiers are likely to have a negative or a very large positive treatment effect. If that sounds unlikely, CD is likely to hold. Similarly, when their first stage is large, they can be more confident that their results are robust to defiers than when it is weak. To conclude this section, it is worth noting that results presented in this paper generalize to the local IV approach introduced in Heckman and Vytlacil (1999) and Heckman and Vytlacil (2005). These authors show that with a continuous instrument Zsatisfying Assumptions 1and 2,if(17) is satisfied with (i) Vz=Vfor every zin the support of Zand (ii) vzdecreasing in z, then under some regularity conditions ∂E(Y |P(D=1|Z=z)=p) ∂p is equal to the average treatment effect of units at the 1−pth quantile of the distribution of V. This result can be extended to selection equations where Vzis allowed to vary across values of z, under a generalization of the CD condition. For instance, if for every z1in the support of Zthere is a z0<z 1such that for every z∈[z0z1]there is a subset of the {Vz1≥vz1Vz<v z}subpopulation accounting for the same percentage of the total population and with the same average treatment effect as the {Vz1<v z1Vz≥vz}
Quantitative Economics 8 (2017) Tolerating defiance? 383 subpopulation, then ∂E(Y|P(D=1|Z)=p) ∂p is equal to the average treatment effect of units at the 1−pth quantile of the distribution of Vzp,wherezpis the unique solution of P(D=1|Z=z) =p. 5. Applications In this section, I show how one can use the previous results in various applications where it is likely that defiers are present. Maestas, Mullen, and Strand (2013) and French and Song (2014) Maestas, Mullen, and Strand (2013) study the effect of receiving disability insurance (DI) on labor market participation. They use average allowance rates of randomly assigned examiners as an instrument for receipt of DI. In this context, Y1≤Y0is a plausible restriction.9It is, for instance, satisfied in a static labor supply model under standard restrictions on agents’ utility functions. Assume agents’ utilities depend on consumption Cand leisure L. To simplify, assume agents can only work fulltime or not work at all, which is denoted by a dummy Y. To choose Y, agents maximize U(CL) subject to C=YW +Iand L=T−HY,whereW,I,H,andT, respectively, denote agents’ wages, their nonlabor income, the amount of time spent on a fulltime job, and the total amount of time available. Let UCC,ULL,andUCL, respectively, denote the second order and cross derivatives of U, and assume that UCC ≤0,ULL ≤0,andUCL ≥0, a property satisfied by most standard utility functions. Let I0<I 1denote agents’ nonlabor income without and with disability insurance, and let Y0and Y1denote their corresponding labor market participation decisions. As is well known, UCC ≤0,ULL ≤0,andUCL ≥0 imply that U(W +IT −HY) −U(IT) is increasing in I, which in turn implies that Y1≤Y0. The 2SLS coefficient in this study is significantly negative. Following the discussion in the previous paragraph, Assumption 7is plausible in this context: it will hold if E(Y1−Y0|F) is not strictly greater than 0, something which will be automatically satisfied if Y1≤Y0.10 Therefore, one can invoke Theorems 2.3 and 2.1 to claim that this coefficient consistently estimates the LATE of surviving-compliers, even though it might not be consistent for the LATE of compliers because of defiers. Moreover, Y1≤Y0also implies that Assumption 12 is satisfied. One could then use Theorem 2.6 to estimate bounds for the ATE in this application.11 9Ex ante restrictions on the sign of the treatment effect are usually called monotone treatment response assumptions and were first introduced by Manski (1997). 10The instrument used in Maestas, Mullen, and Strand (2013) is multivariate. Theorem 2.3 can easily be extended to this type of setting, assuming that Assumption 7holds within the sample of cases dealt with by each pair of judges. In the Supplemental Material, I cover in more details the case of multivariate instruments. 11The Disability Operational Data Store (DIODS) data archives used in this paper contain personally identifiable information. It is only possible to access them at a secure location, after having signed an agreement with the U.S. Social Security Administration.
384 Clément de Chaisemartin Quantitative Economics 8 (2017) Finally, French and Song (2014) also study the effect of disability insurance on labor supply and find a strictly negative 2SLS coefficient. Following the same line of argument as in the previous paragraph, the CD condition should also hold in this study. Aizer and Doyle (2015) Aizer and Doyle (2015) study the effect of juvenile incarceration on high school completion. They use average sentencing rates of randomly assigned judges as an instrument for incarceration. Here as well Y1≤Y0sounds like a plausible restriction. Being incarcerated disrupts schooling and increases the chances the youth form relationships with nonacademically oriented peers. This should increase the chances of dropout. Their 2SLS coefficient is significantly negative, so Assumption 7is also plausible in this context. Therefore, one can invoke Theorems 2.3 and 2.1 to claim that this coefficient consistently estimates the LATE of surviving-compliers. Angrist and Evans (1998) Angrist and Evans (1998) study the effect of having a third child on mothers’ labor supply. In their study, P(F)=372%,andthe95% confidence upper bound for P(F)constructed using Theorem 1 in Andrews and Soares (2010) is 374%. The left axis of Figure 3shows the sample counterpart of Δ(P(F)) for all values of P(F) included between 0and 374%. The right axis shows the same quantity normalized by the standard deviation of the outcome. Assumption 9is satisfied for values of P(F)and |E(Y1−Y0|C)−E(Y1−Y0|F)|below the black solid line. For instance, Δ(005)=0072.12 Therefore, Assumption 9holds if there are less than 5% of defiers and the LATEs of compliers and defiers differ by less than 72percentage points, or 145% of a standard deviation of the outcome. The limited evidence available suggests that 5% is a conservative upper bound for the share of defiers in this application. In the 2012 Peruvian wave of the Demographic and Health Figure 3. For all values of P(F) and |E(Y1−Y0|C) −E(Y1−Y0|F)|below the green line, the compliers–defiers condition is satisfied in Angrist and Evans (1998). 12The 95% confidence interval of Δ(005)is [00440100]. It can be estimated using standard Stata commands. A code is available upon request.
Quantitative Economics 8 (2017) Tolerating defiance? 385 Surveys, women were asked their ideal sex sibship composition. Among women whose first two offspring are a boy and a girl, 18% had three children or more and retrospectively declare that their ideal sex sibship composition would have been two boys and no girl, or no boy and two girls. These women seem to have been induced to having a third child because their first two children were a boy and a girl. To my knowledge, similar questions have never been asked in a survey in the United States: 18% could underor overestimate the share of defiers in the U.S. population. But this figure is, as of now, the best piece of evidence available to assess the percentage of defiers in Angrist and Evans (1998). Therefore, 5% sounds like a reasonably conservative upper bound. 15% of a standard deviation is also a reasonably conservative upper bound for |E(Y1−Y0|C) −E(Y1−Y0|F)|in this application. Compliers are couples with a preference for diversity, while defiers are sex-biased couples. Preference for diversity and sex bias are probably correlated with some of the variables entering into mothers’ decisions towork(mothers’potentialwages,preferencesforleisure,...),buttheyareunlikelyto enter directly into that decision. As a result, 15% of a standard deviation is arguably a conservative upper bound for |E(Y1−Y0|C)−E(Y1−Y0|F)|, because selection into being a complier or a defier is not directly based on gains from treatment. Duflo and Saez (2003) Duflo and Saez (2003) conduct a randomized experiment with an encouragement design to study the effect of an information meeting on take-up of a retirement plan. To encourage the treatment group to attend, subjects were given a financial incentive upon attendance. Unless it is poorly designed, the meeting should not reduce take-up. In this context, Y1≥Y0sounds like a plausible restriction. The authors’ 2SLS coefficient is significantly positive, so Assumption 7is also plausible in this context. Therefore, one can invoke Theorems 2.3 and 2.1 to claim that this coefficient consistently estimates the LATE of surviving-compliers. 6. Conclusion Applied economists often use instruments affecting the take-up of a treatment to estimate its effect. When doing so, the methods they use rely on a monotonicity assumption. In many instances, this assumption is not applicable. In this paper, I show that these methods are still valid under a weaker condition than monotonicity. Doing so, I extend the applicability of these methods. Specifically, I show that researchers can confidently use them in applications where one can reasonably assume that defiers’ LATE has the same sign as the reduced form effect of the instrument on the outcome, or that compliers’ and defiers’ LATEs do not differ too much. My weaker condition is also more likely to hold when the instrument has a strong first stage. I put forward examples where my weaker condition is likely to hold, while monotonicity is likely to fail. Appendix A: The CD algorithm In this section, I present the CD algorithm used in Section 4to assess whether a joint distribution of (D0D1Y1−Y0)satisfies Assumption 5.
386 Clément de Chaisemartin Quantitative Economics 8 (2017) Theorem A.1. Assume that Y1−Y0|Cis dominated by the Lebesgue measure on R,and that its density relative to this measure is strictly positive on the support of Y1−Y0|C.13 If RF ≥0,one can use the following algorithm to assess whether Assumption 5is satisfied: (i) If E((Y1−Y0)1{Y1−Y0≥0}1{C})<RF, Assumption 5is violated. (ii) Else,let δ0≥0solve E((Y1−Y0)1{Y1−Y0≥δ}1{C})=RF. If P(Y1−Y0≥δ0C)> FS, Assumption 5is violated. (iii) Else,if E((Y1−Y0)1{Y1−Y0≤δ0}1{C})≤0,let δ1solve E((Y1−Y0)× 1{Y1−Y0∈[δδ0]}1{C})=0. 1. If P(Y1−Y0≥δ1C)≥FS, Assumption 5is satisfied. 2. Else,Assumption 5is violated. (iv) Else,if E((Y1−Y0)1{Y1−Y0≤δ0}1{C})>0,let δ2solve E((Y1−Y0)× 1{Y1−Y0≤δ}1{C})=RF. 1. If P(Y1−Y0≤δ2C)≥FS, Assumption 5is satisfied. 2. Else,Assumption 5is violated. If RF <0,one can substitute −(Y1−Y0)to Y1−Y0in the previous algorithm. The intuition for this theorem is as follows. Assume RF ≥0. If CD holds, there must be a subpopulation of compliers such that P(CV)=FS and E((Y1−Y0)1{CV})=RF. If E((Y1−Y0)1{Y1−Y0≥0}1{C})<RF, CD must be violated, because for any subpopulation of compliers, E((Y1−Y0)1{CV})≤E((Y1−Y0)1{Y1−Y0≥0}1{C}). Even summing the treatment effects for all compliers who gain from treatment is not enough to reach the numerator of the 2SLS coefficient. Similarly, if P(Y1−Y0≥δ0C)>FS, CD must be violated: even the smallest subpopulation of compliers such that E((Y1−Y0)1{CV})= RF is already too large. The following steps of the algorithm follow from similar arguments. Appendix B: Proofs In the proofs, I assume the probability distributions of Y1−Y0,Y1−Y0|C,andY1−Y0|F are all dominated by the same measure λ.LetfY1−Y0,fY1−Y0|C,andfY1−Y0|Fdenote the corresponding densities. I also adopt the convention that 0 0×0=0. Lemma B.1. (i) A subpopulation of compliers CFsatisfies (6)and (7)if and only if there is a realvalued function gdefined on S(Y1−Y0)such that 0≤g(δ) ≤fY1−Y0|C(δ)P(C) for λ-almost every δ∈S(Y1−Y0) (19) 13This ensures that the numbers δ0,δ1, and δ2introduced hereafter are uniquely defined.
Quantitative Economics 8 (2017) Tolerating defiance? 387 S(Y1−Y0) g(δ)dλ(δ) =P(F) (20) S(Y1−Y0) δg(δ) P(F) dλ(δ) =E(Y1−Y0|F) (21) (ii) A subpopulation of compliers CVsatisfies (8)and (9)if and only if there is a realvalued function hdefined on S(Y1−Y0)such that 0≤h(δ) ≤fY1−Y0|C(δ)P(C) for λ-almost every δ∈S(Y1−Y0) (22) S(Y1−Y0) h(δ)dλ(δ) =FS(23) S(Y1−Y0) δh(δ) FS dλ(δ) =W (24) Proof.InviewofTheorem2.1, the proof will be complete if I can show the if part of the first statement, the only if part of the second statement, and finally that if a function h satisfies (22), (23), and (24), then a function gsatisfies (19), (20), and (21). I start by proving the if part of the first statement. Assume a function gsatisfies (19), (20), and (21). Densities being uniquely defined up to 0probability sets, I can assume without loss of generality that those three equations hold everywhere. Let p(δ) =g(δ) fY1−Y0|C(δ)P(C)1fY1−Y0|C(δ) > 0 It follows from (19) that p(δ) is always included between 0and 1.ThenletBbe a Bernoulli random variable such that P(B =1|CY1−Y0=δ) =p(δ).Finally,letCF= {CB =1}.Then P(CF)=EP(CF|Y1−Y0) =EP(C|Y1−Y0)P(B =1|CY1−Y0) =EP(C|Y1−Y0)g(Y1−Y0) fY1−Y0|C(Y1−Y0)P(C)1fY1−Y0|C(Y1−Y0)>0 =Eg(Y1−Y0) fY1−Y0(Y1−Y0) =S(Y1−Y0) g(δ)dλ(δ) =P(F) The first equality follows from the law of iterated expectations, the second from the definition of CFand Bayes, the third from the definition of B, the fourth from the fact that under (19), fY1−Y0|C(δ)P(C) =0⇒g(δ) =0,andthelastfrom(20). This proves that CF satisfies (6).
388 Clément de Chaisemartin Quantitative Economics 8 (2017) Then E(Y1−Y0|CF)=E(Y1−Y0)1{CF} P(CF) =E(Y1−Y0)P(CF|Y1−Y0) P(CF) = E(Y1−Y0)g(Y1−Y0) fY1−Y0(Y1−Y0) P(CF) =S(Y1−Y0) δg(δ) P(F) dλ(δ) =E(Y1−Y0|F) The fourth equality follows from (6) and the fifth follows from (21). This proves that CF satisfies (7). I now prove the only if part of the second statement. Assume a subset of Cdenoted CVsatisfies (8)and(9). Then h=fY1−Y0|CVP(CV)must satisfy (22); otherwise we would not have CV⊆C. It must also satisfy (23)and(24); otherwise CVwould not satisfy (8) and (9). I finally show the last point. Assume hsatisfies (22), (23), and (24). Then it follows from (1)and(2) that g=fY1−Y0|CP(C)−hsatisfies (19), (20), and (21). Proof of Theorem 2.2. Under Assumption 6,g1=fY1−Y0|FP(F) satisfies (19), (20), and (21). Proof of Theorem 2.3. I only prove the result when RF >0. The proof follows from a symmetric reasoning when RF <0.WhenRF=0, proving the equivalence and the first implication becomes trivial. To prove the second implication, if E(Y1−Y0|F) ≥0,one can use the same reasoning as that used for RF >0, while if E(Y1−Y0|F) ≤0,onecan use the same reasoning as that used for RF <0. I first prove that Assumption 9⇒Assumption 7. As I have assumed 0<RF, Assumption 7implies that 0≤E(Y1−Y0|F). Rearranging (2) yields E(Y1−Y0|C)−E(Y1−Y0|F)=FS FS +P(F)W−E(Y1−Y0|F) Assumption 9is therefore equivalent to W−E(Y1−Y0|F)≤W which implies that 0≤E(Y1−Y0|F). This proves the result. Then I prove that Assumption 7⇔Assumption 8. Let Assumption 7be satisfied with 0=E(Y1−Y0|F). As I have assumed 0<RF, Assumption 7implies that 0<E(Y 1−Y0|F). Then it follows from (2) that E(Y1−Y0|C) must also be strictly positive. Finally, rearranging (2) yields E(Y1−Y0|F) E(Y1−Y0|C) ≤P(C) P(F). This proves that Assumption 8is satisfied. If Assumption 7is satisfied with E(Y1−Y0|F) =0, Assumption 8is also trivially satisfied. Conversely, if Assumption 8is satisfied with E(Y1−Y0|F)=0,onehas1≤P(C)E(Y1−Y0|C) P(F)E(Y1−Y0|F).
Quantitative Economics 8 (2017) Tolerating defiance? 389 Using (2), this in turn implies that 0≤RF P(F)E(Y1−Y0|F), thus proving that either RF =0or E(Y1−Y0|F) has the same sign as RF. This proves that Assumption 7is satisfied. If Assumption 8is satisfied with E(Y1−Y0|F) =0, Assumption 7is also trivially satisfied. This proves the result. Finally, I prove that Assumption 7⇒Assumption 5. To do so, I show that if Assumption 7is satisfied, there is a function h1satisfying (22), (23), and (24). In view of Lemma B.1, this will prove the result. As I have assumed 0<RF, Assumption 7implies that 0≤E(Y1−Y0|F). With binary potential outcomes, this is equivalent to 0≤P(Y1−Y0=1F)−P(Y1−Y0=−1F). With binary potential outcomes, (2) simplifies to P(Y1−Y0=1C)−P(Y1−Y0=−1C) =RF +P(Y1−Y0=1F)−P(Y1−Y0=−1F) (25) Once combined with (25), Assumption 7implies RF ≤P(Y1−Y0=1C) (26) Then notice that FS −RF −P(Y1−Y0=0C) (27) =2P(Y1−Y0=−1C)−2P(Y1−Y0=−1F)+P(Y1−Y0=0F) FS +RF −P(Y1−Y0=0C) (28) =2P(Y1−Y0=1C)−2P(Y1−Y0=1F)+P(Y1−Y0=0F) Now, consider the function h1defined on {−101}and such that h1(−1)=max0FS −RF −P(Y1−Y0=0C) 2 h1(0)=minP(Y1−Y0=0C)FS −RF h1(1)=maxRFFS +RF −P(Y1−Y0=0C) 2 If FS −RF ≤P(Y1−Y0=0C), then h1(−1)=0 h1(0)=FS −RF h1(1)=RF The term h1(−1)is trivially included between 0and P(Y1−Y0=−1C). The inequality 0≤h1(0)follows from the fact that by assumption |W|≤1. By assumption, we also have h1(0)≤P(Y1−Y0=0C) and 0≤h1(1). The inequality h1(1)≤P(Y1−Y0=1C) follows from (26). This proves that h1satisfies (22). It is easy to see that it also satisfies (23) and (24).
390 Clément de Chaisemartin Quantitative Economics 8 (2017) If FS −RF >P(Y 1−Y0=0C), then h1(−1)=FS −RF −P(Y1−Y0=0C) 2 h1(0)=P(Y1−Y0=0C) h1(1)=FS +RF −P(Y1−Y0=0C) 2 The term h1(−1)is greater than 0by assumption; h1(−1)≤P(Y1−Y0=−1C) follows from (27); h1(0)is trivially included between 0and P(Y1−Y0=0C);h1(1)is greater than 0because it is greater than h1(−1);h1(1)≤P(Y1−Y0=1C) follows from (28). This proves that h1satisfies (22). It is easy to see that it also satisfies (23)and(24). Proof of Theorem 2.4. Following the same steps as those used by Angrist, Imbens, and Rubin (1996) to prove (1)and(2), one can show that under Assumptions 10 and 2, for every xin the support of X, E(D|Z=1X =x) −E(D|Z=0X =x) =P(C|X=x) −P(F|X=x) E(Y|Z=1X=x) −E(Y|Z=0X =x) =E(Y1−Y0|CX =x)P(C|X=x) −E(Y1−Y0|FX =x)P(F|X=x) Therefore, EE(D|Z=1X)−E(D|Z=0X)=P(C)−P(F) EE(Y|Z=1X)−E(Y|Z=0X)=E(Y1−Y0|C)P(C) −E(Y1−Y0|F)P(F) Under Assumption 5, one can apply to the right-hand side of the previous display the same steps as in the proof of Theorem 2.1. One finally obtains EE(D|Z=1X)−E(D|Z=0X)=P(CV) EE(Y|Z=1X)−E(Y|Z=0X)=E(Y1−Y0|CV)P(CV) This proves the result. Proof of Theorem 2.5.InviewofTheorem2.1, it is sufficient to show that if a subpopulation of compliers CFsatisfies (6), (7), and (15), then CV=C\CFsatisfies (16). Using the same steps as those used in Angrist, Imbens, and Rubin (1996) to prove (2), one can show that WXD =P(C)E[X|C]−P(F)E[X|F] P(C)−P(F) Then it follows from (6)and(15) that E[X|C]=P(C)−P(F) P(C) E[X|CV]+P(F) P(C)E[X|F] Plugging this equation into the previous one yields the result.