This　paper　is　concerned　with　the　specification　test　of　the　nonparametric　component　in　partially　nonlinear　models　with　response　missing　at　random.　Two　quadratic　conditional　moment　test　statistics　with　their　asymptotic　properties　are　developed.　Our　test　methods　possess　the　virtue　that　p-values　can　be　easily　determined　and　Type　I　error　can　be　asymptotically　exact　controlled.　Further,　the　tests　can　also　detect　the　alternatives　distinct　from　the　null　hypothesis　at　the　optimal　nonparametric　rate　for　local　smoothing-based　methods.　Thorough　Monte　Carlo　simulations　and　an　application　to　one　real　data　set　are　provided　to　demonstrate　excellent　finite-sample　performance　of　the　proposed　methods.　(C)　2019　Elsevier　B.V.　All　rights　reserved.