Refined instrumental variable estimation: Maximum likelihood optimization of a unified Box–Jenkins model

For many years, various methods for the identification and estimation of parameters in linear, discrete-time transfer functions have been available and implemented in widely available Toolboxes for Matlab™. This paper considers a unified Refined Instrumental Variable (RIV) approach to the estimation of discrete and continuous-time transfer functions characterized by a unified operator that can be interpreted in terms of backward shift, derivative or delta operators. The estimation is based on the formulation of a pseudo-linear regression relationship involving optimal prefilters that is derived from an appropriately unified Box–Jenkins transfer function model. The paper shows that, contrary to apparently widely held beliefs, the iterative RIV algorithm provides a reliable solution to the maximum likelihood optimization equations for this class of Box–Jenkins transfer function models and so its en bloc or recursive parameter estimates are optimal in maximum likelihood, prediction error minimization and instrumental variable terms.