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Optimal instrumental variable estimation and approximate implementations

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2 Author(s)
Stoica, Petre ; Polytechnic Institute of Bucharest, Bucharest, Romania ; Soderstrom, T.

The accuracy properties of instrumental variables (IV) methods are investigated. Extensions such as prefiltering of data and use of additional instruments are included in the analysis. The parameter estimates are shown to be asymptotically Gaussian distributed. An explicit expression is given for the covariance matrix of their distribution. The covariance matrix is then taken as a (multivariable) measure of accuracy. It is shown how it can be optimized by an appropriate selection of instruments and prefilter. The so obtained optimal instrumental variable estimates cannot be used directly since the true system and the statistical properties of the disturbance must be known in order to compute the optimal instruments and prefilters. A multistep procedure consisting of three or four simple steps is then proposed as a way to overcome this difficulty. This procedure includes modeling of the disturbance as an ARMA process using a statistically efficient method such as a prediction error method. The statistical properties of the estimates obtained with the multistep procedure are also analyzed. Those estimates are shown to be asymptotically Gaussian distributed as well. The covariance matrix of the estimation errors is compared to that corresponding to a prediction error method. For some model structures these two approaches give the same asymptotic accuracy. The conclusion is that the multistep procedure, which is quite easy to implement and also has nice uniqueness properties, can be viewed as an interesting alternative to prediction error methods.

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Automatic Control, IEEE Transactions on  (Volume:28 ,  Issue: 7 )