Distribution of Residual Autocorrelations in Autoregressive-Integrated Moving Average Time Series Models

Abstract

JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.. American Statistical Association is collaborating with JSTOR to digitize, preserve and extend access to Journal of the American Statistical Association.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.1509

The effect of experience on Internet auction bidding dynamics

“…Results from the Box-Pierce test (cf. Box and Pierce, 1970) indicate that one lag is sufficient to remove the serial correlation in the error terms in our data.…”

Is Pakistan Stock Market Moving Towards Weak-Form Efficiency? Evidence from the Karachi Stock Exchange and the Random Walk Nature of Free-Float of Shares of KSE 30 Index.

“…Box-Pierce (1970) Q statistic is a portmanteau test that is used to examine the whole set of return series for correlation up to k lags. For instance, return series for 10 lags will examine r 1 to r 10 all at once.…”

Purchasing power parity and uncovered interest parity: another look using stable law econometrics

“…This result is in contrast with the results showed by Cumby and Obstfeld (1984). Cumby and Obstfeld (1981) showed indirect evidence of a time-varying risk premium by finding serial correlation in , by using weekly data on Eurocurrency rates denominated in six major currencies over the period 1974-80 and the Box-Pierce (1970) and log likelihood tests.…”