Treatment response with social interactions: Partial identification via monotone comparative statics

Lazzati, Natalia Article Treatment response with social interactions: Partial identification via monotone comparative statics Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Lazzati, Natalia (2015) : Treatment response with social interactions: Partial identification via monotone comparative statics, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 6, Iss. 1, pp. 49-83, https://doi.org/10.3982/QE308 This Version is available at: https://hdl.handle.net/10419/150380 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. 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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/3.0/ Quantitative Economics 6 (2015), 49–83 1759-7331/20150049 Treatment response with social interactions: Partial identification via monotone comparative statics Natalia Lazzati University of Michigan, Ann Arbor This paper studies (nonparametric) partial identification of treatment response with social interactions. It imposes conditions motivated by economic theory on the primitives of the model, that is, the structural equations, and shows that they imply shape restrictions on the distribution of potential outcomes via monotone comparative statics. The econometric framework is tractable and allows for counterfactual predictions in models with multiple equilibria. Under three sets of assumptions, we identify sharp distributional bounds on the potential outcomes given observable data. We illustrate our results by studying the effect of police per capita on crime rates in New York state. Keywords. Treatment effects, social interactions, nonparametric bounds, supermodular games, monotone comparative statics, first order stochastic dominance. JEL classification. C31, D71. 1. Introduction This paper studies partial identification of treatment response in environments with endogenous social interdependencies. During the last three decades, social interactions have become an essential component of economic analysis. Activities we suspect are subject to strong social pressure include crimes, schooling, and fertility decisions.1Models of network goods and two-sided markets display this feature as well. Despite the attention received by this kind of interdependence in many areas of economics, just a few studies of treatment response incorporate the social dimension.2That is, standard models assume that each person’s outcome varies only with her own treatment. To accommodate the applications above, we allow individual outcomes to also depend on Natalia Lazzati: [email protected] The author is deeply indebted to Keisuke Hirano and Rabah Amir, and the other members of her dissertation committee: Stanley Reynolds, Mark Walker, and John Wooders. She especially thanks Charles Manski, Lawrence Blume, Steven Durlauf, Dan Ackerberg, Amilcar Menichini, Francesca Molinari, Joerg Stoye, Dan Silverman, Christopher Flinn, Gautam Gowrisankaran, Ivana Komunjer, and the other participants at the Conference on Econometric Analysis of Social Interactions at Northwestern University, ESAM (Singapore), NYU Stern, University of Connecticut, University of Warwick, UCSD, Caltech, Cornell University, University of Arizona, University of Kansas, and University of Michigan. Last, but not least, she is very thankful to three anonymous referees and the editor, Orazio Attanasio, for their helpful suggestions. 1See Blume, Brock, Durlauf, and Ionnides (2010) for many other applications. 2See, for example, Shaikh and Vytlacil (2011). Copyright ©2015 Natalia Lazzati. Licensed under the Creative Commons Attribution-NonCommercial License 3.0. Available at http://www.qeconomics.org. DOI: 10.3982/QE308 50 Natalia Lazzati Quantitative Economics 6 (2015) the outcomes of other people in the population. Under this setup, we use monotone comparative statics to provide distributional bounds on treatment effects. Models of endogenous interactions often start with the outline of a system of structural equations that define the outcome of each individual as a function of observable characteristics and the outcomes of the other group members. The solution to the system of equations (when it exists) is the predicted outcome or behavior of the group members. When there are multiple equilibria, observed behavior also depends on the mechanism by which people select among them. We assume the analyst observes a vector of realized outcomes and treatments for a sample of groups. The objective of the researcher is to learn about the potential outcome distribution that would occur in the study population if the groups were to receive a specific treatment. Our approach for identification consists of two steps. We first impose monotone restrictions on the primitives of the model (i.e., the structural equations) and derive their implications on the predicted outcomes of each group. We then translate the monotone comparative statics results into sharp distributional bounds for the potential outcomes. Our approach is robust to the possibility of multiple equilibria. Most of the research in econometrics is concerned with the identification of the structural equations. This is indeed the case of recent results in the econometric analysis of game theoretic models; see Bajari, Hahn, Hong, and Ridder (2011) for an updated overview.3Following Manski (2013), our objective is (partial) identification of the potential outcome distributions under alternative treatment rules, not the structural functions per se. To achieve this goal, Manski (2013) proceeds by imposing the identification restrictions directly on the equilibrium behavior of the agents. Our approach differs from his in that we impose all shape conditions on the primitives of the structural model and derive their implications on the solution sets. Once this connection is established, our final propositions follow from Manski (2013) with the sole difference that we need to adapt his proofs to multivariate distributions. We thereby provide the microfoundations for some of his novel results and make explicit the strength of the identifying assumptions. The first two conditions we impose are as follows: the outcome of each individual increases with the outcomes of the others and it varies monotonically with the treatment to be received by the group. These conditions imply clear restrictions on the predicted outcomes: The system of structural equations leads to an increasing function that maps possible outcomes into itself, so that the set of solutions of the model coincides with the set of fixed points of this artificial function. By Tarski’s fixed point theorem, the first assumption guarantees the system has a minimal and a maximal solution, that is, the model is coherent (see Chesher and Rosen (2012) for identification of models that may have no solution for some covariate values). The second restriction shifts the function up or down, inducing the extremal solutions to vary monotonically with the potential treatments. These two implications are akin to the main results in the literature of supermodular games (see, e.g., Milgrom and Roberts (1990), Topkis (1979), and Vives (1990)). We motivate coordination on extremal equilibria via simple arguments and propose an 3We elaborate on this connection at the end of Section 3.3. Quantitative Economics 6 (2015) Treatment response with social interactions 51 alternative restriction that leads to the same comparative statics. Our results extend the routes for inference in Manski (1990,1997) to models with endogenous interactions by using an approach that relates to game theory. In doing this, we bridge the econometric theory on nonparametric identification with the recent literature on supermodular games. So far, the monotone assumptions discussed above restrict the response function of each member of the group but are silent with respect to the process of treatment selection. Many studies have established identification results by assuming the outcome functions are statistically independent of realized treatments. Manski and Pepper (2000) weaken that restriction by assuming the outcome functions are stochastically increasing in the realized treatments.4We extend this result to interactions-based models by developing an approach that allows for comparisons of equilibrium outcomes for different sets of groups. Our results relate to the methodology proposed by Amir (2008)to contrast Nash equilibria of different games. One important difference between the latter and our result is that we need to compare the distribution of equilibrium points for two sets of games, not only two games. This distributional comparative statics has no direct precedent in the theoretical literature. We provide identification results for two models of social interactions. In the first setup we study small groups, where each member has a distinctive role, for example, men and women in married couples. The second setup is appropriate to study large neighborhoods with anonymous social interactions, for example, crimes and infectious diseases. Finally, we use the latter framework to study crime rates in New York state for different levels of enforcement that we measure by police per capita, showing the discussed monotone conditions provide quite valuable information. The rest of the paper is organized as follows. Section 2presents an initial example that motivates our main results. Sections 3and 4study identification of treatment effects for small and large groups, respectively. Section 5uses our results to study crime rates in New York state. Section 6concludes and we collect all the proofs in Appendixes A and B. Replication files are available in a supplementary file on the journal website, http://qeconomics.org/supp/308/code_and_data.zip. 2. An initial example This section uses a model similar to the one in Becker (1991) to highlight the objective of our analysis and to differentiate our identification strategy from existing results. Each person j∈Gdecides whether to go to a popular restaurant. Here the treatment is the price she would pay for the service and the outcome (yj) is a yes/no indicator taking the value 1or 0. Consumer jmaximizes her utility, taking as given the behavior of the others. Her demand for the good is given by yj=fjt1/|G| i∈G yij∈G (1) 4Brock and Durlauf (2007) use a similar idea to address partial identification of a model with social interactions in a semiparametric framework. 52 Natalia Lazzati Quantitative Economics 6 (2015) where tis the price for the service and (1/|G|)i∈Gyiis the average of decisions in neighborhood G. This is the source of endogeneity in our model. An equilibrium in this market is a solution to the system of structural equations (1). We indicate such a solution by y(t) ≡yj(t)j ∈G(2) The empirical evidence consists of vectors of prices and individual demands for a set of neighborhoods in the population. The analyst wants to learn about the distribution of individual demands that would occur if groups were to receive a price t,thatis,P[y(t)]. Theliteratureongamesrefersto(1) as the best-reply function of person jto the profile of actions of other people in her group. Recent results in the econometrics of games provide conditions for identification of the best-reply functions. As in Manski (2013), our purpose is identification of P[y(t)], not the best replies per se. Our approach differs from Manski (2013) in that while we impose all our assumptions on the primitives of the model (1), his restrictions are placed directly on the solution set (2). Thus, for instance, while Manski (2013) directly assumes that y(t),asdefinedin(2), increases in t, we provide conditions on the primitives (i.e., (1)) that validate this assumption. The main restrictions we impose, which are subsequently elaborated, are as follows: A 1 (Positive Interactions). Individual j’s demand increases with the decisions of the other group members. A2 (Monotone Treatment Response). Individual j’s demand decreases with the market price. A3 (Monotone Treatment Selection). The owner of the restaurant chain is more likely to set higher prices to those neighborhoods with stronger demands. We then use monotone comparative statics to show that the conditions we place on (1) imply clear restrictions on the solution set (2). Within the analysis, these results are captured by a set of lemmas. We finally exploit the shape restrictions we derive on the equilibrium sets to provide nonparametric bounds for the distribution of potential outcomes, or demands, P[y(t)]. We next formalize and extend all these ideas to frameworks where groups are small and large, respectively. The restaurant example is closer to the second setup. 3. Identification of treatment response for small groups This section studies identification of treatment response in situations where groups are small and each group member has a distinctive role. Models of decisions of married couples, small teams of co-workers, and pairs of patients and doctors fit in here. Quantitative Economics 6 (2015) Treatment response with social interactions 53 3.1 The model and the analyst’s problem The population Jis partitioned into a finite set of classes L=12|L|,thatis, J=(J1J2J|L|).EachgroupGis composed of one individual from each class. We indicate by t∈Ta potential treatment and allow tto specify policies that may vary across classes. The behavior or achievement of agent jl ∈G,yjl ∈Y⊆Rwith jl ∈Jl, depends on the treatment and the behavior of the other group members, y−jl =(yjmjm∈G m =l). That is, yjl =fjl(t y−jl)with jl ∈G (3) We assume social interactions occur within groups and group membership is known to the econometrician. If the underlying model is a complete information game, then we can think of (3) as the best-reply function of player jl to the profile of actions of the other players in G. A simultaneous solution to the system (3) is a vector of potential outcomes y(t) =yjl(t) jl ∈Gwith y(t) ∈Y|L|(4) We typify groups through the outcome functions of the group members and relate the distribution of types of groups to the underlying process of group formation. The next example illustrates the probabilistic approach we describe afterward. Example 1. We model a scenario of a tobacco prevention program. The population has six people, three boys and three girls. Thus, J=(J1J2),whereJl=(1l2l3l) with l=12 is the set of boys and girls, respectively. Each group is a dating couple and its members decide whether to smoke. The treatment is 1if the couple receives information about the risks of smoking and is 0otherwise. An individual’s decision about smoking depends on the treatment received and the smoking decision of his or her partner. Thus, T={01}, Y={01},andfjl :T×Y→Ywith j=123and l=12. Couples differ with respect to their outcome functions, that is, in terms of the way they react to the treatments and partners’ decisions. In this example, we have 24(or 16) possible types of people—for each of the four possible configurations of treatments and partner’s decisions, there are two possible actions—and then 162(or 256)typesof couples. Let us assume the population has outcome functions as those in Figure 1.In this figure, the 1in square brackets means the second boy will smoke if he does not receive information and his girlfriend smokes. The population described in Figure 1involves only two types of girls and boys (out of 16) and then four types of couples (out of 256). The distribution of types of couples depends on both the smoking and the dating preferences of the individuals. Suppose these individuals prefer to date rather than remaining alone and to engage with individuals with similar smoking tastes. Thus, two out of the three couples will have members who smoke if and only if they are uninformed and their partners smoke, and the other couple will have members who smoke if and only if their partners smoke irrespective of the treatment. The other two types of groups have no chance of being formed given the assumed dating preferences. 54 Natalia Lazzati Quantitative Economics 6 (2015) Treatments and Partners’ Decisions Outcomes (00)(01)(10)(11) Boys y11 0100 y21 0[1]00 y31 0101 Girls y12 0100 y22 0100 y32 0101 Figure 1. Structural functions of the population. In our model, individuals may differ with respect to the way they react to the treatments and the decisions of the other group members. We let Tbe countable and Ybe finite with Y=min(Y) and Y=max(Y).5Thus, there are countably many different systems of structural equations (3)thatcandescribeagroup.Wesayagroupisoftypekif its members have structural equations fk(ty)≡[fkl(t y−l) l ∈L]with fkl :T×Y|L|−1→Y. We let Kindicate the set of possible types of groups. Since random group formation is often hard to motivate, we do not impose such a restriction. Instead, we think that the relative proportion of different agents within each class and the underlying matching process define the distribution of types of groups in the universe of groups U.Letπkdenote the fraction of type-kgroups, so that π≡(πkk∈K) is the distribution of types in U.InExample1,K={12256}and π=(2/31/300). We assume group formation is independent of t, in the sense that potential treatments do not affect the distribution of types of groups, π.6Nevertheless, realized treatments may provide rich information about the type of group that could have generated the data (see condition A3 in Section 2). In summary, [(fkπk) k ∈K]describes the universe of groups, U, in the population J. Groups have observable realized treatments τmand outcomes ym≡(ym 1ym 2 ym |L|), so that the available data are [(τmym) m ∈M]. The researcher wants to learn about the joint outcome distribution that would occur in the population Jif the groups were to receive a treatment t,thatis,P[y(t)],wherey(t) ≡[y1(t) y2(t)    y|L|(t)]is a random vector. 5These two restrictions are introduced for simplicity. We conjecture that it may be possible to extend the results to uncountable sets by taking appropriate care of measurability conditions. Manski (2007)usesa similar approach to develop partial identification of counterfactual choice probabilities. 6Manski (2013) describes a similar condition by saying that reference groups are treatment-invariant, and thus nonmanipulable (i.e., the social planner cannot use the treatments to change a person’s influence group). Quantitative Economics 6 (2015) Treatment response with social interactions 55 Remark. Though the primitives in our model, fk(ty), do not incorporate covariate information, we can assume that this information has been taken into account. We just need to interpret the primitives as conditional on some covariate values. The next example provides an application of the model we just presented. Example 2. This model can be used to study retirement decisions of husbands and wives. Let J1be the set of men and J2be the set of women, with Udefined as the set of all married couples in J. Let the outcome of interest be the retirement age and let the treatment be the income tax. Many studies argue that endogenous interactions are important within couples as spouses will obtain more pleasure from retirement if they retire together. 3.2 Monotone assumptions and their implications We next impose three sets of conditions on the structural equations and derive their implications in terms of equilibrium behavior. We then use these results to provide bounds for P[y(t)]. Coherence of the model This section shows equilibrium existence and relates observed and equilibrium outcomes with the primitives of the model. Without further restrictions, the system of equations (3) might have no solution. When this happens our equilibrium concept has no predictive power. We next provide a restriction that precludes this possibility. Throughout, we indicate by ≥the standard coordinatewise order. A1 (Positive Interactions). For each k∈K,l∈L,andt∈T,fkl(ty)≥fkl(ty)for all yy∈Y|L|−1with y≥y. Condition A1 requires the outcome of each individual to increase with the outcomes of the other group members. In Example 2, this condition requires that each member of the married couple be more willing to retire whenever his or her partner retires. Among the models that satisfy A1,wehavethesupermodulargames.7Let Ukl(ylty) be the payoff function of a type-lagent in a type-kgroup who gets treatment tand let fkl(ty)=argmaxyl∈YUkl(ylty).ThenA1 holds if, for all yl≥y land y≥y, Ukl(ylty)−Ukly lty≥Uklylty−Ukly lty(5) This condition states that the extra payoff of selecting a high over a low action increases with the action profile of the others.8This is the distinctive requirement of any supermodular game.9 Condition A1 is also satisfied by models with positive externalities, for example, peer effects in the classroom. 7See, for example, Milgrom and Roberts (1990), Vives (1990), and Topkis (1979). 8This analysis implicitly assumes the maximizer of Ukl(ylty)exists and is unique. 9Condition (5) could be relaxed by using the results in Milgrom and Shannon (1994). 56 Natalia Lazzati Quantitative Economics 6 (2015) Let φ(tk) denote the solution set of the system of structural equations for a given t,k. The next result states the existence of extremal equilibria. Lemma 1. If A1 holds,then φ(tk) has a least and a greatest solution for all k∈Kand t∈T. We next offer a sketch of proof. Let Mtk :Y|L|→Y|L|be defined as Mtk(y1y2y|L|)=y k1y k2y k|L|(6) with y kl =fkl(ty−l)for all l∈L By construction, the set of fixed points of Mtk coincides with φ(tk). Then this proof reduces to showing that the set of fixed points of (6) has a least and a greatest element. Under positive interactions, Mtk is increasing. Thus, the result follows by Tarski’s fixed point theorem.10 While condition A1 guarantees equilibrium existence, it does not eliminate the possibility of multiple solutions. Thus, to complete the model, we need to introduce an equilibrium selection mechanism. In this study, we restrict attention to deterministic equilibrium selection rules that depend on kand t. These types of selection mechanisms are widely used in economic theory. One of the reasons is that they can often be motivated by either Pareto dominance or simple learning processes in which the individuals adjust their choices based on the observed behavior of the others.11 The experimental literature on coordination in games offers substantial support to our approach; see, forexample,Cooper,DeJong,Forsythe,andRoss(1992). In contrast to our restriction, stochastic selection rules have been used for identification and testability in previous studies; see, for example, Bajari et al. (2011)andEchenique and Komunjer (2009). Let yk(t) be the element of φ(tk) that is selected by a type-kgroup that gets treatment t. Thus, the probability that the vector of potential outcomes falls in B⊆R|L|is given by Py(t) ∈B= k∈K 1yk(t) ∈Bπk(7) Let πk|τindicate the fraction of groups in Uthat are of type kconditional on τ,and let P(τ) denote the distribution of realized treatments across groups. We institute the convention that πk|τ≡0if the conditioning event does not hold. The probability that the joint vector of realized outcomes falls in a set B⊆R|L|is given by P(y∈B) = s∈T P(y∈B|τ=s)P(τ =s) (8) with P(y∈B|τ=s) =k∈K1[yk(s) ∈B]πk|τ=s.ThenP(y)is a mixture of realized outcome distributions conditional on observed treatments, with P(τ) as the mixing probability function. We next evaluate the implications of two other monotone restrictions. 10A similar approach has been used by Ackerberg and Gowrisankaran (2006), Jia (2008), and De Paula (2009). 11See, for example, Amir and Lazzati (2011), Monderer and Shapley (1996), and Oyama, Sandholm, and Tercieux (2012). Quantitative Economics 6 (2015) Treatment response with social interactions 63 us assume that the monotone treatment response is negative and the monotone treatment selection is positive. Then, for each s>s ,allwecansayis P(y|τ=s) ≥st Py(s) |τ=sand Py|τ=s≥st Py(s) |τ=s where the right inequality follows from negative treatment response and the left inequality follows from monotone treatment selection. This information does not suffice to compare the observable distributions P(y|τ=s) and P(y|τ=s). All our identified bounds can be consistently estimated from sample data of a subset of groups under regularity conditions. The analyst just needs to substitute all the empirical distributions with their sample analogs. The techniques developed by Andrews and Soares (2010), Imbens and Manski (2004), Rosen (2008), and Stoye (2009), among others, could be adapted to construct confidence sets for the identified regions. The asymptotic properties of the estimates depend on sampling a large number of groups. 4. Identification of treatment response for large groups In this section, groups are large (but finite) and social interactions are anonymous. Models of crimes, schooling, infectious diseases, and addictions fit well here. An important difference between this section and the previous one is that groups may differ in the number of people. This feature makes the comparative statics harder. 4.1 The model and the analyst’s problem For each treatment t∈Tto be received by group Gin the population J, the vector of potential outcomes y(t) ≡[yj(t)j ∈G]∈Y|G|solves the system of structural equations yj=fjtPG(y)with j∈G (12) where PG(y) is the distribution of outcomes induced by vector y≡(yjj∈G).22 We say a group is of type kif its members have structural equations fk≡[fkl(·) l∈Gk],wherefkl :T×ΔY→Y.Letπkdenote the fraction of individuals who belong to type-kgroups, so that π≡(πkk∈K) is the discrete distribution of types. Since groups are allowed to differ with respect to the number of members, πkdepends on both the fraction of groups that are of type kand the relative size of this type of group. Then [(fkπk) k ∈K]characterizes the universe of groups in U. Groups have realized treatments and outcomes given by [(τmym)m ∈M]. The objective of analysis is to learn about the counterfactual distribution of individual outcomes in the population for treatment t,thatis,P[y(t)]. Example 3. This model can be used to study crime rates and social interactions. Let J be the citizens of a given state and let us define a group as the set of people who live in one of its cities. The outcome of interest is the decision to commit a crime at the 22Formally, for all sets B⊂R,PG(y ∈B) =j∈G1(yj∈B)(1/|G|). 64 Natalia Lazzati Quantitative Economics 6 (2015) individual level and the treatment is police per capita at the city level. Distributional endogenous interactions are important in this model, as a higher crime participation rate by members of a city leads to fewer resources being spent on apprehending each criminal, which lowers his probability of punishment and further increases his incentives to commit a crime (see Sah (1991)). 4.2 Monotone assumptions and their implications This section imposes three alternative sets of conditions that are naturally satisfied in various social interactions models and derives their main implications in terms of equilibrium behavior. Coherence of the model The first assumption imposes a monotone condition on (fkk∈K),which,asA1, guarantees equilibrium existence. A1(Positive Interactions). For each k∈K,l∈Gk,andt∈T,fkl[tP(y)]≥fkl[tP(y)] for all P(y)P(y)∈ΔYwith P(y)≥st P(y). Let ϕ(tk) indicate the solution set for t,k. Since the analyst’s objective is to perform inference on the individual outcome distributions in J,welettheelementsofϕ(tk) be the distribution functions induced by the solution vectors. In the next lemma, least and greatest are with respect ≥st. Lemma 9. If A1holds,then ϕ(tk) has a least and a greatest solution for all k∈Kand t∈T. We write P[yk(t)]for the element of ϕ(tk) that is selected by a type-kgroup that receives treatment t. The probability that the potential outcome falls in a set B⊆Ris given by Py(t)∈B= k∈K Pyk(t) ∈Bπk(13) and it is linear in πk.Letπk|τbe the proportion of individuals in Jwhobelongtotype-k groups conditional on the realized treatment τ.WeletP(τ) denote the distribution of realized treatments in J. The probability that the realized outcome falls in a set B⊆Ris given by P(y ∈B) = s∈T Py(s) ∈B|τ=sP(τ =s) (14) where P[y(s)∈B|τ=s]=k∈KP[yk(s) ∈B]πk|τ=s. We next elaborate on the power of two extra monotone restrictions. As we explained earlier, though we assume that the monotone treatment response and selection restrictions are positive, we can easily accommodate the opposite cases. Quantitative Economics 6 (2015) Treatment response with social interactions 65 Monotone treatment response We now assume that individual outcomes increase in t on T. A2(Monotone Treatment Response). For each k∈K,l∈Gk,andP(y) ∈ΔY,fkl[t P(y)]≥fkl[tP(y)]for all tt∈Twith t≥t. If A1and A2hold, then the smallest and the largest distributions in ϕ(tk) increase in twith respect to ≥st. Thus, here again, extremal equilibrium selection rules are enough to make counterfactual predictions as we vary the treatment. We introduce an alternative assumption that leads to the same result. To this end, let us define Fkfor all B⊆Ras Fky∈B|tP(y)=1/|Gk| l∈Gk 1fkltP(y)∈B(15) Thus, Fkis an aggregate response function that indicates the fraction of people in a typekgroup whose outcomes would lie in the set Bfor some tand some initial P(y). S1(Equilibrium Selection). One of the following conditions holds: (i) each group selects either the smallest or the largest element of ϕ(tk) and the selection rule (weakly) increases in tor (ii) Fk(y |tinf φ(tk)) ≥st sup φ(tk) ∀tt∈Tsuch that t>t and ∀k∈K. We can motivate S1(i) in the same way as we did with S1(i). The following result is the analog to Lemma 2. Lemma 10. If A1,A2,and S1hold,then P[yk(t)]≥st P[yk(t)]for all tt∈Twith t>t  and k∈K. We finally study monotone treatment selection. Monotone treatment selection This section provides sufficient conditions to validate the use of realized treatments as monotone instrumental variables. To this end, we introduce a partial order on (Fkk∈K),asdefinedin(15). Definition 11. We say Fk≥Fkif Fk[y|tP(y)]≥st Fk[y|tP(y)]∀[tP(y)]∈T×ΔY. According to the last definition, Fkis greater than Fkif the outcome distribution of the type-kgroup is stochastically higher than the outcome distribution of the typekgroup for any conditioning event. Let Fdenote a random function with support (Fkk∈K),anddefineP(F =Fk|τ) ≡πk|τfor all k∈K. The next condition is similar to condition A3. A3(Monotone Treatment Response). We have P(F |τ=s) ≥st P(F |τ=s)for all s≥s. 66 Natalia Lazzati Quantitative Economics 6 (2015) Condition A3states that the proportion of individuals who belong to groups with weakly higher structural functions increases with realized treatments. The motivation for this constraint is well captured by Example 3: in the study of crime rates, it is reasonable to think that the public authority is more likely to invest more (per capita) on the criminal apprehension system of those cities where people’s willingness to commit crimes is believed to be higher. (As we explained earlier, our framework can easily accommodate the case of negative treatment selection.) Here again, before elaborating on the identification power of adding the monotone treatment selection condition, we need to introduce another restriction. S2(Equilibrium Selection). One of the following conditions holds: (i) groups select either the smallest or the largest element of the solution set and (for each t∈T) the selection rule is the same ∀k∈Kor (ii) Fk[y|tinf ϕ(t k)]≥st supϕ(tk)∀k k∈Ksuch that Fk≥Fk. The next lemma is the analog to Lemma 4. Lemma 12. Assume A1,A3,and S2hold.Let ss∈T.Then,for all t∈T, s≥s⇒ Py(t)|τ=s≥st Py(t) |τ=s(16) The next section uses all of our previous results to provide bounds for P[y(t)]. 4.3 Identification region for P[y(t)] The bounds for P[y(t)]are quite similar to those of P[y(t)]. We only need to substitute y(t) by y(t) and A1,(A1,A2,S1), and (A1,A3,S2)byA1,(A1,A2,S1), and (A1,A3,S2) in Propositions 5,6,and7to obtain Propositions 5,6,and7, respectively. Appendix A formalizes these claims. Thus, instead of repeating these results, we elaborate on two interesting differences between the two models. Quantiles are often parameters of interest in applied studies. For α∈(01),the α-quantile of P[y(t)]is defined as Qα[y(t)]≡infy{E{1[y(t) ≤y]} ≥ α}. The characterization of the standard stochastic order in terms of the expectations of increasing functions induces a partial order on the quantiles of random variables: if a distribution function stochastically dominates another distribution function, then all the quantiles of the former are larger than the corresponding quantiles of the latter. Hence, the analogs of Propositions 5,6,and7for the case of large groups can be easily reformulated in terms of the quantiles of the pertinent distributions, for example, the medians. Manski (1997) provides functional forms to construct bounds for quantiles in an individualistic model. The lack of objective basis for ordering multivariate observations is a major difficulty in extending the previous definition to random vectors. There are several attempts in the statistical literature toward multidimensional generalizations of univariate quantiles, each of which captures distinct aspects of interest. We remained silent about quantiles in Section 3.3 as it is not immediate that the multivariate standard stochastic order has clear monotone predictions for all the quantiles according to all the existing definitions. Quantitative Economics 6 (2015) Treatment response with social interactions 67 We mentioned in Section 3.3 that all our identified bounds can be consistently estimated from sample data of a subset of groups under regularity conditions. When groups are large, the same result holds if we sample a subset of individuals from each group in the random sample of groups. The reason is that, in this second case, we are not interested in obtaining the joint distribution of outcomes of the group members; see, for example, Brock and Durlauf (2000). 5. Application:Crime rates and social interactions This section illustrates our previous results by applying them to the analysis of crime rates in New York state. Becker (1968) studies individual decisions to commit crimes from an economic perspective. He develops a cost–benefit analysis and argues that a key ingredient in an individual’s choice of whether to become a criminal is his perceived probability of punishment. Subsequent work emphasizes the importance of positive social interactions in motivating criminal behavior (see Glaeser, Sacerdote, and Scheinkman (1996) and the literature therein). We use the model in Section 4to study crimes across cities. Let each person in a given city decide whether to commit a crime. We define the treatment as police per capita at the city level and define the outcome as a yes/no indicator that takes the value 1or 0, that is, Y={01}.Sah (1991) presents a model where one individual’s choice to commit a crime lowers the probability that any other individual ends up arrested. Since the police cannot be in two places at the same time, the higher is the criminal activity in a given city, the lower is the probability of being punished. His argument justifies A1.23 In addition, each individual’s decision to commit a crime decreases with the amount of police per capita in his own city. (This is a natural direct effect of the treatment.) Then the dual version of A2holds here as well. It is also reasonable to think that the public authority is more likely to invest more on the criminal apprehension system of those cities where people’s willingness to commit crimes is believed to be higher.24 The last statement validates condition A3. We will also assume that the dual of S1and S2hold. The analyst wants to learn about the fraction of people who would commit a crime in New York state if all its cities were to be assigned a given level of police per capita. The next subsection describes the data we use and the subsequent subsection shows our findings. 5.1 Data set Our data source is the Uniform Crime Reporting (UCR) program of the Federal Bureau of Investigation (FBI) for the year 2009. The UCR program informs crimes reported and verified. The data set also provides information about the number of police at the city 23This assumption is justified in the criminology literature by using many different arguments (for example, the theory of differential association by Sutherland (1974)). Our results do not depend on the particular mechanism by which the interactions arise. Thus, any of the existing justifications can be used to validate A1. 24Lazzati and Menichini (2014) provide a theoretical model of optimal police allocation that could be used to justify this restriction. 68 Natalia Lazzati Quantitative Economics 6 (2015) Figure 2. Distribution of realized levels of police per capita. level and the population of each city. We decided to eliminate New York City from the sample, as its features (e.g., number of people) are markedly different from the characteristics of the other cities. Our data cover 47% of the remaining population of New York state (314 cities). To perform the analysis, we discretized the level of police per capita in multiples of 00001. Figure 2displays the data for 994% of the sampled population: for expositional ease, the figure does not include a few observations with extremely high levels of police per capita, but these observations were taken into account in the estimation. For levels of police per capita between 0and 0005,P(τ) indicates the fraction of individuals in the sample who received treatment τduring the year 2009. Since a large part of the sampled population—specifically, 9102%—received treatments between 0001 and 0004, the next section estimates the outcomes of interest for levels of police per capita in that range of values. The asymptotic properties of our estimators should be thought of as the number of sampled cities approaches the whole universe of cities in New York state. In Figure 3,P(y =1|τ) indicates the fraction of people who committed a crime in 2009 conditional on living in a city with a level of police per capita τ. Throughout this application, we assume each individual has committed at most one crime during 2009. Remark. The assumption thateach individual has committed at most one crime during 2009 is surely strong. This condition could be relaxed by a richer data set that incorporates information on offender reincidence. This issue could also be fixed by interpreting the potential outcome as the decision of committing at least one crime, but results should correspondingly be adapted to this alternative interpretation.25 25I thank one of the referees for this suggestion. Quantitative Economics 6 (2015) Treatment response with social interactions 69 Figure 3. Criminal activity conditional on realized treatments. 5.2 Findings Since the outcome of interest is binary (i.e., Y={01}),thefractionofpeoplewhowould commit a crime for a given level of police per capita t(i.e., P[y(t) =1]) contains all the information we need to describe the distribution of potential outcomes, P[y(t)]. We consider four levels of police per capita (i.e., four treatments): 0001,0002,0003, and 0004. For each of them, Figure 4reports the lower and the upper estimations of the identified bounds for P[y(t) =1]under three sets of assumptions.26 These bounds are reported in the third and the fourth columns respectively, and are expressed in percentage points. They are the sample analogs of Propositions 5,6,and7, respectively. In addition, we provide (in the last column) confidence sets for P[y(t) =1]at the 95% of confidence level. To compute them, we follow the methodology in Imbens and Manski (2004)andStoye (2009); see Appendix Bfor further details. We next highlight our findings.27 We can clearly see in Figure 4that A1alone is practically uninformative in the four cases. The mere additionof either (dual of A2,S1)or(A3,S2) substantially improves all the predictions. For instance, for t=0004,A1alone predicts 100×P[y(t)=1]∈(0100), while the introduction of (dual of A2,S1) reduces that interval to (0504).Thereasonis that a very small fraction (i.e., 025%) of the population has realized treatment τ=0004 26In an individualistic model, Manski and Pepper (2000) study returns to schooling assuming both the monotone treatment response and the monotone treatment selection restrictions are satisfied. They show that the bounds under the joint restriction are much tighter than the simple intersection of the bounds under the individual assumptions. We computed the sharp bounds under the joint restriction for the crime application. In our case, they were almost identical to the intersection of the bounds under the individual restrictions. We speculate that this result, which contrasts with that in Manski and Pepper (2000), relates to the fact that, in our case, the monotone treatment response and selection restrictions work in opposite directions. 27For t=0004, one of the estimators takes a value slightly below 0and a second estimator takes a value slightly above 100.Wewrote0and 100 instead, as the true parameter of interest will always take values between 0and 100. 70 Natalia Lazzati Quantitative Economics 6 (2015) Police Per Capita Assumptions Lower Bound Upper Bound 95% Confidence Set t=0001 A1006 9674 (0039837) A1,dualofA2,S1290 8925 (2729210) A1,A3,S2179 8926 (1749211) t=0002 A1012 9497 (0079698) A1,dualofA2,S1218 5494 (1955947) A1,A3,S2139 5504 (1295956) t=0003 A1050 8974 (0359248) A1,dualofA2,S1123 1842 (1022174) A1,A3,S2126 2003 (1072329) t=0004 A1000 998(0001000) A1,dualofA2,S1002 041(0000504) A1,A3,S2001 0147 (0000242) Source: FBI, UCR (Uniform Crime Reporting) program for year 2009. Figure 4. Lower and upper bounds for P[y(t) =1]in percent. and then the empirical evidence alone is ineffective to identify the potential outcomes. However, by adding monotone treatment response, all the data are used in the estimation: part of the observations improve the upper bound, another segment helps to estimate the lower bound, and a third group of observations is used to construct both. For this treatment level, (A3,S2) provides even more information as the interval shrinks to (0242). The reason for this result is that the groups that received this large treatment displayed very low criminal activity. The bounds we just provided can be improved if we select less conservative, though still credible, upper bounds for the largest possible fraction of people who could commit a crime at any treatment level. We refer to the latter probability as Pmax. Figure 5 provides bounds for the fraction of people who would commit a crime at a level of police per capita t=0002 for three different values of Pmax.28 Our results show that the bounds become substantially more informative as we reduce Pmax. Similar results hold for the other treatment levels. Welfare analysis Though the previous results show that the estimated intervals for criminal activity decrease as police per capita increases, increasing police per capita has an opportunity cost in terms of other social policies that could be pursued with the budget assigned to police resources.29 Thus, to decide whether to increase the police resources, the public authority may want to compare the overall cost of the available alternatives. We next show that our previous bounds can help the public authority to make a decision, even when some of the bounds are quite wide. Our model is based on Becker (1968). 28I thank one referee for this suggestion. 29I thank a referee for suggesting this welfare analysis. Quantitative Economics 6 (2015) Treatment response with social interactions 71 Values of Pmax Assumptions Lower Bound Upper Bound 95% Confidence Set Pmax =1A1012 9497 (0079698) A1,dualofA2,S1218 5494 (1955947) Pmax =050 A1,A3,S2139 5504 (1295956) A1012 4755 (0074852) A1,dualofA2,S1218 2796 (1953018) A1,A3,S2139 2806 (1293027) Pmax =025 A1012 2383 (0072430) A1,dualofA2,S1218 1447 (1951553) A1,A3,S2139 1457 (1291562) Source: FBI, UCR (Uniform Crime Reporting) program for year 2009. Figure 5. Lower and upper bounds for P[y(t =0002)=1]in percent. Let O(t) be the number of offences in New York state when its cities receive a level of police per capita t. If the number of people is N,thenO(t) ≈NP[y(t) =1].LetH(O(t)) be the corresponding social harm. We assume H is linear so that H(O(t)) =hO(t),where h>0is the social cost per crime committed. In addition, we let c>0be the opportunity cost per unit of police officer. The public authority wants to compare the overall cost of two levels of police per capita tt∈Twith t>t ; that is, it is interested in estimating Δtt=hNPy(t)=1+cNt−hNPyt=1+cNt To provide bounds for Δ(tt),weletPl[y(t) =1]and Pu[y(t) =1]be the lower and the upper bounds for P[y(t)=1]. The bounds for Δ(t t)can be obtained as hNPuy(t)=1−Plyt=1+cNt−t ≥Δtt≥hNPly(t) =1−Puyt=1+cNt−t Though they may not be sharp, these bounds can be easily implemented. The lower bound can be either positive or negative. When it is positive, our analysis suggests that it is not worth it to increase the level of police even if it reduces the criminal activity. To illustrate how useful this analysis might be, we next estimate the lower bound of Δ(tt)for two alternative changes in treatment levels. We assume that his $400 and the opportunity cost of a police officer cis $105,000.30 To estimate the lower bounds, we use the results in Figure 4under the restriction (A1,dualofA2,S1). We get Δt=0003t=0002≥−N1288and Δt=0004t=0003≥N1804 This result shows that increasing police per capita in all cities of New York state from 0002 to 0003 might reduce the social cost. It also states that further increments from 30Estimates of the average social cost of crimes substantially differ across studies. Most of them agree that the estimates for most of the crimes are very low. However, they also agree that the average is pushed up by a small number of crimes that involve extremely large monetary losses. 72 Natalia Lazzati Quantitative Economics 6 (2015) 0003 to 0004 will never do so. As we mentioned earlier, this outcome is interesting as it shows that even when our bounds are wide, they can still be useful to guide policy decisions. 6. Concluding remarks This paper provides identification results for treatment response models with endogenous social interactions by means of monotone comparative statics. In doing so, we bridge the theory of identification of treatment effects that exploits monotone restrictions with recent results on games with strategic complementarities. The approach to partial identification is nonparametric and allows for counterfactual predictions under multiple equilibria. Moreover, it relies neither on random treatment assignment nor on random assignment of individuals to the groups. Our results derive from shape restrictions on the primitives of the model that lead to monotone comparative statics of the equilibrium sets. The identification regions have the form of intervals, that is, we identify two extreme distributions that are functions of the observable data such that the true distribution must lie between them in terms of stochastic dominance. The bounds we provide are sharp and, by applying them to the study of crimes in New York state, we show that they can also be very informative. An additional contribution of this study is to develop a flexible and tractable probabilistic framework for the model that typifies groups through the relevant features of the group members and can accommodate various processes of group formation. Last, this article also contributes to the literature on identification of nonparametric simultaneous equations models.31 Appendix A: Proofs To make this paper self-contained, we provide the three theorems we invoke. Tarski’sFixed Point Theorem (TFP). If Xis a complete lattice and f:X→Xis an increasing function,then fhasafixedpoint.Moreover,the set of fixed points of fhas a smallest and a largest element (Tarski (1955)). Milgrom and Roberts’Theorem (MR). If Xis a complete lattice,Sis a partially ordered set,and f:X×S→Xis an increasing function,then the least and greatest fixed points of fare increasing in son S(Milgrom and Roberts (1990)). The concept of first order (or standard) stochastic dominance is based on upper sets. Let us consider (Ω ≥),whereΩis a set and ≥defines a partial order on it. A subset U⊂Ω is an upper set if and only if x∈Uand x≥ximply x∈U. First Order Stochastic Dominance Theorem (FOSD). Let XX∈Rnbe two random vectors.The following two statements are equivalent: 31See, for example, Matzkin (2008) and the literature therein. Quantitative Economics 6 (2015) Treatment response with social interactions 79 II. Positive interactions and (negative) monotone treatment response Proposition 20. Assume A1and the duals of (A2and S1)hold.Then,for all t∈T, HPy(t)=1=δ∈[01]:P(y =1|τ≤t)P(τ ≤t)+P(τ t) (32) ≥δ≥P(y =1|τ≥t)P(τ ≥t) The estimators for the bounds are just the sample analogs of (32); that is, ∀t∈T,  Puy(t)=1= m∈Mdm1(τm≤t)+1(τm>t) nm  Ply(t)=1= m∈M dm1(τm≥t)nm III. Positive interactions and monotone treatment selection Proposition 21. Assume A1,A3,and S2hold.Then,for all t∈T, HPy(t)=1=δ∈[01]:P(y =1|τ=t)P(τ ≤t)+P(τ t) (33) ≥δ≥P(y =1|τ=t)P(τ ≥t) The estimators for the bounds are just the sample analogs of (33); that is, ∀t∈T,  Puy(t)=1= m∈M ⎡ ⎢ ⎢ ⎣⎛ ⎜ ⎜ ⎝ m∈M dm1(τm=t)nm  m∈M 1(τm=t)nm ⎞ ⎟ ⎟ ⎠1(τm≤t)+1(τm>t) ⎤ ⎥ ⎥ ⎦nm  Ply(t)=1= m∈M ⎡ ⎢ ⎢ ⎣⎛ ⎜ ⎜ ⎝ m∈M dm1(τm=t)nm  m∈M 1(τm=t)nm ⎞ ⎟ ⎟ ⎠1(τm≥t)⎤ ⎥ ⎥ ⎦nm Remark. Since all the previous estimators take the form of weighted averages, then the corresponding vectors of variances and correlation, (σ2 lσ2 uρ), can be easily estimated. B.2 Confidence intervals for P[y(t) =1] We estimate confidence regions by following the approaches of Imbens and Manski (2004)andStoye (2009). Imbens and Manski (2004) show uniform validity of their confidence region under the following assumption (Assumption 1adapts their condition to our problem). 80 Natalia Lazzati Quantitative Economics 6 (2015) Assumption 1. (i) There exist estimators  Pl[y(t)=1]and  Pu[y(t)=1]that satisfy √N Ply(t)=1−Ply(t) =1  Puy(t)=1−Puy(t) =1d →N0 0σ2 lρσlσu ρσlσuσ2 u uniformly in P∈Ψ,and there are estimators (σ2 lσ2 uρ) that converge to their population values uniformly in P∈Ψ. (ii) For all P∈Ψ,σ2≤σ2 l≤σ2and σ2≤σ2 u≤σ2for some positive and finite σ2and σ2,and Pu[y(t)=1]−Pl[y(t) =1]≤Δ<∞. (iii) For all ε>0,there are ν>0,K,and N0such that N≥N0implies Pr(√N|Δ−Δ|> KΔν)<εuniformly in P∈Ψ. Stoye (2009) explains that Assumption 1(iii) requires  Δ(t) to be superefficient at 0, a quite strong condition. Thus, he imposes conditions (i) and (ii), and shows that the results of Imbens and Manski (2004) are still valid if we substitute condition (iii) by the next requirement. Assumption 1(iii). There exists a sequence {aN}such that aN−→ 0,aN√N→∞,and √N|Δ−ΔN|p →0for all sequences of distributions PN⊆Ψwith ΔN≤aN. Though condition (iii) is often hard to validate, Stoye (2009) provides a sufficient condition that clearly holds in our setup. Lemma 22. Let Assumption 1(i) and (ii) hold,and assume Pr Puy(t)=1≥ Ply(t)=1=1 Then Assumption 1(iii)is implied. We next introduce the confidence intervals we use in the analysis of crimes. 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