Designing Bivariate Auto-Regressive Timeseries with Controlled Granger Causality

Area of Science:

• Time Series Analysis
• Statistical Modeling
• Information Theory

Background:

• Understanding statistical inter-relationships in time-series data is crucial for accurate modeling.
• Vector Auto-Regressive (VAR) models are widely used for analyzing multivariate time-series.
• Controlling specific statistical properties like covariance and Granger causality independently presents challenges.

Purpose of the Study:

• To establish design principles for generating bivariate time-series with controlled statistical inter-relationships.
• To analyze the interplay between covariance and Granger causality in VAR models.
• To identify constraints on the independent controllability of these statistical measures.

Main Methods:

• Analysis of a bivariate vector auto-regressive (VAR) model.
• Development of methods to generate bivariate time-series with specified covariance.
• Generation of time-series with specified Granger causality (transfer entropy).

Main Results:

• Demonstration of how to generate bivariate time-series with given covariance and Granger causality.
• Characterization of the trade-off relationship between covariance and Granger causality.
• Identification of constraints on the feasible ranges of covariance and Granger causality for VAR model stability and properness.

Conclusions:

• Covariance and Granger causality are not independently controllable in bivariate VAR models.
• A tri-lemma exists between VAR model stability, covariance control, and Granger causality control.
• The findings provide essential design principles for constructing time-series with desired statistical properties.