# UA ECON 696A - Topics in Econometrics (5 pages)

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## Topics in Econometrics

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Economics 696 Topics in Econometrics Lecture Note 13 Social Interactions updated May 3 2010 Based on Manski 1993 Graham and Hahn 2005 and Bramoulle Djebbari and Fortin 2009 1 Introduction As a concrete example suppose that we want to determine how a child i s achievement in school depends on his or her own characteristics the characteristics of the child s classmates and the performance of other classmates We could imagine various stories why all of these effects might be present Following Manski 1993 we could start by writing down an equation for performance of child i in classroom g as yi Eg y Eg x 0 xi0 ui Here yi is an outcome e g test score and xi is a measured characteristic of the child e g parent s income The notation Eg means average within classroom g So Eg y is the average 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 Xg contains all the x values for children in classroom g not just i This is a linear in means model for social interactions The parameter measures how the average outcome in i s classroom affects i s individual outcome This effect is called an endogenous social effect The measures how the average characteristics of children in i s classroom affects i s outcome This is called an exogenous social effect There may also be a correlated effect in that ui and u j may be correlated for children i and j in the same classroom Then the identification question is from knowledge of the joint distribution of yi xi can we recover and and possibly This does not follow immediately from standard OLS arguments because of the presence of terms like Eg y on the right hand side of the main equation of interest 2 Sampling Framework The model in 1 is somewhat heuristic Let s try to put it in a formal probability framework Intuitively we want to have many classes or groups drawn from some joint distribution and some number of individuals within each group This puts the social interaction model into the framework of panel data as pointed out by Graham and Hahn 2005 1 Let g denote a given group Group g has individuals i 1 Ng For each individual i in group g their outcome y gi satisfies y gi 1 Ng Ng ygj j 1 1 Ng Ng xgj 0 x 0gi g egi 1 j 1 We assume that the egi are mutually uncorrelated and have mean zero conditional on Xg x g1 x gNg and g 1 Comparing this to the original equation in 1 we have explicitly spelled out Eg y and Eg x and we have rewritten ui g egi The term g captures both an overall mean and the part of u gi that is correlated between individuals In general we allow g to be correlated with Xg So for each group g there is a set of variables not all observed g Ng Xg Yg Eg where Yg y g1 y gNg Eg eg1 egNg In turn each group g is a draw from the population distribution of groups i i d g Ng Xg Yg Eg Fg where Fg is such that Equation 1 and the conditions on egi are satisfied with probability one We could relax the i i d assumption on groups somewhat but it s easy to see now that random sampling over groups and observability of y gi x gi will identify the joint distribution of Ng Yg Xg 3 Model with Endogenous Interactions Let s first consider a simple version of Equation 1 Suppose that there are no exogenous effects Then using y g for the group average of y gi we have y gi g y g x 0gi egi 2 Assume this equation holds for all i 1 Ng so we can take the average over i to get y g g y g x 0g e g with x g and e g defined in analogy with y g Let s assume 6 1 so we can solve for y g yg 1 We g 1 x 0g eg 1 1 1 could go further and say that the egi are i i d but we will not need this for the later arguments 2 Then substitute the solution for y g back into 2 to get the reduced form equations y gi g y g x 0gi egi g 1 0 e g x 0gi egi xg g 1 1 1 g g x 0gi v gi x 0g 1 1 g x 0g x 0gi v gi 1 1 where v gi e g egi 1 Hence using g g 1 E y gi g Xg g x 0g x 0gi 1 Clearly if 0 then is not identified But even if 6 0 there is an issue because g is expected to be correlated with the x gi and hence x g For example suppose we try to estimate this equation using a fixed effects estimator The FE estimator sweeps out regressors that do not vary within the group hence we cannot estimate the compound coefficient 1 using the FE estimator A possible solution is to try to find and instrumental variable for x g See Graham and Hahn 2005 4 Interactions in Other Network Structures The 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 and not at all with other individuals outside the group Bramoulle Djebbari and Fortin consider more general types of social networks and show that in some cases one can identify interaction effects without additional instrumental variables As an example suppose that each group g consists of three individuals g1 g2 g3 but that 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 y gi g y g i 1 x gi x g i 1 egi with E egi g Xg 0 for all i Then y gi y g i 1 x gi x g i 1 egi 3 and E egi Xg 0 Hence the x gi x g i 1 x g i 2 are valid instruments for the model More generally we can think of each individual i having a group of other individuals Pi who influence him or her Then the linear in means model can be written 1 1 y j xi x j ui yi Pi j Pi j P P i i where E ui X 0 and the ui are allowed to be correlated across individuals Here X refers to all x j For i 1 N and j 1 N let Gij 1 Pi if j Pi and 0 otherwise …

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