Inference on Social Effects When the Network Is Unknown Using Convex or Linear Programming

The paper considers panel data models with social interactions when the network is unobserved and there are endogenous and exogenous social effects. The system of equations is estimated jointly by a convex program. The method does not require the knowledge of the variances of the errors nor a subgaussian assumption. It is possible to incorporate linear and sign restrictions and to impose that the matrices of endogenous and exogenous social effects have same nonzeros. The confidence sets are obtained by solving convex programs. Because the network is unknown, it is not known which exogenous variable does not have a direct effect and can be a valid excluded instrument for the endogenous variables having a direct effect. Therefore we extend the framework of Gautier and Tsybakov (2011, 2014) which handles unknown exclusion restrictions to systems of simultaneous equations. The confidence sets are robust to identification and can be infinite when there is not enough sparsity/exclusion restrictions, when, for some included endogenous regressor, all instruments are too weak or when the number of time periods is too small. The second half of the paper presents an alternative approach that only relies on linear programming. This is numerically very attractive for the study of large networks.