AbstractExtended least squares (ELS) algorithms are proposed for ARMAX model identification with the objective of avoiding the positive real condition associated with standard equation error and output error algorithms. This is achieved by an overparametrization at the cost of additional richness requirements on excitation signals, but without introducing ill‐conditioning or infinite‐dimensional calculations as in earlier methods. Results for the case ofD‐step‐ahead prediction ELS algorithms for ARMAX models are also explored in the paper. Some simulation studies are included to assess the relative performance characteristics of the proposed algorithms, and the nature of the relaxed positive real condition for different degrees of overparametrizationNis investigated in detail for second‐order noi