A Robust Lagrange-Multiplier Test for Spatial Autoregression Versus (Unobserved) Spatially Correlated Shocks

One of the central challenges to inference in the context of potentially interdependent observations, known as Galton's Problem, is the difficulty distinguishing spatially correlated observations due to observed units exposure to spatially correlated shocks from spatial correlation in outcomes due to contagion (spillovers) between units. The applied researcher's first, and to date only, defense against confusing these substantively importantly different processes empirically has been to control as best possible with observable regressors and/or fixed effects for correlated-shocks processes when estimating contagion (spatial-autoregression). While specifying empirical models & measures as precisely and powerfully as possible remains as always optimal practice, these extant strategies cannot guard fully against the possibility of exposure to 'unobserved' exogenous shocks that are distributed spatially in manner not fully common to some set of units (fixed effects) or fully controlled by observable exogenous factors (control variables), but rather distributed across units more similarly to the pattern by which the outcome is contagious. Following the robust Lagrange-multiplier test strategy of Anselin, Bera, Florax, & Yoon (1996), which offered tests of spatial-autoregressive lag or of error against independence, robust to the presence of the other autoregressive process, we derive and evaluate the performance of a robust Lagrange-multiplier test for spatial-autoregression (contagion) against independence, which is robust to the presence of unobserved (uncontrolled/unmodeled) correlated-shocks distributed across units identically to the pattern of contagion (along with the symmetric robust test for spatially correlated shocks robust to autoregressive contagion). The test results are constructive and can be highly informative in offering direct & more-definitive answer than heretofore possible to the question posed by Galton's Problem, common shocks or contagion?