Quantification of the variance of estimated transfer functions in the presence of undermodeling

We study the effect of undermodeling on the parameter variance for prediction error time-domain identification in open loop. We consider linear time-invariant discrete time single-input-single-output systems with known noise model. We examine asymptotic expressions for the variance for large number of data. This quantity depends in general on the fourth order statistical properties of the applied input. However, we establish a sufficient condition under which the asymptotic variance depends on the input power spectrum only. For this case, we deliver exact expressions. For a stochastic input the undermodeling contributes to the parameter variance due to the correlation between the prediction errors and its gradients, while for a deterministic input it has no influence. As an additional contribution, we investigate the parameter variance under the assumptions of the stochastic embedding procedure.