Long Range Dependence

TL;DR

Correlations in a time series persist across long time lags, so past values can influence present (and future) values.
Common analysis tools include autocorrelation and the Hurst exponent (H = 0.5 indicates no correlation; H > 0.5 indicates positive long-range dependence).
Observed in domains such as finance, meteorology, and telecommunications and can affect model performance and predictability.

Definition

Long-range dependence is a phenomenon where the values of a time series are correlated over long periods of time. This means that the behavior of a time series in the present can be influenced by its behavior in the past, and vice versa.

Explanation

Long-range dependence describes persistent correlation structure across long lags in a time series: current values are affected by past values and past behavior can affect future observations. It is commonly studied with statistical measures such as autocorrelation, which quantifies the relationship between a series and its lagged values, and the Hurst exponent, which measures the degree of long-range dependence. A Hurst exponent of 0.5 indicates the time series is uncorrelated, while a Hurst exponent greater than 0.5 indicates positive long-range dependence.

Examples

Stock market

In the stock market, prices of stocks fluctuate and are influenced by factors like changes in the economy, company earnings, and investor sentiment. Even after accounting for these factors, stock prices often exhibit long-range dependence, so the present behavior of a particular stock can be influenced by its past behavior.

Weather patterns

Weather is influenced by factors such as the sun’s radiation, Earth’s rotation, and the atmosphere. Even after accounting for these factors, weather often exhibits long-range dependence. For example, a period of drought can lower river and lake water levels, which in turn can lead to decreased precipitation in the future.

Use cases

Finance: can affect the performance of financial models used for risk management and portfolio optimization.
Meteorology: can affect the predictability of weather patterns.
Telecommunications: can affect the performance of communication networks.

Notes or pitfalls

Long-range dependence can have significant implications across fields because persistent correlations change the statistical properties that many models assume (for example, independence or short-range correlation).
Detecting and measuring long-range dependence requires appropriate statistical tools (e.g., autocorrelation measures, Hurst exponent); mischaracterizing dependence can lead to poor model performance or incorrect conclusions.

Autocorrelation
Hurst exponent
Time series