This paper contributes significantly to time series analysis by discussing the empirical properties of white noise and their implications for model selection. This paper illustrates the ways in which the standard assumptions about white noise typically fail in practice, with a special emphasis on striking differences in sample ACF and PACF. Such findings prove particularly important when assessing model adequacy and discerning between residuals of different models, especially ARMA processes. This study addresses issues involving testing procedures, for instance, the Ljung–Box test, to select the correct time series model determined in the review. With the improvement in understanding the features of white noise, this work enhances the accuracy of modeling diagnostics toward real forecasting practice, which gives it applied value in time series analysis and signal processing.

In this article, we study the autocorrelation function (ACF), which is a crucial element in time series analysis. We compare the distribution of the ACF, both from a theoretical and empirical point of view. We focus on white noise processes (WN), i.e., uncorrelated, centered, and identically distributed variables, whose ACFs are supposed to be asymptotically independent and converge towards the same normal distribution. But, the study of the sum of the sample ACF contradicts this property. Thus, our findings reveal a deviation of the sample ACF from normality beyond a specific lag. Note that this phenomenon is observed for white noise of varying lengths, and evenforn the residuals of an ARMA($p,q$) model. This discovery challenges traditional assumptions of normality in time series modeling. Indeed, when modeling a time series, the crucial step is to validate the estimated model by checking that the associated residuals form white noise. In this study, we show that the widely used portmanteau tests are not completely accurate. Box–Pierce appears to be too conservative, whereas Ljung–Box is too liberal. We suggest an alternative method based on the ACF for establishing the reliability of the portmanteau test and the validity of the estimated model. We illustrate our methodology using money stock data in the USA.

The two main puzzles of this study are the onset and then sudden stopping of severe epidemics in western Provence (a highly malaria-endemic region of Mediterranean France) without any deliberate counter-measures and in the absence of significant population flux.

The identification of the orders of time series models plays a crucial role in their accurate specification and forecasting. The Autocorrelation Function (ACF) is commonly used to identify the order q of Moving Average (MA(q)) models, as it theoretically vanishes for lags beyond q. This property is widely used in model selection, assuming the sample ACF follows an asymptotic normal distribution for robustness. However, our examination of the sum of the sample ACF reveals inconsistencies with these theoretical properties, highlighting a deviation from normality in the sample ACF for MA(q) processes. As a natural extension of the ACF, the Extended Autocorrelation Function (EACF) provides additional insights by facilitating the simultaneous identification of both autoregressive and moving average components. Using simulations, we evaluate the performance of q-order identification in MA(q) models, which is based on the properties of ACF. Similarly, for ARMA($p,q$) models, we assess the ($p,q$)-order identification relying on EACF. Our findings indicate that both methods are effective for sufficiently long time series but may incorrectly favor an ARMA($p,q-1$) model when the $a_q$ coefficient approaches zero. Additionally, if the cumulative sums of ACF (SACF) behave consistently and the Ljung–Box test validates the proposed model, it can serve as a strong candidate. The proposed models should then be compared based on their predictive performance. We illustrate our methodology with an application to wind speed data and sea surface temperature anomalies, providing practical insights into the relevance of our findings.