To improve the identification efficiency of the auxiliary model based gradient iterative algorithm, this article derives a couple of interval error correction auxiliary model based gradient iterative (IEC-AM-GI) algorithms for multirate AutoRegressive eXogenous (ARX) models. The unmeasurable outputs between two measured outputs are estimated by two interval error correction auxiliary models. Unlike the AM-GI algorithm, the IEC-AM-GI algorithms can estimate the missing outputs by the auxiliary model and further fine tune through the measurable outputs, which thus are more efficient and accurate than those by only using the auxiliary model. In addition, this article makes a comparison between these two interval error correction auxiliary model methods and through which the optimal interval error correction auxiliary model method for the unmeasurable outputs can be chosen. The effectiveness of the proposed algorithms is demonstrated by the simulation results.