GS2SLS / SARAR Panels

For panels carrying a spatial lag in y and spatial dependence in the disturbance, with regressors that are themselves endogenous. Maximum likelihood is not available here, so estimation is by generalized spatial two-stage least squares: instruments for the endogenous regressors, spatial instruments for Wy, and a GMM step for the error parameter.

Specification

\[y_{it} = \rho W y_{it} + X_{it} \beta + u_{it}, \qquad u_{it} = \lambda W u_{it} + \varepsilon_{it}\]

Two parameters, and they mean different things: rho is the spatial lag in the dependent variable, so it induces spillovers in y; lambda is dependence among the disturbances, which affects efficiency and standard errors but carries no spillover interpretation.

Estimation

m = fit_gs2sls_panel(TwoWayFE(), W, X_exog, X_endog, Z_iv, y, N, T;
    exog_names = ["x1", "x2"],
    endog_names = ["tax_rate"])
# m.rho:    spatial lag parameter
# m.lambda: spatial error parameter

Z_iv holds the external instruments for the columns of X_endog, and must have at least as many columns as X_endog for the model to be identified.

Formula DSL

endo(x, z1 + z2) marks x as endogenous and names its instruments, so the exogenous block, the endogenous block and the instrument block do not have to be assembled by hand:

m = fit_gs2sls_panel(
    @formula(y ~ x1 + endo(tax, z1 + z2) + fe(region) + fe(year)),
    table, W; panel = :region, time = :year
)

See Formula Interface for the spatial terms, and Spatial Error Models for the case where there is no lag in y.