Validity of the standard cross-correlation test for model structure validation

The standard prediction error framework provides many theoretical results under the assumption that the true system is in the model class. An important example is the expression for the parameter covariance matrix which is used to derive model uncertainty regions. An essential step in a system identification procedure is the (in)validation of this assumption that the model structure is rich enough to contain the true system. The standard test for this purpose is the sample cross-correlation test between the output residuals and the input. It turns out that this standard test itself is valid only under exactly those assumptions it is meant to verify. As a result considerable undermodelling errors can remain undetected. Besides suggesting caution to users of the standard test, methods are presented to adapt the test adequately.