A revisit to least-squares parameter estimation of ARMAX systems

The standard least-squares (LS) method gives biased parameter estimates when applied to identify ARMAX systems. By exploiting the unique structure of the ARMAX system, it is shown that extra delayed outputs can be used to evaluate the noise-induced bias in the LS parameter estimates. This paves the way for establishing an extended version of bias-eliminated least-squares (BELS) method. The distinctive features of the new BELS method also lie in its use of a weighting matrix so that various forms of the previous BELS types methods fall in the family classified by the new BELS method in terms of the choice of the weighting matrix. The relationship between the new BELS method and the instrumental variable (IV) methods is explored. The application of the IV methods is greatly expanded since the proposed method presents an efficient way to compose the instrumental variables. It is also clarified that the prefiltering based BELS method is substantially different from the new BELS method in terms of construction of the weighting matrix. Numerical results are presented to validate the theoretical analysis and findings.