Uncovering Hidden Insights with Long-Memory Process Detection: An In-Depth Overview

Abstract

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 −12 theorem demonstrates that the sum of the sample ACF is always −12 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.

Keywords

• long-memory process
• sum of sample autocorrelation function
• − 1 2 theorem%22%2C%22default_operator%22%3A%22AND%22%7D%7D%7D"> Hassani’s <named-content content-type="inline-formula"><inline-formula> <mml:math id="mm300"> <mml:semantics> <mml:mfrac bevelled="true"> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> <mml:mn>2</mml:mn> </mml:mfrac> </mml:semantics> </mml:math> </inline-formula></named-content> theorem
• spectral density
• time series
• autocorrelation