Distribution of Residual Autocorrelations in the Regression Model with Autoregressive–Moving Average Errors

Summary In examining the adequacy of a statistical model an analysis of the residuals is often carried out; and for models involving a time series structure the serial correlations of the residuals are natural statistics to consider. In this paper it is shown that asymptotically the first s sample autocorrelations of the residuals, calculated from a least-squares fit of a regression model in which the errors follow a stationary autoregressive–moving average time series, possess a multivariate normal distribution. For moderately sized s this is nearly a singular distribution, with the “singularity” concentrated heavily on the autocorrelation coefficients of lowest order or lag. This distribution was first obtained by Box and Pierce (1970) for time series with no deterministic component; that it is independent of the regression structure of the model is shown to follow from general specification results of Durbin (1970), utilizing the large sample distribution of the parameter estimates recently obtained by the author (1971). Application of these results to testing model adequacy is also considered.