Clarification on residuals_ and statistical significance in VARLiNGAM

[aaautumnwind]

Hello,
Thank you for this wonderful library!

I’m studying the VARLiNGAM implementation and have two questions about its practical application:

1. Interpretation of residuals_
In the tutorial, you show that after calling fit(), one can obtain model.residuals_ and then apply DirectLiNGAM to recover the instantaneous connections matrix B0. My understanding is that residuals_ contains the residuals of the VAR part only (i.e., after removing the lagged influences), before the LiNGAM step. Is that correct?

2. Statistical significance of causal effects
The original 2010 paper "Estimation of a Structural Vector Autoregression Model Using Non-Gaussianity" discusses testing the significance of causal relations. In the current library, the bootstrap functionality provides probabilities for each causal edge (e.g., via get_probabilities()).

Is this bootstrap-based probability the recommended way to assess significance, or are there plans to implement classical p-values (like those from the likelihood ratio tests mentioned in the paper)?

If I want to obtain p-values for the null hypothesis that a specific coefficient is zero (or several coefficients are zero), how can I derive them? Is there any way to do this in practice?

Thank you for your time and for maintaining such a useful tool!

[sshimizu2006]

1. Yes, that is correct. residuals_ contains the residuals obtained after fitting the VAR part only.
2. The bootstrap probability in the package is not a well-estimated p-value in a strict statistical sense, but it can be used as a rough indicator of the statistical reliability (or stability) of each causal edge, which we believe is sufficiently useful in practice. If one wishes to obtain a more accurately estimated p-value, methods such as multiscale bootstrap can be used (see Komatsu, Shimizu & Shimodaira, ICANN 2010). However, multiscale bootstrap is computationally much more expensive, so the package currently provides the ordinary bootstrap probabilities instead.

[aaautumnwind]

It's all clear now. Thank you very much!