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 objective of this paper is to assess the distribution of the Partial Autocorrelation Function (PACF), both theoretically and empirically, emphasizing its crucial role in modeling and forecasting time series data. Additionally, it evaluates the deviation of the sum of sample PACF from normality: identifying the lag at which departure occurs. Our investigation reveals that the sum of the sample PACF, and consequently its components, diverges from the expected normal distribution beyond a certain lag. This observation challenges conventional assumptions in time series modeling and forecasting, indicating a necessity for reassessment of existing methodologies. Through our analysis, we illustrate the practical implications of our findings using real-world scenarios, highlighting their significance in unraveling complex data patterns. This study delves into 185 years of monthly Bank of England Rate data, utilizing this extensive dataset to conduct an empirical analysis. Furthermore, our research paves the way for future exploration, offering insights into the complexities and potential revisions in time series analysis, modeling, and forecasting.

Long-memory models are frequently used in finance and other fields to capture long-range dependence in time series data. However, correctly identifying whether a process has long memory is crucial. This paper highlights a significant limitation in using the sample autocorrelation function (ACF) to identify long-memory processes. While the ACF establishes the theoretical definition of a long-memory process, it is not possible to determine long memory by summing the sample ACFs. Hassani’s $\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ theorem demonstrates that the sum of the sample ACF is always $\raisebox{1ex}{$-1$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$ for any stationary time series with any length, rendering any diagnostic or analysis procedures that include this sum open to criticism. The paper presents several cases where discrepancies between the empirical and theoretical use of a long-memory process are evident, based on real and simulated time series. It is critical to be aware of this limitation when developing models and forecasting. Accurately identifying long-memory processes is essential in producing reliable predictions and avoiding incorrect model specification.

Geospatial–statistical integration remains a persistent bottleneck for official statistics and applied spatial analysis. The GISINTEGRATION R package provides a modular, reproducible workflow for preprocessing, harmonizing, and linking heterogeneous GIS and non-GIS datasets, with export utilities that are compatible with common desktop GIS. This paper outlines the package architecture and demonstrates its use in two applications. The first integrates population statistics with newly introduced statistical output geographies for Northern Ireland, enabling rapid preparation of analysis-ready layers such as all usual residents and population density at Super Data Zones. The second links daily PM_2.5 measurements from the U.S. EPA Air Quality System with county boundaries for California (July 2020) to produce policy-relevant indicators; spatial aggregation yielded valid monthly means for 46 of 58 counties (79.31%) and reduced variance from 40.716 (monitor level) to 5.777 (county means), improving signal stability and comparability. Across both cases, the workflow standardizes variable names, supports user-controlled overrides, identifies common keys, and performs quality checks, thereby reducing manual effort while increasing transparency and reproducibility. The results illustrate how standardized integration facilitates official statistical production, environmental monitoring, and evidence-based decision-making.