Granger Causality: A Review and Recent Advances (2105.02675v2)

Abstract: Introduced more than a half century ago, Granger causality has become a popular tool for analyzing time series data in many application domains, from economics and finance to genomics and neuroscience. Despite this popularity, the validity of this notion for inferring causal relationships among time series has remained the topic of continuous debate. Moreover, while the original definition was general, limitations in computational tools have primarily limited the applications of Granger causality to simple bivariate vector auto-regressive processes or pairwise relationships among a set of variables. Starting with a review of early developments and debates, this paper discusses recent advances that address various shortcomings of the earlier approaches, from models for high-dimensional time series to more recent developments that account for nonlinear and non-Gaussian observations and allow for sub-sampled and mixed frequency time series.

• The paper provides a comprehensive review of Granger causality, detailing its evolution and methodological challenges in time series analysis.
• The paper introduces advanced techniques including network-based VAR models and sparsity regularization to improve causal inference in multivariate settings.
• The paper overcomes traditional issues such as linearity and fixed lag assumptions by incorporating methods for non-stationary, nonlinear, and irregularly sampled data.

Overview of Granger Causality: A Review and Recent Advances

In this paper by Shojaie and Fox, the concept and application of Granger causality, particularly in time series analysis, are meticulously reviewed and expanded upon with recent advances. The historical context provided highlights that while the original Granger causality definition was influential, it was limited by computational constraints and assumptions of linearity and Gaussian distributions. Granger causality is traditionally used to determine if one time series is predictive of another by analyzing whether past values of a series reduce the variance of predictions of another. This methodology, however, has faced scrutiny over its validity in truly inferring causality due to several underlying assumptions which are often unrealistic in complex real-world scenarios.

Historical Developments and Limitations

The paper begins by discussing the fundamental definition and assumptions originally made by Granger. These include assumptions of continuous-valued observations, linear relationships, discrete-time representation at fixed sampling rates, known lags, stationary processes, perfect measurement, and complete information systems. Such assumptions are rarely met, limiting the application of traditional Granger causality. The authors discuss early methods that primarily used bivariate analyses to infer Granger causality, which proved inadequate due to ignoring potential confounders in multivariate settings.

Recent Advances

To address these limitations, the paper highlights several recent advances:

1. Network Granger Causality: Techniques have evolved to accommodate multivariate settings and network-based interpretations. This involves studying a large set of endogenous variables or considering numerous exogenous variables through methodologies like factor-augmented VAR (FAVAR) models. High-dimensional approaches now often use sparsity-inducing regularization techniques, such as group lasso penalties, to estimate VAR models effectively even when the number of variables is large relative to the number of observations.
2. Lag Selection and Non-Stationary Models: Traditional models require predetermined lag lengths and assume stationarity, which are not always practical. Recent methods incorporate strategies for automatic lag determination and cater to non-stationary processes through piece-wise stationary and time-varying VAR models.
3. Generalized Models for Nonlinear and Discrete-Valued Series: Advances have been made in extending Granger causality analysis to nonlinear models and discrete-valued time series. Approaches using multivariate point processes like Hawkes processes, and models for categorical time series, handle non-linear dynamics beyond traditional VAR frameworks.
4. Subsampled and Mixed-Frequency Time Series: The paper discusses how sub-sampling or irregular sampling can lead to biased or spurious conclusions in Granger causality analysis. Recent approaches provide methodologies to work with mixed and subsampled frequency data without missing causal interactions due to mismatched sampling rates.

Practical and Theoretical Implications

These advancements in Granger causality methodology have broadened its applicability across diverse fields such as neuroscience, genomics, economics, and climate science. They enable more accurate modeling of complex systems, accounting for nonlinearity, high-dimensionality, and irregular sampling. This evolution posits Granger causality as a tool for hypothesizing causal interactions rather than proving them, facilitating systems-level understanding.

Future Developments

The paper suggests further exploration into nonparametric approaches that can seamlessly handle various real-world complexities like unmeasured variables and non-stationarity. The authors emphasize the growing potential with emerging data from interventions and controlled experiments, advancing towards more definitive causal inference.

Overall, Shojaie and Fox’s paper offers a comprehensive review and expansion of Granger causality, setting a foundation for future research to tackle its longstanding challenges.