The relation between Granger causality and directed information theory: a review <s> C. Instantaneous coupling <s> A general definition of causality is introduced and then specialized to become operational. By considering simple examples a number of advantages, and also difficulties, with the definition are discussed. Tests based on the definitions are then considered and the use of post-sample data emphasized, rather than relying on the same data to fit a model and use it to test causality. It is suggested that a bayesian viewpoint should be taken in interpreting the results of these tests. Finally, the results of a study relating advertising and consumption are briefly presented. <s> BIB001 </s> The relation between Granger causality and directed information theory: a review <s> C. Instantaneous coupling <s> We introduce graphical time series models for the analysis of dynamic relationships among variables in multivariate time series. The modelling approach is based on the notion of strong Granger causality and can be applied to time series with non-linear dependences. The models are derived from ordinary time series models by imposing constraints that are encoded by mixed graphs. In these graphs each component series is represented by a single vertex and directed edges indicate possible Granger-causal relationships between variables while undirected edges are used to map the contemporaneous dependence structure. We introduce various notions of Granger-causal Markov properties and discuss the relationships among them and to other Markov properties that can be applied in this context. Examples for graphical time series models include nonlinear autoregressive models and multivariate ARCH models. <s> BIB002 </s> The relation between Granger causality and directed information theory: a review <s> C. Instantaneous coupling <s> 1. Introduction to probabilities, graphs, and causal models 2. A theory of inferred causation 3. Causal diagrams and the identification of causal effects 4. Actions, plans, and direct effects 5. Causality and structural models in the social sciences 6. Simpson's paradox, confounding, and collapsibility 7. Structural and counterfactual models 8. Imperfect experiments: bounds and counterfactuals 9. Probability of causation: interpretation and identification Epilogue: the art and science of cause and effect. <s> BIB003
The definitions given so far concern the influence of the past of one process on the present of another one. This is one reason that justifies the use of the term 'causality', when the definitions are actually based on statistical dependence. For an extensive discussion on the differences between causality and statistical dependence, we refer to BIB003 . There is another influence between the processes that is not taken into account by definitions 1 and 2. This influence is referred to as 'instantaneous causality'  , BIB001 . However, we will use our preferred term of 'instantaneous coupling', specifically to insist that it is not equivalent to a causal link per se, but actually a statistical dependence relationship. The term 'contemporaneous conditional independence' that is used in BIB002 could also be chosen. Instantaneous coupling measures the common information between x A (n+1) and x B (n+1) that is not shared with their past. A definition of instantaneous coupling might then be that x A (n+1) and x B (n+1) are not instantaneously ∀n. This definition makes perfect sense if the observation set is reduced to A and B, a situation we refer to as the bivariate case. However, in general, there is also side information C, and the definition must include this knowledge. However, this presence of side information then leads to two possible definitions of instantaneous coupling. Definition 3. x A and x B are not conditionally instantaneously coupled relative to V if and only if x A (n + 1) The second possibility is the following: Inclusion of all of the information up to time n + 1 in the conditioning variables allows the dependence or independence to be tested between x A (n + 1) and x B (n + 1) conditionally to x C (n + 1). We end up here with the same differences as those between correlation and partial correlation, or dependence and conditional independence for random variables. In graphical modeling, the usual graphs are based on conditional independence between variables  ,  . These conditional independence graphs are preferred to independence graphs because of their geometrical properties ( e.g., d-separation, BIB003 ), which match the Markov properties possibly present in the multivariate distribution they represent. From a physical point of view, conditional independence might be preferable, specifically to eliminate 'false' coupling due to third parties. In this respect, conditional independence is not the panacea, as independent variables can be conditionally dependent. The well-known example is the conditional coupling of independent x and y by their addition. Indeed, even if independent, x and y are conditionally dependent to z = x + y.