Unraveling Time Series Dynamics: Evaluating Partial Autocorrelation Function Distribution and Its Implications

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.