Moneta Chlass Entner HoyerStep 3 Apply a causal search algorithm to recover the causal structure among u 1t ,...,u kt ,which is equivalent to the causal structure among Y 1t ,...,Y kt (cfr. section 1.2 and seeMoneta 2003). In case of acyclic (no feedback loops) and causally sufficient (no latentvariables) structure, the suggested algorithm is the PC algorithm of Spirtes et al. (2000,pp. 84-85). Moneta (2008) suggested few modifications to the PC algorithm <strong>in</strong> orderto make the orientation of edges compatible with as many conditional <strong>in</strong>dependencetests as possible. This <strong>in</strong>creases the computational time of the search algorithm, butconsider<strong>in</strong>g the fact that VAR models deal with a few number of time series variables(rarely more than six to eight; see Bernanke et al. 2005), this slow<strong>in</strong>g down does notcreate a serious concern <strong>in</strong> this context. Table 1 reports the modified PC algorithm. Incase of acyclic structure without causal sufficiency (i.e. possibly <strong>in</strong>clud<strong>in</strong>g latent variables),the suggested algorithm is FCI (Spirtes et al. 2000, pp. 144-145). In the caseof no latent variables and <strong>in</strong> the presence of feedback loops, the suggested algorithmis CCD (Richardson and Spirtes, 1999). There is no algorithm <strong>in</strong> the literature whichis consistent for search when both latent variables and feedback loops may be present.If the goal of the study is only impulse response analysis (i.e. trac<strong>in</strong>g out the effectsof structural shocks ε 1t ,...,ε kt on Y t ,Y t−1 ,...) and neither contemporaneous feedbacksnor latent variables can be excluded a priori, a possible solution is to apply only steps(A) and (B) of the PC algorithm. If the result<strong>in</strong>g set of possible causal structures (representedby an undirected graph) conta<strong>in</strong>s a manageable number of elements, one canstudy the characteristics of the impulse response functions which are robust across allthe possible causal structures, where the presence of both feedbacks and latent variablesis allowed (Moneta, 2004).Step 4 Calculate structural coefficients and impulse response functions. If the outputof Step 3 is a set of causal structures, run sensitivity analysis to <strong>in</strong>vestigate therobustness of the conclusions under the different possible causal structures. Bootstrapprocedures may also be applied to determ<strong>in</strong>e which is the most reliable causal order(see simulations and applications <strong>in</strong> Demiralp et al., 2008).3. Identification via <strong>in</strong>dependent component analysisThe methods considered <strong>in</strong> the previous section use tests for zero partial correlation onthe VAR-residuals to obta<strong>in</strong> (partial) <strong>in</strong>formation about the contemporaneous structure<strong>in</strong> an SVAR model with Gaussian shocks. In this section we show how non-Gaussianand <strong>in</strong>dependent shocks can be exploited for model identification by us<strong>in</strong>g the statisticalmethod of ‘Independent Component Analysis’ (ICA, see Comon (1994); Hyvär<strong>in</strong>enet al. (2001)). The method is aga<strong>in</strong> based on the VAR-residuals u t which can be obta<strong>in</strong>edas <strong>in</strong> the Gaussian case by estimat<strong>in</strong>g the VAR model us<strong>in</strong>g for example ord<strong>in</strong>aryleast squares or least absolute deviations, and can be tested for non-Gaussianity us<strong>in</strong>gany normality test (such as the Shapiro-Wilk or Jarque-Bera test).To motivate, we note that, from equations (3) and (4) (with matrix Γ 0 ) or theCholesky factorization <strong>in</strong> section 1.2 (with matrix PD −1 ), the VAR-disturbances u t and114
Causal Search <strong>in</strong> SVARTable 1: Search algorithm (adapted from the PC Algorithm of Spirtes et al. (2000:84-85); <strong>in</strong> bold character the modifications).Under the assumption of Gaussianity conditional <strong>in</strong>dependence is tested by zero partialcorrelation tests.(A): (connect everyth<strong>in</strong>g):Form the complete undirected graph G on the vertex set u 1t ,...,u kt so that each vertex isconnected to any other vertex by an undirected edge.(B)(cut some edges):n = 0repeat :repeat :select an ordered pair of variables u ht and u it that are adjacent <strong>in</strong> G such thatthe number of variables adjacent to u ht is equal or greater than n + 1. Select aset S of n variables adjacent to u ht such that u ti S . If u ht ⊥ u it |S delete edgeu ht — u it from G;until all ordered pairs of adjacent variables u ht and u it such that the number ofvariables adjacent to u ht is equal or greater than n + 1 and all sets S of n variablesadjacent to u ht such that u it S have been checked to see if u ht ⊥ u it |S ;n = n + 1;until for each ordered pair of adjacent variables u ht , u it , the number of adjacent variablesto u ht is less than n + 1;(C)(build colliders):for each triple of vertices u ht ,u it ,u jt such that the pair u ht ,u it and the pair u it ,u jt are eachadjacent <strong>in</strong> G but the pair u ht ,u jt is not adjacent <strong>in</strong> G, orient u ht — u it — u jt as u ht −→ u it
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