Residual Correlations and Diagnostic Checking in Dynamic-Disturbance Time Series Models

In examining adequacy of fit of a statistical model the stochastic properties of the residuals are often studied. In particular, for an important class of time series models involving an exogenous or input variable and correlated errors or disturbances, Box and Jenkins [1] recommend an examination of the autocorrelations of the residuals and the cross correlations between the residuals and the independent variable. It is shown in this article that the residual auto- and cross correlations to a close approximation for large samples possess singular normal distributions. Implications of this for diagnosing model inadequacy are considered.