Separating Common-Factor from Covariate-Specific Spillovers: An Exact SLX Score-Test Decomposition

This paper studies the score test for spatially lagged regressors (WX) in SLX modelswhen the data may also exhibit spatial dependence in the lagged dependent variable or theerror term. Building on the exact score identity of Koley (2024), Sρ = Sλ + ˆ β′ncSθ, we show that the unadjusted SLX score test admits an exact, sample-by-sample decompositionRSθ = PSθ + OSθ. PSθ equals numerically, in every sample, the classical robust lag testRS^*_ρ of Anselin et al. (1996); the orthogonal remainder OSθ, on k − 2 degrees of freedom,is the only component orthogonal to the common-factor direction of Burridge (1981). Itis asymptotically χ2 k−2 without normality and, under normality, has an exact finite-samplenull law, a rescaled Beta with an equivalent F statistic at every n. OSθ tests whetherthe lagged regressors enter only through the aggregated column WXncβnc; non-rejectionbounds, rather than rules out, spillovers along that direction. At k = 2 the remainderis undefined and the battery reduces to the classical tests. Across five cross-sections, thedecomposition separates common-factor effects in Ertur–Koch, Columbus, and Georgia fromcovariate-specific spillovers in St. Louis and Baltimore.