A new discussion paper by Guy Tchuente, KDPE 1607, July 2016

**Non-technical summary**

This paper considers the estimation of social interaction models with network structures and the presence of endogenous, contextual, correlated and group fixed effects. In network models, an agent's behavior may be influenced by peers' choices (the endogenous effect), by peers' exogenous characteristics (the contextual effect), and/or by the common environment of the network (the correlated effect) (see Manski (1993) for a description of these models).

As discussed in Manski (1993) work on reflection problem in network model, identification and estimation of the endogenous interaction effect are of major interest in social interaction models. Following Manski, recent works have shown that identification of the parameters of the model is based on the structure or the size of the group in the network. For example, identification of the network effect can be achieved by using individuals' Bonacich (1987) centrality as an instrumental variable. However, the number of such instruments increases with the number of groups; leading to the many instruments problem. Identification can also be achieved using the friend of a friend exogenous characteristics. Unfortunately, if the network is very dense, the identification is weakened.

The variation in group size is another source of identification of the network effect. However, if the groups are very large the identification power is lowered. This paper uses high-dimensional estimation techniques, also know as regularization methods, to estimate network models. The regularization is proposed as a solution to the weak identification problem in network models.

The proposed regularized two stage least square and generalized method of moments based on three regularization methods help to deal with many moments or/and weak identification problems. We show that these estimators are consistent and asymptotically normal. Moreover, the regularized two stage least square estimators are asymptotically unbiased and achieve the asymptotic efficiency bound. The regularized estimators all involve the use of a regularization parameter. An optimal data-driven selection method for the regularization parameter is derived.

A Monte Carlo experiment shows that the regularized estimator performed well. The regularized two stage least square and generalized method of moments procedure substantially reduce the many instruments bias for both the two stage least square and generalized method of moments estimators, specifically in a large sample. Moreover, the qualities in term of bias and precision of the regularized estimator improves with the increase of the network density and the number of groups. These results show that regularization is a valuable solution to the potential weak identification problem existing in network models estimation.