On covariance function tests used in system identification. (English) Zbl 0714.93056

Summary: Tests on the autocovariance function of the residuals of an estimated model, or on the cross-covariance function between the model residuals and the past values of the input signal are frequently used in system identification for model validation purposes. Usually, the threshold of such a test is derived under the assumption that the model and the identified system coincide.
The present paper shows that this simplifying assumption leads to a risk of overfitting less than expected and, accordingly, to a greater risk of underfitting. The correct test threshold is problem dependent and cannot be determined easily in the general case. However, it can be shown to lie in a certain interval; and to be equal to one of the bounds defining this interval under certain (sufficient) conditions. The implication of these results for model validation based on covariance tests is discussed and illustrated by means of some numerical examples.


93E12 Identification in stochastic control theory
62M10 Time series, auto-correlation, regression, etc. in statistics (GARCH)
Full Text: DOI


[1] ()
[2] Bohlin, T., On the problem of ambiguities in maximum likelihood identification, Automatica, 7, 199-210, (1971) · Zbl 0216.27904
[3] Box, G.E.P.; Jenkins, G.W., ()
[4] Box, G.E.P.; Pierce, D.A., Distribution of residual autocorrelation in autoregressive-integrated moving average time series model, J. am. statist. ass., 65, 1509-1526, (1970) · Zbl 0224.62041
[5] Davies, N.; Newbold, P., Some power studies of a portmanteau test of time series model specification, Biometrika, 66, 153-155, (1979)
[6] Davies, N.; Triggs, C.M.; Newbold, P., Significance level of the box-pierce portmanteau statistic in finite samples, Biometrika, 64, 517-522, (1977) · Zbl 0391.62066
[7] Hosking, J.R.M., The multivariate portmanteau statistic, J. amer. statist. ass., 75, 602-607, (1980) · Zbl 0444.62104
[8] Jensen, D.R.; Solomon, H., A Gaussian approximation to the distribution of a definite quadratic form, J. am. statist. ass., 67, 898-902, (1972) · Zbl 0254.62013
[9] Ljung, G.M., Diagnostic testing of univariate time series models, Biometrika, 73, 725-730, (1986) · Zbl 0656.62098
[10] Ljung, G.M.; Box, G.E.P., On a measure of lack of fit in time series models, Biometrika, 65, 297-303, (1978) · Zbl 0386.62079
[11] Ljung, L., Some limit results for functionals of stochastic processes, ()
[12] Ljung, L., ()
[13] Ljung, L., ()
[14] McLeod, A.I., On the distribution of residual autocorrelations in box-Jenkins models, J. R. statist. soc., 40, 296-302, (1978), Ser B · Zbl 0407.62065
[15] Moustakides, G.; Benveniste, A., Detecting changes in the AR parameters of a nonstationary ARMA process, IEEE trans. inf. theory, IT-30, 137-155, (1986) · Zbl 0598.62105
[16] Söderström, T.; Stoica, P., ()
[17] Wieslander, J., ()
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.