We　propose　a　new　test　for　the　specification　of　linear　spatial　autoregressive　models　where　the　spatial　weights　matrix　is　pre　specified.　Our　test　is　built　on　the　difference　of　two　estimates　of　the　spatial　parameter　where　the　two　estimates　are　obtained　by　the　parametric　and　nonparametric　GMM　estimation　methods,　respectively.　Under　mild　assumptions,　we　derive　the　limiting　null　distribution　and　show　consistency　for　our　test.　Unlike　the　general　nonparametric　test,　our　test　can　detect　the　local　alternatives　that　approach　the　null　at　a　rate　n-1/2,　where　n　is　the　sample　size.　Monte　Carlo　simulations　are　conducted　to　study　the　finite　sample　performance　of　our　test.　Finally,　we　apply　our　test　to　check　the　model　specification　for　the　economic　growth　rate　example.(c)　2023　Elsevier　B.V.　All　rights　reserved.