Trivariate Granger causality analysis seeks to distinguish between true causality and spurious causality results from the topology of the system. However, this analysis is sensitive both to the choice of test criteria and the presence of noise, and this can lead to incorrect inference of causality: either to infer causality that does not exist (spurious causality), or to fail to infer causality that does exist (unidentified causality). Here we analyse the effects of the choice of test criteria and the presence of noise and give general conditions under which incorrect inference is likely to occur. By studying the test criteria (likelihood ratio, Lagrange multiplier, Rao efficient scoring and Wald), we demonstrate that Rao efficient scoring and Wald tests are statistically indistinguishable and that for small sample sizes they offer a the lowest likelihood of spurious causality, with the likelihood ratio test offering the lowest likelihood of unidentified causality. We also show the sample size at which convergence between these tests occurs. We also give empirical results for intrinsic noise (in a variable) and extrinsic noise (between an variable and a observer), with a varying signal-to-noise ratio for each variable, showing that for intrinsic noise a strong dependence on the signal-to-noise ratio of the last variable exists, and for extrinsic noise no dependence the true topology exists.
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