  We study linear peer effects models where peers interact in groups,
individual's outcomes are linear in the group mean outcome and characteristics,
and group effects are random. Our specification is motivated by the moment
conditions imposed in Graham 2008. We show that these moment conditions can be
cast in terms of a linear random group effects model and lead to a class of GMM
estimators that are generally identified as long as there is sufficient
variation in group size. We also show that our class of GMM estimators contains
a Quasi Maximum Likelihood estimator (QMLE) for the random group effects model,
as well as the Wald estimator of Graham 2008 and the within estimator of Lee
2007 as special cases. Our identification results extend insights in Graham
2008 that show how assumptions about random group effects as well as variation
in group size can be used to overcome the reflection problem in identifying
peer effects. Our QMLE and GMM estimators accommodate additional covariates and
are valid in situations with a large but finite number of different group sizes
or types. Because our estimators are general moment based procedures, using
instruments other than binary group indicators in estimation is straight
forward. Our QMLE estimator accommodates group level covariates in the spirit
of Mundlak and Chamberlain and offers an alternative to fixed effects
specifications. Monte-Carlo simulations show that the bias of the QMLE
estimator decreases with the number of groups and the variation in group size,
and increases with group size. We also prove the consistency and asymptotic
normality of the estimator under reasonable assumptions.
