VLTimeCausality: Variable-Lag Time Series Causality Inference Framework

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A framework to infer causality on a pair of time series of real numbers based on Variable-lag Granger causality (VL-Granger) and transfer entropy (VL-Transfer Entropy).

Typically, Granger causality and transfer entropy have an assumption of a fixed and constant time delay between the cause and effect. However, for a non-stationary time series, this assumption is not true. For example, considering two time series of velocity of person A and person B where B follows A. At some time, B stops tying his shoes, then running to catch up A. The fixed-lag assumption is not true in this case.

We propose a framework that allows variable-lags between cause and effect in Granger causality and transfer entropy to allow them to deal with variable-lag non-stationary time series.


You can install our package from CRAN


For the newest version on github, please call the following command in R terminal.


This requires a user to install the “remotes” package before installing VLTimeSeriesCausality.

Example: Inferred VL-Granger causality time series

In the first step, we generate time series TS\(X and TS\)Y where TS\(X causes TS\)Y with variable-lags.

# Generate simulation data
TS <- VLTimeCausality::SimpleSimulationVLtimeseries()

We can plot time series using the following function.


A sample of generated time series pair that has a causal relation is plotted below:

We use the following function to infer whether X causes Y.

# Run the function

The result of VL-Granger causality is below:

[1] 0.8882051

[1] TRUE

If out\(XgCsY is true, then it means that X VL-Granger-causes Y. The value out\)BICDiffRatio is a BIC difference ratio. If out\(BICDiffRatio>0, it means that X is a good predictor of Y behaviors. The closer out\)BICDiffRatio to 1, the stronger we can claim that X VL-Granger-causes Y.

For the comparison between normal Granger test with our VL-Granger test, we recommend to use the F-test decision criterion the same as typical Granger test criterion.

Below are the results of VL-Granger Causality using F-test.

ChickEgg <- as.data.frame(ChickEgg)

#============ The the egg causes chicken. 
out_test1 <- VLTimeCausality::VLGrangerFunc(X=ChickEgg$egg,Y=ChickEgg$chicken)
[1] 0.004980847
[1] TRUE 

#============ The reverse direction has no causal relation
out_test2 <- VLTimeCausality::VLGrangerFunc(Y=ChickEgg$egg,X=ChickEgg$chicken)
[1] 1


Chainarong Amornbunchornvej, Elena Zheleva, and Tanya Berger-Wolf (2021). Variable-lag Granger Causality and Transfer Entropy for Time Series Analysis. ACM Transactions on Knowledge Discovery from Data (TKDD), 15(4), 1-30. https://doi.org/10.1145/3441452