Relating Granger causality to long-term causal effects.

Area of Science:

• Complex Systems Analysis
• Stochastic Processes
• Time Series Analysis

Background:

• Granger causality quantifies short-term predictive relationships between time series.
• Evaluating long-term coupling effects on system statistics (e.g., variance) is crucial but lacks established links to short-term measures.
• Existing methods do not connect short-term Granger causality with long-term statistical impacts.

Purpose of the Study:

• To establish a rigorous relationship between short-term Granger causality and long-term coupling effects in stochastic systems.
• To quantify how short-term predictive improvements relate to changes in the driven process variance.
• To provide a framework for understanding coupling dynamics across different time scales.

Main Methods:

• Developed analytical derivations for overdamped linear oscillators.
• Quantified short-term effects using prediction improvement (PI) in autoregressive models.
• Derived a proportionality coefficient linking short-term PI to long-term variance changes.

Main Results:

• A direct proportionality was rigorously derived between long-term coupling effects and short-term Granger causality for linear oscillators.
• The proportionality coefficient depends on prediction interval and system relaxation times.
• This coefficient is typically greater than one, implying small PI values can represent substantial long-term impacts.

Conclusions:

• The derived relationship bridges the gap between short-term and long-term causality measures.
• This finding is critical for interpreting Granger causality in systems where long-term effects are paramount.
• The study offers a new perspective for analyzing causal relationships in complex systems, with applications in climate science.