Economics 696, Topics in EconometricsLecture Note 13: Social Interactions (updated May 3, 2010)Based on Manski (1993), Graham and Hahn (2005), and Bramoulle, Djebbari, and Fortin(2009).1 IntroductionAs a concrete example, suppose that we want to determine how a child i’s achievementin school depends on his or her own characteristics, the characteristics of the child’s class-mates, and the performance of other classmates. We could imagine various stories why allof these “effects” might be present.Following Manski (1993), we could start by writing down an equation for performance ofchild i in classroom g as:yi= α + Eg[y]β + Eg[x]0γ + x0iδ + ui,Here, yiis an outcome (e.g. test score), and xiis a measured characteristic of the child(e.g. parent’s income). The notation Egmeans average within classroom g. So Eg[y] is theaverage of outcome y among all children in the child’s class g (including child i), and Eg[x]is the average of the regressor x among the child’s class.To complete the model we need to say something about ui. Let’s assume that E[ui|Xg] = 0,where Xgcontains all the x values for children in classroom g (not just i).This is a “linear-in-means” model for social interactions. The parameter β measures howthe average outcome in i’s classroom affects i’s individual outcome. This effect is calledan endogenous social effect. The γ measures how the average characteristics of children ini’s classroom affects i’s outcome. This is called an exogenous social effect. There may alsobe a correlated effect, in that uiand ujmay be correlated for children i and j in the sameclassroom.Then the identification question is: from knowledge of the joint distribution of {yi, xi}, canwe recover β and γ (and possibly α)? This does not follow immediately from standardOLS arguments, because of the presence of terms like Eg[y] on the right hand side of themain equation of interest.2 Sampling FrameworkThe model in §1 is somewhat heuristic. Let’s try to put it in a formal probability frame-work. Intuitively, we want to have many classes (or groups), drawn from some joint distri-bution, and some number of individuals within each group. This puts the social interactionmodel into the framework of panel data, as pointed out by Graham and Hahn (2005).1Let g denote a given group. Group g has individuals i = 1, . . . , Ng. For each individual iin group g, their outcome ygisatisfies:ygi="1NgNg∑j=1ygj#β +"1NgNg∑j=1xgj#0γ + x0giδ + αg+ egi. (1)We assume that the egiare mutually uncorrelated and have mean zero conditional on Xg:=(xg1, . . . , xgNg) and αg.1Comparing this to the original equation in §1, we have explicitly spelled out Eg[y] andEg[x], and we have rewritten α + ui= αg+ egi. The term αgcaptures both an overall meanα and the part of ugithat is correlated between individuals. In general, we allow αgto becorrelated with Xg.So for each group g, there is a set of variables (not all observed):( αg, Ng, Xg, Yg, Eg), where Yg:= (yg1, . . . , ygNg), Eg:= (eg1, . . . , egNg).In turn, each group g is a draw from the population distribution of groups:( αg, Ng, Xg, Yg, Eg)i.i.d.∼ Fg,where Fgis such that Equation (1) and the conditions on egiare satisfied with probabilityone.We could relax the i.i.d. assumption on groups somewhat, but it’s easy to see now that ran-dom sampling over groups and observability of ygi, xgiwill identify the joint distributionof (Ng, Yg, Xg).3 Model with Endogenous InteractionsLet’s first consider a simple version of Equation (1). Suppose that there are no exogenouseffects. Then, usingygfor the group average of ygi, we haveygi= αg+ ygβ + x0giδ + egi. (2)Assume this equation holds for all i = 1, . . . , Ng, so we can take the average over i to get:yg= αg+ ygβ + x0gδ + eg,with xgand egdefined in analogy with yg. Let’s assume β 6= 1, so we can solve for yg:yg=αg1 − β+ x0gδ(1 − β)+11 − βeg.1We could go further and say that the egiare i.i.d., but we will not need this for the later arguments.2Then substitute the solution for ygback into (2) to get the reduced form equationsygi= αg+ ygβ + x0giδ + egi= αg+αg1 − β+ x0gδ(1 − β)+11 − βegβ + x0giδ + egi=αg+αgβ1 − β+ x0gδβ1 − β+ x0giδ + vgi=αg1 − β+ x0gδβ1 − β+ x0giδ + vgi,wherevgi=β1 − βeg+ egi.Hence (using α∗g= αg/(1 − β)),E[ygi|αg, Xg] = α∗g+ x0gδβ1 − β+ x0giδ.Clearly, if δ = 0 then β is not identified. But even if δ 6= 0, there is an issue becauseα∗gis expected to be correlated with the xgi, and hence xg. For example, suppose we tryto estimate this equation using a fixed effects estimator. The FE estimator sweeps outregressors that do not vary within the group, hence we cannot estimate the compoundcoefficient δβ/( 1 − β) using the FE estimator. A possible solution is to try to find andinstrumental variable for xg. See Graham and Hahn (2005).4 Interactions in Other Network StructuresThe setup above imposes a particular structure to the interactions between individuals:individuals within a group interact with all of each other (through the group means), andnot at all with other individuals outside the group. Bramoulle, Djebbari, and Fortin con-sider more general types of social networks and show that in some cases, one can identifyinteraction effects without additional instrumental variables.As an example, suppose that each group g consists of three individuals (g1, g2, g3), butthat the influence of individuals within a group runs in a particular direction:g1 → g2 → g3 · · ·Thus each group is an intransitive directed network.Then a model (following our previous linear specification) could have, for i > 1,ygi= αg+ βyg,i−1+ δxgi+ γxg,i−1+ egi,with E[egi|αg, Xg] = 0 for all i. Then∆ygi= β∆yg,i−1+ δ∆xgi+ γ∆xg,i−1+ ∆egi,3andE[∆egi|Xg] = 0.Hence the ∆xgi, ∆xg,i−1, ∆xg,i−2, . . . are valid instruments for the model.More generally, we can think of each individual i having a group of other individuals Piwho influence him or her. Then the linear in means model can be writtenyi= α + β"1|Pi|∑j∈Piyj#+ δxi+ γ"1|Pi|∑j∈Pixj#+ ui,where E[ui|X] = 0 and the uiare allowed to be correlated across individuals. Here X refersto all xj.For i = 1, . . . , N and j = 1, . . . , N, let Gij= 1/Piif j ∈ Pi, and 0 otherwise. Then let GNbethe N × N matrix with elements Gij. Then we can write the model in stacked form asYN= αι + βGNYN+ δXN+ γGNXN+