Many statistical models, and in particular autoregressive-moving average time series models, can be regarded as means of transforming the data to white noise, that is, to an uncorrelated sequence of errors. If the parameters are known exactly, this random sequence can be computed directly from the observations; when this calculation is made with estimates substituted for the true parameter values, the resulting sequence
is referred to as the "residuals," which can be regarded as estimates of the errors.
If the appropriate model has been chosen, there will be zero autocorrelation in the
errors. In checking adequacy of fit it is therefore logical to study the sample autocorrelation
function of the residuals. For large samples the residuals from a correctly fitted
model resemble very closely the true errors of the process; however, care is needed
in interpreting the serial correlations of the residuals. It is shown here that the
residual autocorrelations are to a close approximation representable as a singular
linear transformation of the autocorrelations of the errors so that they possess
a singular normal distribution. Failing to allow for this results in a tendency to
overlook evidence of lack of fit. Tests of fit and diagnostic checks are devised
which take these facts into account.