Nonlinear Granger Causality Test
Also known as: nonlinear causality test, BDS-based causality, Diks-Panchenko test, nonparametric Granger causality
Nonlinear Granger causality extends the classic linear Granger causality framework to detect predictive relationships that operate through nonlinear dynamics. Using nonparametric or semi-parametric statistics based on correlation integrals or kernel density estimation, it identifies whether past values of one variable improve forecasts of another beyond what any linear model can capture.
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When to use it
Use nonlinear Granger causality when linear Granger tests show no significant relationship but theory or exploratory analysis suggests a causal link may exist through nonlinear channels — common in financial markets, energy markets, and macroeconomic regime-switching settings. It is also appropriate when residuals from a linear causality test display nonlinear serial dependence (detected by a BDS test). Do not use it as a first-line test when a well-specified linear VAR already captures the dynamics adequately; the added complexity and lower power in small samples make it unnecessary. Avoid when the time series are very short (fewer than ~100 observations), as nonparametric statistics require sufficient data for reliable density estimation.
Strengths & limitations
- Detects causal relationships that are invisible to linear Granger causality, including threshold, regime-switching, and volatility-driven channels.
- Model-free: does not impose a specific functional form on the causal mechanism.
- The Diks-Panchenko correction avoids the over-rejection bias present in the earlier Hiemstra-Jones statistic.
- Applicable to stationary residuals after pre-whitening, making it compatible with standard econometric workflows.
- Identifies asymmetric causality: X may cause Y nonlinearly without Y causing X.
- Requires relatively large samples for nonparametric density estimation to be reliable; power degrades sharply with fewer than 100 observations.
- Results are sensitive to the choice of bandwidth, lag length, and embedding dimension; no universally accepted selection rule exists.
- Pre-whitening by a misspecified linear model may fail to remove all linear dynamics, contaminating the nonlinear test.
- Does not identify the structural form or direction of the nonlinear relationship — only its existence.
- Computationally intensive, especially when bootstrap critical values are used.
Frequently asked
How does nonlinear Granger causality differ from the standard Granger causality test?
The standard test fits linear VAR models and uses F-tests. Nonlinear Granger causality uses nonparametric statistics that can detect any form of predictive dependence, not just linear. It is typically applied to residuals from a linear VAR to isolate the genuinely nonlinear component.
Why is the Diks-Panchenko statistic preferred over Hiemstra-Jones?
Hiemstra and Jones (1994) showed that the statistic over-rejects the null of no causality even when none exists, leading to spurious findings. Diks and Panchenko (2006) identified the source of bias and derived a corrected statistic with correct asymptotic size.
How do I choose the bandwidth and embedding dimension?
Common practice follows Diks and Panchenko (2006): set the embedding dimension m to 1 or 2 and choose the bandwidth as a fraction of the sample standard deviation (typically 0.5 to 2 times T^{-2/7}). Sensitivity analysis across a range of bandwidths is strongly recommended.
Does rejection mean I have found a nonlinear causal model?
No. Rejection indicates that some form of nonlinear predictive dependence exists from X to Y, but it does not specify the functional form. Follow-up modelling with threshold VARs, Markov-switching models, or smooth transition models is needed to characterise the mechanism.
What sample size is needed for reliable results?
As a practical guideline, at least 100 observations are recommended for nonparametric statistics; 200 or more are preferable. In very small samples the kernel density estimates are unreliable and the test has low power.
Sources
- Diks, C., & Panchenko, V. (2006). A new statistic and practical guidelines for nonparametric Granger causality testing. Journal of Economic Dynamics and Control, 30(9-10), 1647-1669. DOI: 10.1016/j.jedc.2005.08.008 ↗
- Hiemstra, C., & Jones, J. D. (1994). Testing for linear and nonlinear Granger causality in the stock price-volume relation. Journal of Finance, 49(5), 1639-1664. DOI: 10.1111/j.1540-6261.1994.tb04776.x ↗
How to cite this page
ScholarGate. (2026, June 3). Nonlinear Granger Causality Test. ScholarGate. https://scholargate.app/en/econometrics/nonlinear-granger-causality
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