The regression model consisted of exogenous characteristics chosen based on expected fit with the key employment and earnings outcomes.
The availability of such characteristics offers more possible IVs: in particular, the total and average exogenous characteristics of direct and indirect neighbours can be used as instruments.
The model parameters may in principle thus be identified even if there are no node-level exogenous characteristics, X, in the model, as long as β1≠0.
2, we select plausibly exogenous characteristics from the individuals' backgrounds and communities to predict the three E i life-cycle stage outcomes.
Hence, we use data on the average age of new mothers for the period 1972 2001 and exogenous characteristics, from the March CPS, for the period 1977 2001.
Node degree (GL), along with the sum of the exogenous characteristics of the node's direct neighbours (GX), and sum of the average exogenous characteristics of its second-degree neighbours ((boldsymbol {G}tilde {boldsymbol {G}}boldsymbol {X})) can be used as instruments for the total outcome of the node's neighbours (GY).
X is a matrix of observable characteristics of nodes, w x (G, X) interacts network statistics with exogenous characteristics of nodes, and Z and (bar {boldsymbol {X}}) are network-level observable characteristics.
However, if β1=0, the model excluding exogenous characteristics, X, is tautological in this case one is simply regressing individuals' outcomes on the mean of the outcomes (see Angrist 2014, for further discussion).
As outlined above, this model implies that a node's outcome is affected by the average outcome of its network neighbours, its own and network-level exogenous characteristics (the latter may be subsumed into a network fixed effect), and the average characteristics of its network neighbours.
The exogenous characteristics of k, xk,g, directly affect j's outcome, but not (directly) that of i, hence forming valid instruments for the outcome of i's neighbours (i.e. j's outcome) in the equation for node i. Intuitively, this method uses the characteristics of second-degree neighbours who are not direct neighbours as instruments for outcomes of direct neighbours.
