Abstract: SUMMARY Autocorrelation coefficients calculated from n observations are known to have variances approxi- mately equal to 1/n, for a series of independent and identically distributed variables. The variances can be higher for a general uncorrelated process. Estimates of the variances are derived, assuming only that the process is uncorrelated with symmetric distributions. Results are presented for 17 financial time series. Most estimates exceed 2.5/n for daily returns from commodities, 1.6/n for currencies and 1.3/n for a share index. Standard tests for zero autocorrelation are therefore un- reliable. Suitably rescaled data have autocorrelation variances close to 1/n. Daily changes in the prices of a stock, currency or commodity are either uncorrelated or very weakly autocorrelated. Tests for zero autocorrelation are needed to help discover how quickly and how accurately prices respond to relevant information. This is an important issue for all users of financial markets. Tests are usually derived from an asymptotic theorem about the distributions of sample autocorrelations, proved by Anderson and Walker (1964). The theorem implies that the variance of a sample autocorrelation is approximately 1 /n for n observations from a finite variance, strict white noise process. A process is strict white noise if its variables are independent and identically distributed (i.i.d.). Zero autocorrelation does not imply a strict white noise process. Indeed the variances of price-changes appear to fluctuate. Consequently, l/n may not be an appropriate sampling variance for autocorrelation coefficients. This paper presents a method for estimating the sampling variance. Results are given for 17 financial time series. The median estimate of the sampling variance is about 2.5/n. Therefore standard tests, based on an assumed variance 1 /n, are most unreliable. It is also shown that certain rescaled data have autocorrelation variances close to 1/n. Thus reliable tests can be performed by using the rescaled data.

Abstract: The inverse autocorrelation function of a weakly stationary stochastic process Xt at lag h, γi h, is shown to equal the negative of the partial correlation between random variables Xt and Xt+h after elimination of the influence of random variables Xk, k≠t5,t+h.

Abstract: : A new definition of generalized autocorrelation function is proposed. It has properties useful for identification of ARMA processes. The advantages of the proposed procedure include simple asymptotic theory and quick recursive computation. These are natural generalizations of corresponding results for autocorrelation and partial autocorrelation functions.

Abstract: Summary This paper deals with the problem of analysing the change over design in the context of a first order autoregressive process for the error terms. The method of maximum likelihood has been adopted for estimating treatment effects. The conditions derived for obtaining a balanced change over design show that a change over design balanced in the absence of autocorrelation is not necessarily balanced in the presence of autocorrelation. Also, it is observed that the autocorrelation co-efficient and the treatment effect when p≠0 can be tested as usual with the likelihood ratio test criterion.

Abstract: In general, procedures for the analysis of interrupted time series are quite sophisticated and powerful. However, procedures for identifying the intervention component of interrupted time-series models remain relatively primitive. In this article we demonstrate how exponential smoothing can play a function in the identification of the intervention component of an interrupted time-series model that is analogous to the function that the sample autocorrelation and partial autocorrelation functions serve in the identification of the noise portion of such a model.

Abstract: In several disciplines time series analysis is of increasing importance. It is used (1) in a number of applications: Optimal forecast, i.e. the estimation of future values of the known current and past values of the series up to the present time. Parameter estimation, i.e. the estimation of system parameters from time series (signals) generated during a measurement procedure. Transfer function estimation. A transfer function typifies the inertial characteristics of a linear system. Information extraction, i.e. the extraction of relevant information from time series containing much more but not relevant information. The separation of signal and noise (noise reduction, filtering, signal estimation) belongs to this category. Optimal control. A time series of (analytical) results can be used for optimum process control.

Abstract: This note shows that the ordinary least squares estimator of a first-order autoregressive model is always more efficient relative to the Cochrane-Orcutt estimator if the autocorrelation process has a finite past than if its past is infinite. This result cast doubt on the usual suggestion that it might be better to delete the initial observation rather than weight it if the autocorrelation process has a finite past.

Abstract: It can be shown that the mean frequency of a real-valued stochastic signal can be expressed as an integral of the normalized autocorrelation function r(τ) weighted by a function equal to 1/τ2. The fast decline of the weighting function implies that the behavior of the autocorrelation function for small values of τ is the most important portion for estimation of the mean frequency of a signal. It is demonstrated in a simulation study that estimates of the mean frequency with mean squared error equal to the error in estimates obtained via a FFT derived mean frequency estimate can be obtained by using just a few lags of the normalized autocorrelation function with a computational effort substantially less than that required for estimation via FFT. Upper bounds, that can be used as guidelines when implementing the estimator, are given for the bias error introduced by using just a few lag values of the autocorrelation function.

Abstract: This paper discusses the estimation of serial correlation in fixed effects models for longitudinal data Like time series data, longitudinal data often contain serially correlated error terms, but the autocorrelation estimators commonly used for time series, which are consistent as the length of the time series goes to infinity, are not consistent for a short time series as the size of the cross-section goes to infinity This form of inconsistency is of particular concern because a short time series of a large cross-section is the typical case in longitudinal data This paper extends Nickell's method of correcting for the inconsistency of autocorrelation estimators by generalizing to higher than first-order autocorrelations and to error processes other than first-order autoregressions The paper also presents statistical tables that facilitate the identification and estimation of autocorrelation processes in both the generalized Nickell method and an alternative method due to MaCurdy Finally, the paper uses Monte Carlo methods to explore the finite-sample properties of both methods

Abstract: We show that the General Partial Autocorrelation Function (GPAC), which has recently been suggested to be used as one of a set of convenient tools for order identification in ARMA models, has unstable behavior when applied to time series of moderate length. Its use in detecting the order of MA components in real series is very limited and can only be recommended as a means to confirm a pure AR fit to the data.