This paper studies the auxiliary model based stochastic gradient(AM-SG) algorithm,the auxiliary model based multi-innovation stochastic gradient(AM-MISG) algorithm,the auxiliary model based recursive least squares(AM-RLS) algorithm and the auxiliary model based multi-innovation least squares algorithm for output-error systems and output-error autoregressive moving average(OEARMA) systems(namely,Box-Jenkins systems),the modified AM-SG algorithm,the forgetting factor AM-SG algorithm,the interval-varying AM-SG algorithm,the interval-varying AM-MISG algorithm and the interval-varying AM-RLS algorithm for output-error systems,and presents the filtering based auxiliary model generalized stochastic gradient algorithm and the filtering based multi-innovation generalized least squares algorithm for output-error autoregressive systems(namely,OEAR systems).