Estimating local interactions among many agents who observe their neighbors

Canen, Nathan; Schwartz, Jacob; Song, Kyungchul Article Estimating local interactions among many agents who observe their neighbors Quantitative Economics Provided in Cooperation with: The Econometric Society Suggested Citation: Canen, Nathan; Schwartz, Jacob; Song, Kyungchul (2020) : Estimating local interactions among many agents who observe their neighbors, Quantitative Economics, ISSN 1759-7331, The Econometric Society, New Haven, CT, Vol. 11, Iss. 3, pp. 917-956, https://doi.org/10.3982/QE923 This Version is available at: https://hdl.handle.net/10419/253560 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/4.0/ Quantitative Economics 11 (2020), 917–956 1759-7331/20200917 Estimating local interactions among many agents who observe their neighbors Nathan Canen Department of Economics, University of Houston Jacob Schwartz Department of Economics, University of Haifa Kyungchul Song Vancouver School of Economics, University of British Columbia In various economic environments, people observe other people with whom they strategically interact. We can model such information-sharing relations as an information network, and the strategic interactions as a game on the network. When any two agents in the network are connected either directly or indirectly in a large network, empirical modeling using an equilibrium approach can be cumbersome, since the testable implications from an equilibrium generally involve all the players of the game, whereas a researcher’s data set may contain only a fraction of these players in practice. This paper develops a tractable empirical model of linear interactions where each agent, after observing part of his neighbors’ types, not knowing the full information network, uses best responses that are linear in his and other players’ types that he observes, based on simple beliefs about the other players’ strategies. We provide conditions on information networks and beliefs such that the best responses take an explicit form with multiple intuitive features. Furthermore, the best responses reveal how local payoff interdependence among agents is translated into local stochastic dependence of their actions, allowing the econometrician to perform asymptotic inference without having to observe all the players in the game or having to know the precise sampling process. Keywords. Strategic interactions, behavioral modeling, information sharing, games on networks, cross-sectional dependence. JEL classification. C12, C21, C31. Nathan Canen: [email protected] Jacob Schwartz: [email protected] Kyungchul Song: [email protected] This research has benefited from conversations with Qingmin Liu and João Ramos at its initial stage. We also thank Don Andrews, Khai Chiong, Tim Conley, Sokbae Lee, Michael Leung, Xiaodong Liu, and seminar participants at University of Colorado at Boulder, UBC, University of Southern California, and participants at Cemmap Conference in Kyoto and Canadian Econometrics Study Group Meeting for valuable comments and questions. We would like to thank Mike Peters and Li Hao for valuable conversations and suggestions. We also thank three anonymous referees for valuable comments, criticisms, and suggestions. All errors are ours. Song acknowledges that this research was supported by Social Sciences and Humanities Research Council of Canada. ©2020 The Authors. Licensed under the Creative Commons Attribution-NonCommercial License 4.0. Available at http://qeconomics.org.https://doi.org/10.3982/QE923 918 Canen, Schwartz, and Song Quantitative Economics 11 (2020) 1. Introduction Interactions between agents—for example, through personal or business relations— generally lead to their actions being correlated. In fact, such correlated behaviors form the basis for identifying and estimating peer effects, neighborhood effects, or more generally, social interactions in the literature. (See Blume, Brock, Durlauf, and Ioannides (2010)andDurlauf and Ioannides (2010) for a review of this literature.) Empirical modeling becomes nontrivial when one takes seriously the fact that people are often connected directly or indirectly on a large complex network, and observe some of their neighbors’ types. Such strategic environments may be highly heterogeneous across agents, with each agent occupying a nearly “unique” position in the network. Information sharing potentially creates a complex form of cross-sectional dependence among the observed actions of agents, yet the econometrician typically observes only a fraction of the agents on the network, and rarely observes the entire network which governs the cross-sectional dependence structure. The main contribution of this paper is to develop a tractable empirical model of linear interactions among agents with the following three major features. First, assuming a large game on a complex, exogenous network, our empirical model does not require the agents to observe the full network. Instead, we assume that each agent observes only a local network around herself and only part of the type information of those who are local to her.1 Second, our model explains strategic interdependence among agents through correlated observed behaviors. In this model, the cross-sectional local dependence structure among the observed actions reflects the network of strategic interdependence among the agents. Most importantly, unlike most incomplete information game models in the literature, our set-up allows for information sharing on unobservables, that is, each agent is allowed to observe his neighbors’ payoff-relevant signals that are not observed by the econometrician. Third, the econometrician does not need to observe the whole set of players in the game for inference. It suffices that he observe many (potentially) nonrandom samples of local interactions. The inference procedure that this paper proposes is asymptotically valid independently of the actual sampling process, as long as the sampling process satisfies certain weak conditions. Accommodating a wide range of sampling processes is useful because random sampling is rarely used for the collection of network data, and a precise formulation of the actual sampling process is often difficult in practice. A standard approach for studying social interactions is to model them as a game, and use the game’s equilibrium strategies to derive predictions and testable implications. Such an approach is cumbersome in our set-up. Since a particular realization of any agent’s type affects all the other agents’ equilibrium actions through a chain of information sharing, each agent needs to form a “correct” belief about the entire information graph. Apart from such an assumption being highly unrealistic, it also implies 1For example, a recent paper by Breza, Chandrasekhar, and Tahbaz-Salehi (2018) documents that people in a social network may lack substantial knowledge of the network and that such informational assumptions may have significant implications for the predictions of network models. Models assuming that agents possess only local knowledge have drawn interest in the literature on Bayesian learning on networks. For example, see a recent contribution by Li and Tan (2020) and references therein. Quantitative Economics 11 (2020) Estimating local interactions among many agents 919 that predictions from an equilibrium that generate testable implications usually involve all the players in the game, when it is often the case that only a fraction of the players are observed in practice. Thus, an empirical analysis which regards the players in the researcher’s sample as coincident with the actual set of players in the game may suffer from a lack of external validity when the target population is a large game involving many more players than those present in the actual sample. Instead, this paper adopts an approach of behavioral modeling, where it is assumed that each agent, not knowing fully the information sharing relations, optimizes according to simple beliefs about the other players’ strategies. The crucial part of our behavioral assumption is a primitive form of belief projection which says that each agent, not knowing the full set of information-sharing relations, projects his own beliefs about other players onto his payoff neighbors. More specifically, if agent igives more weight to agent jthan to agent k,agentibelieves that each of his payoff neighbors sdoes the same in comparing agents jand k. Here, the “weights” represent the strategic importance of other players, and belief projection can be viewed as a rule-of-thumb for an agent who needs to form expectations of the actions of the players, not knowing who they observe. When the strategic importance of one player to another is based primarily on “vertical” characteristics such as skills or assets, the assumption of belief projection does not seem unrealistic.2 Our belief projection approach yields an explicit form of the best response which has intuitive features. For example, the best response is such that each agent igives more weight to those agents with a higher local centrality to him, where the local centrality of agent jto agent iis said to be high if and only if a high fraction of agents whose actions affect agent i’s payoff have their payoffs affected by agent j’s action. Also, the best response is such that each agent responds to a change in his own type more sensitively when there are stronger strategic interactions, due to what we call the reflection effect. The reflection effect of player icaptures the way that player i’s type affects his own action through his payoff neighbors whose payoffs are affected by player i’s types and actions. The best responses reveal an explicit form of local dependence among the observed actions from which we can derive minimal conditions for feasible asymptotic inference. It turns out that the econometrician does not need to observe all the players in the game, nor does he need to know the precise sampling process. Furthermore, the best response from the belief assumption provides a testable implication for information sharing on unobservables in data. In fact, the cross-sectional correlation of residuals indicates information sharing on unobservables. (See the end of the replication file 2Belief projection in our paper can be viewed as connected, though loosely, to interpersonal projection studied in behavioral economics. A related behavioral concept is projection bias of Loewenstein, O’Donohue, and Rabin (2003) which refers to the tendency of a person projecting his own current taste to his future taste. See also Van Boven, Loewenstein, and Dunning (2003) who reported experimental results on the interpersonal projection of tastes onto other agents. Since an agent’s belief formation is often tied to their information, belief projection is closely related to information projection in Madarász (2012), who focuses on the tendency of a person to project his information to other agents’. The main difference here is that our focus is to formulate the assumption in a way that is useful for inference using observational data on the actions of agents who interact on a network. 920 Canen, Schwartz, and Song Quantitative Economics 11 (2020) (Canen, Schwartz, and Song (2020)) for details on the testing procedure based on the cross-sectional correlation of residuals.) It is instructive to compare the predictions from our behavioral model to those from an equilibrium model. When the payoff graph is comprised of multiple disjoint subgraphs that are complete, the behavioral strategies and equilibrium strategies coincide. Moreover, for a game on a general payoff graph, we show that as the rationality of agents deepens and their information expands, the behavioral strategies converge to the equilibrium strategies of an incomplete information game where each agent observes all the sharable types of every other agent. We provide conditions under which the parameters are locally identified, but propose asymptotic inference in a general setting that does not require such conditions. We also investigate the finite sample properties of our asymptotic inference through Monte Carlo simulations using various payoff graphs. The results show reasonable performance of the inference procedures. In particular, the size and the power of the test for the strategic interaction parameter are good in finite samples. We apply our methods to an empirical application which studies the decision of state presence by municipalities, revisiting Acemoglu, García-Jimeno, and Robinson (2015). We consider an incomplete information game model which permits information sharing on unobservables. The fact that our best responses explicitly reveal the local dependence structure means that it is unnecessary to separately correct for spatial correlation following, for example, the procedure of Conley (1999). The literature on social interactions often looks for evidence of interactions through correlated behaviors. For example, linear interactions models investigate correlation between the outcome of an agent iand the average outcome over agent i’s neighbors. See, for example, Manski (1993), De Giorgi, Pellizzari, and Redaelli (2010), Bramoullé, Djebbari, and Fortin (2009), and Blume, Brock, Durlauf, and Ioannides (2015) for identification analysis in linear interactions models, and see Calvó-Armengol, Pattacchini, and Zenou (2009) for an application to the study of peer effects. Goldsmith-Pinkham and Imbens (2013) considered nonlinear interactions on a social network and discusses endogenous network formation. Such models often assume that the researcher observes many independent samples of such interactions, where each independent sample constitutes a game containing the entire set of the players in the game. In the context of a complete information game, a linear interaction model on a large social network can generally be estimated without assuming independent samples. The outcome equations in such a setting frequently take the form of spatial autoregressive models, which have been actively studied in the spatial econometrics literature (Anselin (1988)). A recent study by Johnsson and Moon (2016) considers a model of linear interactions on a large social network which allows for endogenous network formation. Developing inference on a large game model with nonlinear interactions is more challenging. See Menzel (2016), Xu (2015), Song (2014), Xu and Lee (2015), and Yang and Lee (2016) for a large game model of nonlinear interactions. This large game approach is suitable when the data set does not have many independent samples of interactions. One of the Quantitative Economics 11 (2020) Estimating local interactions among many agents 921 major issues in the large game approach is that the econometrician often observes only a subset of the agents from the original game of interest.3 Our empirical approach is based on a large game model which is close to models of linear interactions in the sense that it attempts to explain strategic interactions through the correlated behavior of neighbors. In our set-up, the cross-sectional dependence of the observed actions is not merely a nuisance that complicates asymptotic inference; it provides the very information that reveals the nature of strategic interdependence among agents. Such correlated behavior also arises in equilibrium in models of complete information games or games with types that are either privately or commonly observable. (See Bramoullé, Djebbari, and Fortin (2009)andBlume et al. (2015).) However, as emphasized before, such an approach can be cumbersome in our context of a large game primarily because the testable implications from the model typically involve the entire set of players, when in many applications the econometrician observes only a small subset of the game’s players. After finishing the first draft of paper, we learned of a recent paper by Eraslan and Tang (2017) who model the interactions as a Bayesian game on a large network with private link information. They do not require the agents to observe the full network, and show identification of the model primitives adopting a Bayesian–Nash equilibrium as a solution concept. One of the major differences of our paper from theirs is that our paper permits information sharing on unobservables, so that the actions of neighboring agents are potentially correlated even after controlling for observables. A departure from the equilibrium approach in econometrics is not new in the literature. Aradillas-Lopez and Tamer (2008) studied implications of various rationality assumptions for identification of the parameters in a game. Unlike their approach, our focus is on a large game where many agents interact with each other on a single complex network, and, instead of considering all the beliefs which rationalize observed choices, we consider a particular set of beliefs that satisfy a simple rule and yield an explicit form of best responses. (See also Goldfarb and Xiao (2011)andHwang (2017)forempirical research adopting behavioral modeling for interacting agents.) This paper is organized as follows. In Section 2, we introduce an incomplete information game of interactions with information sharing. This section derives the crucial result of best responses under simple belief rules. We also show the convergence of behavioral strategies to equilibrium strategies as the rationality of agents becomes higher and their information sets expand. Section 3focuses on econometric inference. It explains the data set-up and a method for constructing confidence intervals. Section 4investigates the finite sample properties of our inference procedure through a Monte Carlo study. Section 5presents an empirical application on state capacity among municipalities. Section 6concludes. Due to the space constraints, the technical proofs of the results are found in the Online Supplemental Material (Canen, Schwartz, and Song (2020)). Further materials including extensions to a model of information sharing among many 3Song (2014), Xu (2015), Johnsson and Moon (2016), Xu and Lee (2015), and Yang and Lee (2016) assumed that all the players in the large game are observed by the researcher. In contrast, Menzel (2016) allowed for observing i.i.d. samples from the many players, but assumes that each agent’s payoff involves all the other agents’ actions exchangeably. 922 Canen, Schwartz, and Song Quantitative Economics 11 (2020) agents over time, testing for information sharing on unobservables, and a model selection procedure for choosing among different behavioral models, are found in the replication file (Canen, Schwartz, and Song (2020)). 2. Strategic interactions with information sharing 2.1 A model of interactions with information sharing Strategic interactions among a large number of information-sharing agents can be modeled as an incomplete information game. Let Nbe the set of a finite yet large number of players. Each player i∈Nis endowed with his type vector (τiηi),whereηiis a private type and τiasharabletype. 4As we will elaborate later, information ηiis kept private to player iwhereas τiis observed by his neighbors in a network which we define below. To capture strategic interactions among players, let us introduce an undirected graph GP=(NEP),whereEPdenotes the set of edges ij ,i j ∈Nwith i=j, and each edge ij ∈EPrepresents that the action of player iaffects player j’s payoff.5We denote NP(j) to be the GP-neighborhood of player j, that is, the collection of players whose actions affect the payoff of player j: NP(j) ={i∈N:ij ∈EP} and let nP(j) =|NP(j)|. We define NP(j) =NP(j) ∪{j}and let nP(j) =|NP(j)|. Player ichoosing action yi∈Ywith the other players choosing y−i=(yj)j=iobtains payoff: ui(yiy−iτηi)=yi(τi+β0yi+ηi)−1 2y2 i(2.1) where τ=(τi)i∈N,and yi=1 nP(i)  k∈NP(i) yk if NP(i) =∅,andyi=0otherwise. Thus the payoff depends on other players’ actions and types only through those of his GP-neighbors. We call GPthe payoff graph. The parameter β0measures the payoff externality among agents. As for β0,wemake the following assumption. Assumption 2.1. −1<β 0<1. This assumption is commonly used to characterize a pure strategy equilibrium in the social interactions literature. (See, e.g., Bramoullé,Djebbari,andFortin(2009)and Blume et al. (2015) for examples of its use.) When β0>0, the game is called a game of strategic complements and, when β0<0, a game of strategic substitutes. 4Later in a section devoted to econometric inference, we specify the sharable type τito be a linear index of (X iεi),whereXiis a covariate vector observed by the econometrician and εi(together with the private type ηi) is not observed. Thus our framework permits information sharing on unobservables in the sense that “neighbors” of an agent iobserve εi. 5AgraphG=(NE) is undirected if ij ∈Ewhenever ji ∈Efor all i j ∈N. Quantitative Economics 11 (2020) Estimating local interactions among many agents 923 Let us introduce information sharing relations in the form of a directed graph (or anetwork)GI=(N EI)on Nso that each ij in EIrepresents the edge from player ito player j, where the presence of edge ij joining players iand jindicates that τiis observed by player j.Hencethepresenceofanedgeij between agents iand jrepresents information flow from ito j. This paper calls graph GIthe information graph. For each j∈N, define NI(j) ={i∈N:ij ∈EI} that is, the set of GI-neighbors observed by player j.6Also let NI(i) =NI(i) ∪{i},and nI(i) =|NI(i)|. In this paper, we do not assume that each agent knows the whole information graph GIand the payoff graph GP. To be precise about each agent’s information set, let us introduce some notation. For each i∈N,wesetNP1(i) =NP(i) and NI1(i) =NI(i), and for m≥2, define recursively NPm(i) = j∈NP(i) NPm−1(j) and NIm(i) = j∈NI(i) NIm−1(j) Thus NPm(i) denotes the set of players which consist of player iand those players who are connected to player ithrough at most medges in GP, and similarly with NIm(i). Also, define NPm(i) =NPm(i) \{i}and NIm(i) =NIm(i) \{i}. For each player i∈N, let us introduce a local payoff graph GPm(i) =(NPm(i) EPm(i)),wherefork1k2∈NPm(i),k1k2∈EPm(i) if and only if k1k2∈EP. Define for m≥1,7 Iim−1=GPm+1(i) NIm(i)τNIm(i)ηi(2.2) where τNIm(i) =(τj)j∈NIm(i).WeuseIim to represent the information set of agent i.For example, when agent ihas Ii0as his information set, it means that agent iknows the payoff subgraph GP2among the agents NP2(i), the set of agents whose types he observes (i.e., NI(i)), and his own private signal ηi. As for the payoff graph and information graph, we make the following assumption. Assumption 2.2. For each i∈Nand m≥1, NPm+1(i) ⊂NIm(i) This assumption requires for example that an agent with information Ii0observes their GPneighbors and their payoff relevant neighbors. The assumption on GIonly requires what each set NIm(i) should at least include but not what it should exclude. Hence all the results of this paper carry through even if we have NIm(i) =Nfor all i∈N, as in a complete information game. In other words, the incomplete information feature of our game is permitted but not required for our framework. 6More precisely, the neighbors in NI(j) are called in-neighbors and nI(j) =|NI(j)|in-degree. Throughout this paper, we simply use the term neighbors and degrees, unless specified otherwise. 7The graph GPm(i) is an induced subgraph of GPinduced by the vertex set NPm(i).Notealsothatwhile NP1(i) ⊂NP2(i), this does not imply that a player who knows the set NP2(i) knows what the set NP1(i) is. Our information set assumption requires them to know the local graph GPm(i) rather than just NPm(i). 924 Canen, Schwartz, and Song Quantitative Economics 11 (2020) 2.2 Predictions from rationality Each player chooses a strategy that maximizes his expected payoff according to his beliefs. Given player i’s strategy, information set Ii, and his beliefs on the strategy of other players si −i=(si k)k∈N\{i}, the (interim) expected payoff of player iis defined as Uisisi −i;Ii=Euisi(Ii)si −i(I−i)τηi|Ii where si −i(I−i)=(si k(Ik))k∈N\{i},I−i=(Ik)k=iand τ=(τi)i∈N.Abest response sBR iof player icorresponding to the strategies si −iof the other players as expected by player iis such that for any strategy si, UisBR isi −i;Ii≥Uisisi −i;Iia.e. The quadratic payoff function and the information structure of the game implies that if player ihas information set Iiand believes that each of her GP-neighbors, say, k,plays a strategy si k(Ik), her best response is given by sBR i(Ii)=τi+β0 nP(i)  k∈NP(i) Esi k(Ik)|Ii+ηi(2.3) This implies that the best responses will be linear in types τjas long as the conditional expectation is. In order to generate predictions, one needs to deal with the beliefs (i.e., si k(Ik))in the conditional expectation. There are three approaches. The first approach is an equilibrium approach, where we take the predicted strategies as a set of best response strategies sBNE isuch that for any strategy si, UisBNE isBNE −i;Ii≥UisisBNE −i;Iia.e. (2.4) Hence in equilibrium strategies, each player believes that the other players’ strategies coincide with the best response strategies by the agents in equilibrium. The second approach, rationalizability, considers all strategies that are rationalizable given some belief. The third approach is a behavioral approach where one considers a set of simple behavioral assumptions on the beliefs and focuses on the best responses to these beliefs. There are pros and cons with each of the three approaches. The equilibrium approach requires that the beliefs of all the players be “correct” in equilibrium. However, since each player igenerally does not know who each of his GP-neighbors observes, a Bayesian player in an incomplete information game with rational expectations would need to know the distribution of the entire information graph GI(or at least have a common prior on the information graph commonly agreed upon by all the players) to form a “correct” belief given his information. Given that the players are only partially observed and GIis rarely observed with precision, producing a testable implication from such an equilibrium model appears far from a trivial task. The rationalizability approach can be used to relax this rational expectations assumption by eliminating the requirement that the beliefs be correct. Such an approach Quantitative Economics 11 (2020) Estimating local interactions among many agents 931 As compared to game Γ0,gameΓ1predicts outcomes with broader network externality. Indeed, when m=1, s[1] i(Ii1)=1+β0 nP(i)  k∈NP(i) w[0] ki τi + j∈NP2(i)β0 nP(i)  k∈NP(i) w[0] kj 1j∈NP(k)τj+ηi(2.10) For example, the types of neighbors whose actions do not affect player i’s payoff can affect his best response. More specifically, note that for j∈NP2(i) \NP(i), ∂s[1] i(Ii1) ∂τj=β0 nP(i)  k∈NP(i) 1j∈NP(k)w[0] kj  The externality from player jto player iis strong when player jhas a high local centrality λkj to a large fraction of player i’s GP-neighbors k. 2.4 Comparing equilibrium strategies and behavioral strategies 2.4.1 Convergence of behavioral strategies to equilibrium strategies We show that as the information set expands and the order of sophistication becomes higher, the behavioral strategies converge to the equilibrium strategies from a game where all players observe all other players’ sharable types. Let Γ∞be the game where players have the same payoff function and the same payoff graph as in Γ0except that the information set for each player iis given by Ii∞=(GPτηi). Thus each player iknows the whole payoff graph GP, all sharable types, τ=(τi)i∈N, and private information ηi. (This information structure is similar to Blume et al. (2015).) Let sBNE =(sBNE i)i∈Nbe the Bayesian–Nash equilibrium strategy profile from the game Γ∞. Below, we give a theorem which shows that the sequence of behavioral strategies s[m] i converges to the equilibrium strategies sBNE ias m→∞. Theorem 2.3. Suppose that the conditions of Theorem 2.1 hold and that max i∈NEτi2<∞(2.11) Then,as m→∞, Emax i∈Ns[m] i(Iim)−sBNE i(Ii∞))2→0 Theorem 2.3 shows that as the order of sophistication deepens, the best response strategies from the behavioral model become closer to the equilibrium strategies. It is nothardtocheckthats[m] i(Iim)=s[m] i(Ii∞), that is, the best response remains the same if we expand the information set Iim to Ii∞. Therefore, the convergence in Theorem 2.3 can be viewed as the convergence of the best responses s[m] i(Ii∞)to equilibrium strategies sBNE i(Ii∞)while the information set is fixed to be Ii∞. 932 Canen, Schwartz, and Song Quantitative Economics 11 (2020) 2.4.2 Comparison in terms of network externality We compare the behavioral strategies and equilibrium strategies in terms of network externality which measures how sensitively an agent’s action responds to a change in her neighbor’s types. We also compare how this network externality changes as the network grows. Let Yibe the observed outcome of player ias predicted from either of the two game models. For simplicity, we remove ηi’s from the models so that the game Γ∞now becomes a complete information game. The complete information game gives the following prediction for action Yiof agent i: Yi=β0 nP(i)  j∈NP(i) Yj+τi where Yidenotes the action of player iin equilibrium. Then the reduced form for Yi’s can be written as y=(I −β0A)−1τ (2.12) where y=(Y1Yn),τ=(τ1τn),andAis a row-normalized adjacency matrix of the payoff graph GP,thatis,the(i j)-th entry of Ais 1/nP(i) if j∈NP(i) and zero otherwise. Thus when β0is close to one (i.e., the local interaction becomes strong), the equilibrium outcome can exhibit extensive cross-sectional dependence. On the other hand, our behavioral model predicts the following: Yi=1+β2 0λi nP(i) −β2 0λiτi+ j∈NP(i) β0λij nP(i)τj which comes from Theorem 2.1 without ηi’s. When we compare this with (2.12), it is clear that the cross-sectional dependence structure of our behavioral model is different from that from the complete information equilibrium model. In the case of the complete information equilibrium model, it is possible that two actions Yiand Yjbetween two agents iand jcan be correlated even if iand jare very far from each other in graph GP. However, the cross-sectional dependence structure of the actions from the behavioral model closely follows the graph GP:Yiand Yjcan be correlated only if their GP neighbors overlap. For comparison purposes, for a given strategy si(Ii)for an agent iwith information set Ii, we introduce the average network externality (ANE): 1 n j∈N i∈N:i=j ∂si(Ii) ∂τj (2.13) The ANE measures the average impact of a change in the neighbors’ type on the actions of the player. The ANE from equilibrium strategies of the complete information game is 1 nj∈Ni∈N:i=j[(I −β0A)−1]ij ,where[(I −β0A)−1]ij denotes the (i j)-th entry of the matrix (I −β0A)−1. We consider the average of the ANE’s over simulated payoff graphs. For the payoff graph GP, we considered two different models for random graph generation. The first Quantitative Economics 11 (2020) Estimating local interactions among many agents 933 Table 1. The characteristics of the payoff graphs. Erd˝ os–Rényi Barabási–Albert Network A Network B Network C Network A Network B Network C n1649 7834 31168 2361 15210 47738 dmx 1114 1274 1412 7000 1244 1354 dav 2046 2307 3198 1563 2057 2568 Note: This table gives average characteristics of the payoff graphs, GP, used in the simulation study, where the average was over 50 simulations. dav and dmx denote the average and maximum degrees of the payoff graphs. kind of random graphs are Erd˝ os–Rényi (ER) random graph with the probability equal to 5/n and the second kind of random graphs are Barabási–Albert (BA) random graph such that beginning with an Erd˝ os–Rényi random graph of size 20 with each link forming with equal probability 1/19 and grows by including each new node with two links formed with the existing nodes with probability proportional to the degree of the nodes. For each random graph, we first generate a random graph of size 10,000,andthen construct three subgraphs A,B,Csuch that network Ais a subgraph of network Band the network Bis a subgraph of network C. We generate these subgraphs as follows. First, we take a subgraph Ato be one that consists of agents within distance kfrom agent i=1. Then network Bis constructed to be one that consists of the neighbors of the agents in network Aand network Cis constructed to be one that consists of the neighbors of the agents in network B. For an ER random graph, we took k=3and for a BA random graph, we took k=2. We repeated the process 50 times to construct an average behavior of network externality as we increase the network. Table 1shows the average network sizes and degree characteristics as we move from Networks A, B to C. The ANEs from the equilibrium strategies from game Γ∞, and the behavioral strategies from games Γmas mbecomeshigherareshowninFigures1and 2.15 First, as m becomes larger, the ANEs from Γ∞and those from Γmget closer, as predicted by Theorem 2.3. Furthermore, the ANEs from the behavioral model are similar to that from the equilibrium model especially when β0is between −05and 05. Finally, the network externalities from the games with simple type players are somewhat sensitive to the size of the networks when β0is very high or very low. This sensitivity is reduced substantially when we consider the game with first-order sophisticated agents. Finally, network externality tends to be much higher for equilibrium models than the behavioral models when β0is high. Hence using our behavioral approach as a proxy for an equilibrium approach makes sense only when strategic interdependence is not too high. 3. Econometric inference 3.1 General overview 3.1.1 Partial observation of interactions A large network data set is often obtained through a nonrandom sampling process (see, e.g., Kolaczyk (2009)). The actual sampling 15We provide conditions for the local identification of βin Section 3.1.3 below. 934 Canen, Schwartz, and Song Quantitative Economics 11 (2020) Figure 1. The average network externality comparison between equilibrium and behavioral models: Erd˝ os–Rényi graphs. process of network data is often unknown to the researcher. Our approach of empirical modeling can be useful in a situation where only a fraction of the players are observed through a certain nonrandom sampling scheme that is not precisely known to the researcher. In this section, we make explicit the data requirements for the econometrician and propose inference procedures. We mainly focus on the game where all the players in the game are of simple type. The inference for games with agents of first-order sophisticated type is found in the replication file of Canen, Schwartz, and Song (2020). Suppose that the original game of interactions consists of a large number of agents whose set we denote by N. Let the set of players be on a payoff graph GPand an information graph GI, facing the strategic environment as described in the preceding section. Denote the best response as an observed dependent variable Yi:fori∈N, Yi=s[0] i(Ii0) where the sharable type τiis specified as τi=X iρ0+εi(3.1) and Xiis a d-dimensional vector of covariates pertaining to agent iobserved by the econometrician, ρ0∈Rdis a coefficient vector, and εiis unobserved heterogeneity. Quantitative Economics 11 (2020) Estimating local interactions among many agents 935 Figure 2. The average network externality comparison between equilibrium and behavioral models: Barabási–Albert graphs. The covariate Xican contain GP-neighborhood averages of individual covariates. Let us make the following additional assumption on this original large game. Let us first define F=σ(XGPGI)∨C that is, the σ-field generated by X=(Xi)i∈N,GP,GIand Cis a given common shock which is to be explained below. Assumption 3.1. (i) εi’s and ηi’s are conditionally i.i.d.across i’s given F. (ii) {εi}n i=1and {ηi}n i=1are conditionally independent given F. (iii) For each i∈N,E[εi|F]=0and E[ηi|F]=0. Condition (i) excludes preexisting cross-sectional dependence of unobserved heterogeneity in the payoffs once conditioned in F. This condition implies that conditional on F, the cross-sectional dependence of observed actions is due solely to the information sharing among the agents. Condition (ii) requires that conditional on F,theunobserved payoff heterogeneities observed by other players and those that are private are independent. Condition (iii) excludes endogenous formation of GPor GI, because 936 Canen, Schwartz, and Song Quantitative Economics 11 (2020) the condition requires that the unobserved type components εiand ηibe conditionally mean independent of these graphs, given X=(Xi)i∈Nand C. However, the condition does not exclude the possibility that GPand GIare exogenously formed based on (XC). For example, suppose that ij ∈EPif and only if fij (XiXjaiajuij )≥0 where airepresents degree heterogeneity, uij ’s errors, and fij a given nonstochastic function. In this set-up, the econometrician does not observe ai’s or uij ’s. This nests the dyadic regression model of Graham (2017) as a special case. Condition (iii) accommodates such a set-up, as long as {ai}n i=1and {uij }n ij=1are conditionally independent of {εi}n i=1and {ηi}n i=1given X. One simply has to take Cto contain ai’s and uij ’s. The econometrician observes only a subset N∗⊂Nof agents and part of GPthrough a potentially stochastic sampling process of unknown form. We assume for simplicity that n∗≡|N∗|is nonstochastic. This assumption is satisfied, for example, if one collects the data for agents with predetermined sample size n∗. We assume that though being a small fraction of N,thesetN∗is still a large set justifying our asymptotic framework that sends n∗to infinity. Most importantly, constituting only a small fraction of N,the observed sample N∗of agents induces a payoff subgraph which one has no reason to view as “approximating” or “similar to” the original payoff graph GP. Let us make precise the data requirements. Condition A. The stochastic elements of the sampling process are conditionally independent of {(τiηi)}i∈Ngiven F. Condition B. For each i∈N∗, the econometrician observes NP(i) and (YiXi),andfor each j∈NP(i), the econometrician observes |NP(i) ∩NP(j)|,nP(j) and Xj. Condition C. Either of the following two conditions is satisfied: (a) For i j ∈N∗such that i=j,NP(i) ∩NP(j) =∅. (b) For each agent i∈N∗, and for any agent j∈N∗such that NP(i) ∩NP(j) = ∅,the econometrician observes Yj,|NP(j) ∩NP(k)|,nP(k) and Xkfor all k∈NP(j). Before we discuss the conditions, it is worth noting that these conditions are trivially satisfied when we observe the full payoff graph GPand N∗=N. Condition Ais satisfied, for example, if the sampling process is based on observed characteristics Xand some characteristics of the strategic environment that is commonly observed by all the players. This condition is violated if the sampling is based on the outcomes Yi’s or unobserved payoff-relevant signals such as εior ηi. Condition Bessentially requires that in the data set, we observe (YiXi)of many agents i, and for each GP-neighbor jof agent i, observe the number of the agents who are common GP-neighbors of iand jand the size of GP-neighborhood of jalong with the observed characteristics Xj.16 As for a GP16Note that this condition is violated when the neighborhoods are top-coded in practice. For example, the maximum number of friends in the survey for a peer effects study can be set to be lower than the actual number of friends for many students. The impact of this top-coding upon the inference procedure is an interesting question on its own which deserves exploration in a separate paper. Quantitative Economics 11 (2020) Estimating local interactions among many agents 937 neighbor jof agent i∈N∗, this condition does not require that the agent j’s action Yjor the full set of his GP-neighbors are observed. Condition C(a) is typically satisfied when an initial sample of agents is randomly selected from a much larger set of agents so that no two agents have overlapping GP-neighbors in the sample, and then their GPneighbors are selected for each agent in the sample to constitute N∗.17 In practice for use in inference, one can take the set N∗to include only those agents that satisfy Conditions A–Cas long as N∗thereof is still large and the selection is based only on (XGP).One can simply use only those agents whose GP-neighborhoods are not overlapping, as long as there are many such agents in the data. 3.1.2 Moment conditions In order to introduce inference procedures for β0and other payoff parameters, let us define for i∈N, Zi=1+β2 0λi nP(i) −β2 0λiXi+β0 nP(i)  j∈NP(i) λij Xj(3.2) (Note that Zirelies on β0although it is suppressed from notation for simplicity as we do frequently below for other quantities.) By Theorem 2.1 and (3.1), we can write Yi=Z iρ0+vi(3.3) where vi=1+β2 0λi nP(i) −β2 0λiεi+β0 nP(i)  j∈NP(i) λij εj+ηi Note that the observed actions Yiare cross-sectionally dependent (conditional on F) due to information sharing on unobservables εi. Suppose that ϕiis M×1vector of instrumental variables (which potentially depend on β0)withM>dsuch that for all i∈N, E[viϕi]=0(3.4) Note that the orthogonality condition above holds for any ϕias long as for each i∈N, ϕiis F-measurable, that is, once Fis realized, there is no extra randomness in ϕi.This is the case, for example, when ϕiis a function of X=(Xi)i∈Nand GP. While the asymptotic validity of our inference procedure admits a wide range of choices for ϕi’s, one needs to choose them with care to obtain sharp inference on the payoff parameters. Especially, it is important to consider instrumental variables which involve the characteristics of GP-neighbors to obtain sharp inference on payoff externality parameter β0. This is because the cross-sectional dependence of observations carries substantial information for strategic interdependence among agents. 17This random selection does not need to be a random sampling from the population of agents. Note that the random sampling is extremely hard to implement in practice in this situation, because one needs to use the equal probability for selecting each agent into the collection N∗, but this equal probability will be feasible only when one has at least the catalog of the entire population N. 938 Canen, Schwartz, and Song Quantitative Economics 11 (2020) 3.1.3 Local identification It is not hard to see that under regularity conditions (such as those preventing multicollinearity in Zi), ρ0is identified up to β0.18 However, the moment function in (3.4)isnonlinearinβ0, and hence even local identification of β0is not guaranteed unless we impose further assumptions. Here we provide conditions for local identification, but for inference we propose later, we pursue asymptotically valid inference allowing the parameters to be only partially identified. Let θ≡[β ρ]and write vi(θ) =Yi−Z i(β)ρ,whereZi(β) is the same as Ziexcept that β0is replaced by β.LetΘbe the parameter space for θ0. Assumption 3.2. (i) For all i=1n,ϕidoes not depend on θ∈Θ,and the parameter space Θis compact,and β∈[−1+ν 1−ν]for all βsuch that [β ρ]∈Θ,for some small ν>0. (ii) There exists C>0such that for all n≥1, 1 n n  i=1 EXin4+1 nP(i)  j∈NP(i) Xj14GP+1 n n  i=1 Eϕi4|GP<C (iii) There exists c>0such that the minimum eigenvalue of the matrix M  m=11 n n  i=1 Him(θ0)1 n n  i=1 Him(θ0) is bounded from below by cfor all n≥2,where Him(θ) ≡E∂vi(θ)/∂β −Zi(β) ϕimGP and ϕim is the mth entry of ϕi. (iv) maxi∈Nn2 P(i)/√n→0,as n→∞. Assumption 3.2(i) in regards to ϕisimplifies the identification arguments and is satisfied when the “instruments” ϕiconsist only of observed variables. Assumption 3.2(ii) is a moment condition for the covariates. Assumption 3.2(iii) is a nontrivial condition and is violated if the parameter space for ρ0includes zero, because we have ∂vi(β0)/∂β =0. Thus this assumption requires the researcher to know that the true parameter ρ0is away from zero. Assumption 3.2(iv) is a mild condition that requires that the payoff graph GP is not overly dense. Under this assumption, in combination with the conditions of Theorem 2.1,wecan show that θ0is locally identified (i.e., consistently estimable over a neighborhood of θ0.) 18A standard identification analysis centers on a “representative probability” from which we observe i.i.d. draws. A parameter is identified if it is uniquely determined under each representative probability. However, in our set-up, there is no such probability, as all observations exhibit heterogeneity and local dependence along a large, complex network. Here, “identification” simply means “consistent estimability” and “local identification” means “consistent estimability around a neighborhood of the true parameter.” Quantitative Economics 11 (2020) Estimating local interactions among many agents 939 Theorem 3.1. Suppose that Assumption 3.2 and the conditions of Theorem 2.1 hold. Then there exists ε>0such that if Θ=B(θ0;ε),θ0is consistently estimable,where B(θ0;ε) is the ε-neighborhood of θ0and B(θ0;ε) is its closure. Since consistent estimability of θ0requires that ρ0be away from zero, it is expected that as ρ0gets close to zero, β0is only “weakly (locally) identified.” As a researcher is rarely aprioricertain that ρ0is away from zero, we pursue inference that does not require this. 3.1.4 Estimation and inference We first estimate ρ0assuming knowledge of β0. Define Sϕϕ =ϕϕ/n∗and ˜ϕ=ϕS−1/2 ϕϕ  where ϕis an n∗×Mmatrix whose i-th row is given by ϕ i,i∈N∗. Define Λ=1 n∗ i∈N∗ j∈N∗ E[vivj|F]˜ϕi˜ϕ j(3.5) where ˜ϕirepresents the transpose of the ith row of ˜ϕ,andlet ˆ Λbe a consistent estimator of Λ. (We will explain how we construct this estimator in Section 3.1.7 below.) Define SZ˜ϕ=Z˜ϕ/n∗and S˜ϕy =˜ϕy/n∗ where Zis an n∗×dmatrix whose ith row is given by Z iand yis an n∗×1vector whose ith entry is given by Yi,i∈N∗.Thenweestimate ˆρ=SZ˜ϕˆ Λ−1S Z˜ϕ−1SZ˜ϕˆ Λ−1S˜ϕy(3.6) Using this estimator, we construct a vector of residuals ˆ v=[ˆ vi]i∈N∗,where ˆ vi=Yi−Z iˆρ (3.7) Finally, we form a profiled test statistic as follows: T(β0)=ˆ v˜ϕˆ Λ−1˜ϕˆ v n∗(3.8) making it explicit that the test statistic depends on β0. Later we show that T(β0)→dχ2 M−das n∗→∞ where χ2 M−ddenotes the χ2distribution with degree of freedom M−d.LetCβ 1−αbe the (1−α)100% confidence set for β0defined as Cβ 1−α≡β∈(−11):T(β)≤c1−α where T(β)is computed as T(β0)with β0replaced by βand the critical value c1−αis the (1−α)-quantile of χ2 M−d. 940 Canen, Schwartz, and Song Quantitative Economics 11 (2020) Let us now construct a confidence set for ρ0. First, we establish that under regularity conditions √n∗ˆ V−1/2(ˆρ−ρ0)→dN(0Id) as n∗→∞,where ˆ V=SZ˜ϕˆ Λ−1S Z˜ϕ−1 (See Section 3.2 below for conditions and formal results.) Using this estimator ˆρ,wecan construct a (1−α)100% confidence interval for aρ0for any nonzero vector a.Forthis, define ˆσ2(a) =aˆ Va Let z1−(α/4)be the (1−(α/4))-percentile of N(01). Define for a vector awith the same dimension as ρ, Cρ 1−(α/2)(β0a)=aˆρ−z1−(α/4)ˆσ(a) √naˆρ+z1−(α/4)ˆσ(a) √n Then the confidence set for aρis given by19 Cρ 1−α(a) = β∈Cβ 1−(α/2) Cρ 1−(α/2)(βa) Notice that since βruns in (−11)and the estimator ˆρhas an explicit form, the confidence interval is not computationally costly to construct in general. Often the eventual parameter of interest is one that captures how strongly the agents’s decisions are interdependent through the network. For this, we can use the average network externality (ANE) introduced in (2.13). Let s[0] i(Ii0)be the best response of agent ihaving information set Ii. Then the ANE with respect to Xir (where Xir represents the rth entry of Xi)isgivenbyθ1(β0ρ0r),where θ1(β0ρ0r )=1 n∗ i∈N∗ j∈NP(i) ∂s[0] i(Ii0) ∂xjr =1 n∗ i∈N∗ j∈NP(i) β0λij nP(i)1+β2 0λi nP(i) −β2 0λiρ0r and ρ0r denotes the rth entry of ρ0.See(2.8). Thus the confidence interval for θ1(β0ρ0r )can be constructed from the confidence interval for β0and ρ0as follows: Cθ1 1−α=θ1(β ρr):β∈Cβ 1−α/2and ρr∈Cρr 1−α/2(3.9) 19Instead of the Bonferroni approach here, one could consider a profiling approach where one uses T(ρ)=supβT(βρ)as the test statistic, where T(βρ)is the test statistic constructed using ρin place of ˆρ. The profiling approach is cumbersome to use here because one needs to simulate the limiting distribution of T(ρ) for each ρ, which can be computationally complex when the dimension of ρis large. Instead, this paper’s Bonferroni approach is simple to use because β0takesvaluesfrom(−11). Quantitative Economics 11 (2020) Estimating local interactions among many agents 947 Table 3. The empirical coverage probability and average length of confidence intervals for β0 at 95% nominal level. β0 Specification 1 Specification 2 m=1m=2m=3λ=1λ=2λ=3 Coverage Probability −05 n=500 09642 09580 09648 09686 09638 09622 n=1000 09638 09634 09574 09650 09644 09604 n=5000 09596 09560 09530 09704 09608 09596 −03 n=500 09540 09536 09612 09608 09546 09568 n=1000 09566 09568 09566 09564 09578 09548 n=5000 09534 09548 09542 09636 09568 09546 0 n=500 09504 09464 09554 09474 09478 09490 n=1000 09486 09508 09514 09498 09510 09526 n=5000 09440 09490 09546 09516 09482 09478 03 n=500 09548 09512 09584 09562 09552 09556 n=1000 09600 09558 09524 09598 09592 09590 n=5000 09524 09536 09574 09604 09544 09522 05 n=500 09648 09574 09618 09640 09610 09620 n=1000 09630 09604 09534 09710 09648 09634 n=5000 09564 09598 09612 09700 09632 09584 Average Length of CI −05 n=500 00834 01307 01947 01089 00751 00750 n=1000 00490 00794 01038 00630 00438 00463 n=5000 00053 00203 00303 00108 00026 00024 −03 n=500 00799 01216 01639 01083 00865 00910 n=1000 00464 00758 00990 00639 00519 00577 n=5000 00034 00187 00296 00116 00060 00075 0 n=500 00785 01212 01572 01070 00970 01087 n=1000 00452 00753 00996 00638 00597 00700 n=5000 00024 00182 00298 00113 00106 00155 03 n=500 00713 01062 01384 00983 00685 00676 n=1000 00404 00640 00872 00562 00389 00412 n=5000 00017 00155 00262 00076 00015 00013 05 n=500 00495 00738 01085 00666 00289 00240 n=1000 00252 00337 00657 00328 00089 00079 n=5000 00001 00055 00147 00004 00000 00000 Note: The first-half of the table reports the empirical coverage probability of the asymptotic confidence interval for β0and the second-half reports its average length. The simulated rejection probability at the true parameter is close to the nominal size of α=005 and the average lengths decrease with n. The simulation number is R=5000. 948 Canen, Schwartz, and Song Quantitative Economics 11 (2020) Table 4. The empirical coverage probability and average length of confidence intervals for aρ0 at 95% nominal level. β0 Specification 1 Specification 2 m=1m=2m=3λ=1λ=2λ=3 Coverage Probability −05 n=500 09848 09802 09860 09862 09834 09740 n=1000 09616 09610 09680 09682 09670 09596 n=5000 09596 09548 09606 09706 09668 09614 −03 n=500 09802 09772 09858 09832 09826 09794 n=1000 09620 09756 09796 09772 09682 09692 n=5000 09544 09510 09562 09588 09568 09556 0 n=500 09740 09754 09868 09828 09792 09786 n=1000 09668 09770 09798 09746 09738 09756 n=5000 09430 09500 09524 09546 09496 09494 03 n=500 09804 09804 09866 09810 09778 09788 n=1000 09698 09794 09812 09816 09718 09758 n=5000 09476 09524 09546 09572 09536 09498 05 n=500 09824 09828 09858 09810 09722 09720 n=1000 09724 09786 09826 09810 09596 09594 n=5000 09552 09536 09596 09626 09596 09542 Average Length of CI −05 n=500 54643 102562 174488 70549 56285 60639 n=1000 33501 59311 81165 42270 34243 38800 n=5000 07489 18254 24970 11165 06337 06588 −03 n=500 42511 68326 95995 56856 49154 54091 n=1000 25915 43335 56775 34711 30685 35331 n=5000 05297 13514 19123 09505 07346 08560 0 n=500 35812 53587 68508 47774 43760 48399 n=1000 21797 34445 44160 29651 27944 32136 n=5000 04238 10993 15392 08365 07964 09882 03 n=500 33664 45527 57822 44438 32458 31962 n=1000 20559 28904 36607 26872 20100 21074 n=5000 04072 09592 12989 06961 04399 04713 05 n=500 30230 41201 58780 37624 21350 19826 n=1000 17684 21613 34957 21027 11718 12075 n=5000 03576 06764 10002 03995 03723 04081 Note:Thetrueaρ0is equal to 14. The first-half of the table reports the empirical coverage probability of the asymptotic confidence interval and the second-half its average length for aρ0. The empirical coverage probability of the confidence interval for aρ0is generally conservative which is expected from the use of the Bonferroni approach. Nevertheless, the length of the confidence interval is reasonably small. The simulation number, R,is5000. Quantitative Economics 11 (2020) Estimating local interactions among many agents 949 5. Empirical application:State presence across municipalities 5.1 Motivation and background State capacity (i.e., the capacity of a country to provide public goods, basic services, and the rule of law) can be limited for various reasons. (See, e.g., Besley and Persson (2009)andGennaioli and Voth (2015)).24 A “weak state” may arise due to political corruption and clientelism, and result in spending inadequately on public goods (Acemoglu (2005)), accommodating armed opponents of the government (Powell (2013)), and war (McBride, Milante, and Skaperdas (2011)). Empirical evidence has shown how these weak states can persist from precolonial times, with higher state capacities apparently related to current level prosperity at the ethnic and national levels (Gennaioli and Rainer (2007)andMichalopoulos and Papaioannou (2013)). Our empirical application is based on a recent study by Acemoglu, García-Jimeno, and Robinson (2015) who investigate the local choices of state capacity in Colombia, using a model of a complete information game on an exogenously formed network. In their set-up, municipalities choose a level of spending on public goods and state presence (as measured by either the number of state employees or state agencies). Network externalities in a municipality’s choice exist because municipalities that are adjacent to one another can benefit from their neighbors’ choices of public goods provisions, such as increased security, infrastructure, and bureaucratic connections. Thus, a municipality’s choice of state capacity can be thought of as a strategic decision on a geographic network. It is not obvious that public good provision in one municipality leads to higher spending on public goods in neighboring municipalities. Some neighbors may freeride and underinvest in state presence if they anticipate others will invest highly. Rentseeking by municipal politicians would also limit the provision of public goods. On the other hand, economies of scale could lend to complementarities in state presence across neighboring municipalities. In our study, we extend the model in Acemoglu, García-Jimeno, and Robinson (2015) to an incomplete information game where information may be shared across municipalities. In particular, we do not assume that all municipalities know and observe all characteristics and decisions of the others. It seems reasonable that the decisions made across the country may not be observed or well known by those municipalities that are geographically remote. 5.2 Empirical set-up Let yidenote the state capacity in municipality i(as measured by the log number of public employees in municipality i)andGPdenote the geographic network, where an edge is defined on two municipalities that are geographically adjacent.25 We assume that GP is exogenously formed. The degree distribution of GPis shown in Figure 3. We study the 24See also an early work by Brett and Pinkse (2000) for an empirical study on the spatial effects on municipal governments’ decisions on business property tax rates. 25This corresponds to the case in of δ1=δ2=0in Acemoglu, García-Jimeno, and Robinson (2015). 950 Canen, Schwartz, and Song Quantitative Economics 11 (2020) Figure 3. Degree distribution of GP. optimal choice of yi,whereyileads to a larger prosperity pi. Prosperity in municipality i is modeled as pi=β¯ yi+x 1iγ+ηi+εi+ςD iyi(5.1) where ςD iis a district specific dummy variable, εiand ηiare our sharable and nonsharable private information, and yi=1 nP(i) j∈NP(i) yj.Thetermx1i represents municipality characteristics. These include geographic characteristics, such as land quality, altitude, latitude, rainfall; and municipal characteristics, such as distance to highways, distance to royal roads and Colonial State Presence.26 The welfare of a municipality is given by ui(yiy−iτηi)=pi(yi¯ yiτηi)−1 2y2 i(5.2) where the second term refers to the cost of higher state presence, and the first term is the prosperity pi. We can rewrite the welfare of the municipality by substituting (5.1) into (5.2): ui(yiy−iτηi)=β¯ yi+x 1iγ+ηi+εi+ςD iyi−1 2y2 i(5.3) We assume that municipalities (or the mayor in charge), wishes to maximize welfare by choosing state presence, given their beliefs about the types of the other municipalities. 26Note that piis only a function of terms that are multiplied by yi. This is a simplification from their specification. We do so because we will focus on the best response equation. The best response equation, derived from the first-order condition to this problem, would not include any term that is not a function of yiitself. Quantitative Economics 11 (2020) Estimating local interactions among many agents 951 In our specification, we allow for incomplete information. This is reflected in the terms εi,ηi, which will be present in the best response function. The municipality, when choosing state presence yi,willbeabletoobserveεiof its neighbors and will use its beliefs over the types of the others to generate its best response. The best response will follow the results from Theorem 2.1. 5.3 Model specification We closely follow Table 3 in Acemoglu, García-Jimeno, and Robinson (2015) for the choice of specifications and variables. First, we will consider the model with simple types.27 Throughout the specifications, we include longitude, latitude, surface area, elevation, annual rainfall, department fixed effects and a department capital dummy (all in X1). We further consider the effect of variables distance to current highways, land quality, and presence of rivers in the municipality. For the choice of instruments, we consider two separate types of instruments. The first is the sum of neighbor values (across GP) of the historical variables (denoted as Ci).28 The historical variables used are Total Crown Employees (also called Colonial State Officials), Distance to Royal Roads, Colonial State Agencies, and Historical Population, as well as Colonial State Presence Index squared and Distance to Royal Roads squared. Using the latter two additionally sharpens inference. We also use the variable ˜ Zi=nP(i)−1j∈NP(i) λij Xj1as part of the instrumental variables, which was shown to perform well in the Monte Carlo Simulations in Section 4. This variable captures cross sectional dependence as a crucial source of variation for inference on the strategic interactions. We use downweighting of our instruments as explained in a preceding section and rescale instruments by multiplying them by S−1/2 ϕϕ . 5.4 Results The results across a range of specifications are presented in Table 5. In these results, we see that the effect is statistically different than 0and stable across specifications. It indicates that there is complementarity in the provision of public goods and state presence (β>0). 27In the replication file of this paper, we consider the empirical application with first-order sophisticated types (game Γ1), as well as the model selection test between simple types and first-order sophisticated types. Since the simple type model is not rejected in the data and it is more parsimonious, we present it in the main text. The results for the first-order sophisticated case are more or less similar except that the confidence intervals of β0are wider. At 5%, the model selection procedure did not reject either of the sets of the moment conditions from the simple type and the first-order sophisticated players. 28For this, we assume the exclusion restriction in Acemoglu, García-Jimeno, and Robinson (2015), namely that historical variables only affect prosperity in the same municipality. This means that although one’s historical variables (Total Crown Employees, Distance to Royal Roads, Colonial State Agencies and Historical Population, as well as functions thereof) can affect the same municipality’s prosperity, it can only affect those of the neighbors by impacting the choice of state capacity in the first, which then impacts the choice of the state capacity in the neighbors. 952 Canen, Schwartz, and Song Quantitative Economics 11 (2020) Table 5. State presence and networks effects across Colombian municipalities. Outcome: The Number of State Employees Baseline Distance to Highway Land Quality Rivers (1) (2) (3) (4) β0[016031][016032][017039][007038] dyi/d(Colonial State Officials)[−00600003][−0048−0001][−00510003][−00340009] Average dyi/d(Colonial State Agencies)[−13234051][−12492793][−09723545][−41862719] Average dyi/d(distance to Royal Roads)[−00100011][−00090011][−00080018][−00100013] n1018 1018 1003 1003 Note: Confidence sets for βare presented in the table, obtained from inverting the test statistic T(β) from Section 3for first-order sophisticated types, with confidence level of 95%. The critical values in the first row come from the asymptotic statistic. Downweighting is used. The average marginal effects for historical variables upon state capacity are also shown. The marginal effect of Colonial State Officials is equal to its γcoefficient. The marginal effect for Distance to Royal Roads for municipality iequals γRoyal Roads +2∗γRoyal Roads2(Royal Roads)i, where γRoyal Roads is the γcoefficient of its linear term, and γRoyal Roads2is the coefficient of its quadratic term, as this variable enters X1as a quadratic form. The analogous expression holds for the variable Colonial State Agencies. We show the average marginal effect for these two variables. We then present the confidence set for these marginal effects, computed by the inference procedure on aγdeveloped in Section 3. All specifications include controls of latitude, longitude, surface area, elevation, rainfall, as well as Department and Department capital dummies. Instruments are constructed from payoff neighbors’ sum of the GPneighbors values of the historical variables Total Crown Employees, Colonial State Agencies, Colonial State Agencies squared, population in 1843, distance to Royal Roads, distance to Royal Roads squared, together with the non-linear function ˜ Zi=nP(i)−1j∈NP(i) λij Xj1. Column (2) includes distance to current highway in X1, Column (3) expands the specification of Column (2) by also including controls for land quality (share in each quality level). Column (4) controls for rivers in the municipality and land quality, in addition to those controls from Column (1). One can see that the results are very stable across specifications. Let us compare our results to those in Acemoglu, García-Jimeno, and Robinson (2015). There, the authors report the average marginal effects over their weighted graph. The (weighted) average degree is 00329, so our results can be compared in an approximation, by considering 00329 ˆ β. In general, our estimates have the same sign and significance as those of Acemoglu, García-Jimeno, and Robinson (2015). Our estimates are in the range of [00020013],after reweighting as mentioned before, somewhat comparable to theirs of [00160022](in the case of the outcome of the number of public employees, in Table 3 in their paper). Hence, we find similar qualitative effects, although a smaller magnitude. Recall that our confidence set is built without assuming that β0is consistently estimable. In Figure 4, we show the results of our estimated network externalities for the estimates from Table 5, for the importance of being a department capital. The average network externality (ANE) is computed as 1 N i∈N j∈NP(i) β0ˆγdc nP(i)(1−β0cij )1+β2 0λi nP(i) −β2 0λi where ˆγdc is the estimated parameter of the X1variable department capital. The parameter is defined in Section 3.1.4, and captures the average effect of a neighbor being a department capital. We construct a confidence interval as in (3.9). Quantitative Economics 11 (2020) Estimating local interactions among many agents 953 Figure 4. Average network externality from being a department capital. The figure shows that there is a strong and increasing network externality from being a department capital over the range of the confidence set of β. This indicates that the effect of being a capital has spillovers on other municipalities: since β>0,andone expects that department capitals have more state presence and resources, being a department capital yields increasing returns the stronger the complementarity. 6. Conclusion This paper proposes a new approach of empirical modeling for interactions among many agents when the agents observe the types of their neighbors in a single large network. The main challenge arises from the fact that the information sharing relations are typically connected among a large number of players whereas the econometrician observes only a fraction of those agents. Using a behavioral model of belief formation, this paper produces an explicit form of best responses from which an asymptotic inference procedure for the payoff parameters is developed. As we showed in our paper, this explicit form gives a reduced form for the observed actions, and exhibits various intuitive features. For example, the best responses show that network externality is heterogeneous across agents depending on the relations of their payoff neighbors. The main advantage of our paper’s approach is two-fold. First, the empirical modeling according to our approach accommodates a wide range of sampling processes. Such a feature is crucial because the econometrician rarely has precise knowledge about the actual sampling process through which data are generated. 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