## Granger-causality graphs for multivariate time series

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@MISC{Eichler_granger-causalitygraphs,

author = {Michael Eichler and Universität Heidelberg},

title = {Granger-causality graphs for multivariate time series},

year = {}

}

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### Abstract

In this paper, we discuss the properties of mixed graphs which visualize causal relationships between the components of multivariate time series. In these Granger-causality graphs, the vertices, representing the components of the time series, are connected by arrows according to the Granger-causality relations between the variables whereas lines correspond to contemporaneous conditional association. We show that the concept of Granger-causality graphs provides a framework for the derivation of general noncausality relations relative to reduced information sets by performing sequences of simple operations on the graphs. We briefly discuss the implications for the identification of causal relationships. Finally we provide an extension of the linear concept to strong Granger-causality. JEL classification: C320 Keywords: Granger-causality, graphical models, spurious causality, multivariate

### Citations

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Citation Context ...every path a −− · · · −− b from an element a in A to an element b in B intersects S. We note that the separation in undirected graphs formally satisfies the properties listed in Proposition A.1 (e.g. =-=Lauritzen, 1996-=-). 3.2 Moralization in causality graphs The essential feature of the graphical modelling approach is to relate the conditional association structure of a multivariate random variable to a graph. One p... |

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Citation Context ...gonal if and only if the corresponding error components εa(t) and εb(t) are conditionally orthogonal given all remaining components εV \{a,b}(t). It then follows from the inverse variance lemma (e.g. =-=Whittaker, 1990-=-, Prop. 5.7.3) that contemporaneous conditional orthogonality between the components of X is given by zeros in the inverse covariance matrix K = Σ −1 . More precisely, we have Xa ≁ Xb [IX] ⇔ εa(t) ⊥ ε... |

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Citation Context ...precisely, we have Xa ≁ Xb [IX] ⇔ εa(t) ⊥ εb(t) | εV \{a,b}(t) ⇔ kab = kba = 0. (2.2) In the case of normally distributed innovations ε(t) these conditions correspond to a covariance selection model (=-=Dempster, 1972-=-) for the innovations. As an example, we consider a five-dimensional VAR(1)-process with parameters ⎛ A(1) = ⎜ ⎝ a11 0 a13 0 0 0 a22 0 a24 0 a31 a32 a33 0 0 0 0 a43 a44 a45 0 0 a53 0 a55 ⎞ ⎛ ⎟ , K = ⎜... |

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Citation Context ... obtained from G by simply keeping all directed edges e ∈ E with both endpoints in S. Likewise the reduction of the information set does not lead to additional spurious instantaneous causalities (cf. =-=Granger, 1988-=-) and thus the contemporaneous conditional association structure of Xs can be determined solely from the undirected edges in the full graph G. However two vertices a and b should be joined by an undir... |

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Citation Context ...sson et al. (2001) in the context of chain graphs satisfying the so-called Alternative Markov property, whereas the definition differs from the concept of moralization commonly used for chain graphs (=-=Frydenberg, 1990-=-). Definition 3.2 Let G = (V, E) be a mixed graph. The moral graph G m = (V, E m ) derived from G is defined as the undirected graph obtained by completing all immoralities, flags, and 2-biflags in G ... |

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Citation Context ...directed acyclic graphs which correspond to factorizations of the joint probability distribution have been associated with concepts for the inference of cause-effect relationships (Pearl, 1995, 2000; =-=Lauritzen, 2000-=-). However, these concepts, which formalize the notion of controlled experiments, often rely on an a priori knowledge of the direction of a possible cause. The essential feature of the proposed graphi... |

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Citation Context ...roduced moralization only for causality graphs, but it can be shown that the augmentation chain graphs satisfies the AMP Markov property, to which this concept of moralization is also applicable (cf. =-=Andersson et al., 2001-=-). This heuristic argument is made rigorous in the following proposition. Proposition 3.7 Let G = (V, E) be the causality graph of X relative to IX. Further let S be a subset of V with B ⊆ S and defin... |

47 |
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Citation Context ...6). More recently, directed acyclic graphs which correspond to factorizations of the joint probability distribution have been associated with concepts for the inference of cause-effect relationships (=-=Pearl, 1995-=-, 2000; Lauritzen, 2000). However, these concepts, which formalize the notion of controlled experiments, often rely on an a priori knowledge of the direction of a possible cause. The essential feature... |

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Citation Context ...s 1 and 4. In the partial correlation graph G pc , on the other hand, an edge a −− b is absent if and only if there is a zero at the corresponding position in the inverse spectral matrix f(λ) −1 (cf. =-=Dahlhaus, 2000-=-) which is equivalent to the following parameter constraints (with K = Σ −1 and A(1) = (aij)) ( kab + 5∑ kjkajaakb j,k=1 ) = 0, 5∑ kakakb = 0, k=1 and 5∑ kbkaka = 0. k=1 Obviously these constraints ar... |

46 |
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Citation Context ...itional orthogonality, additional assumptions are needed to guarantee the composition and the intersection property. The latter holds under the assumption of measurable separability of the variables (=-=Florens et al., 1990-=-) which for finite-dimensional random vectors is satisfied if the joint probability has a positive and continuous density. The composition property is only required to establish the equivalence of the... |

42 | Multivariate dependencies - Cox, Wermuth - 1996 |

32 |
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Citation Context ... d×d matrices and the ε(t) are independent and identically distributed innovations with mean zero and nonsingular covariance matrix Σ. Setting IX(t) = X(−∞, t ] it is well known (cf. Tjøstheim, 1981; =-=Hsiao, 1982-=-) that Xa is noncausal for Xb if and only if the corresponding entries Aba(j) vanish in all matrices A(j), i.e. Xa ↛ Xb [IX] ⇔ Aba(j) = 0 ∀j ∈ {1, . . . , p}. (2.1) Further, Xa and Xb are contemporane... |

31 |
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Citation Context ...od is based on a concept of separation of vertices and can be executed by a sequence of simple operations on the graph. In Section 4 the results are used to characterize noncausality at all horizons (=-=Dufour and Renault, 1998-=-). Furthermore, we briefly discuss the problem of identification of causal effects. Section 5 provides a generalization of the introduced concept to strong Granger-causality which allows the investiga... |

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Citation Context ...) b Figure 2.1: Causality patterns: (a) direct causality, (b) direct feedback, (c) indirect causality, (d) spurious causality of type II, and (e) spurious causality of type I. position property (e.g. =-=Boudjellaba et al., 1992-=-, Corollary 2) XA ↛ XB [IY ] ⇔ Xaj ↛ Xbk [IY ] ∀ j = 1, . . . , m ∀ k = 1, . . . , n. For processes X with more than two variables the pairwise causality structure of X can be visualized by linking th... |

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Citation Context ...ly (v) Intersection: If L1 ⊥ L2 | L3 + L4 and L1 ⊥ L3 | L2 + L4 then L1 ⊥ L2 + L3 | L4. Proof. The first four properties can be proved easily using the properties of ordinary orthogonality in H (e.g. =-=Florens and Mouchart, 1985-=-). For the proof of the last statement we first consider the case where L2 is finite-dimensional and L4 = {0}. By the definition of conditional orthogonality we get L1 ⊥ L⊥ 3 L2 + L⊥ 2 L3. Since L2 ∩ ... |

6 | On the Granger condition for non-causality - Hosoya - 1977 |

4 |
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(Show Context)
Citation Context ...), where A(j) are d×d matrices and the ε(t) are independent and identically distributed innovations with mean zero and nonsingular covariance matrix Σ. Setting IX(t) = X(−∞, t ] it is well known (cf. =-=Tjøstheim, 1981-=-; Hsiao, 1982) that Xa is noncausal for Xb if and only if the corresponding entries Aba(j) vanish in all matrices A(j), i.e. Xa ↛ Xb [IX] ⇔ Aba(j) = 0 ∀j ∈ {1, . . . , p}. (2.1) Further, Xa and Xb are... |

3 | Causality (Cambridge - Pearl - 2000 |

1 | Graphical Models in Time Series Analysis, Doctoral thesis - Eichler - 1999 |

1 | Inference and causality in economic time series - unknown authors - 1984 |