Causality in Time Series - ClopiNet

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 in orderto make the orientation of edges compatible with as many conditional independencetests as possible. This increases the computational time of the search algorithm, butconsidering 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 slowing down does notcreate a serious concern in this context. Table 1 reports the modified PC algorithm. Incase of acyclic structure without causal sufficiency (i.e. possibly including latent variables),the suggested algorithm is FCI (Spirtes et al. 2000, pp. 144-145). In the caseof no latent variables and in the presence of feedback loops, the suggested algorithmis CCD (Richardson and Spirtes, 1999). There is no algorithm in 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. tracing 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 resulting set of possible causal structures (representedby an undirected graph) contains 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 investigate therobustness of the conclusions under the different possible causal structures. Bootstrapprocedures may also be applied to determine which is the most reliable causal order(see simulations and applications in Demiralp et al., 2008).3. Identification via independent component analysisThe methods considered in the previous section use tests for zero partial correlation onthe VAR-residuals to obtain (partial) information about the contemporaneous structurein an SVAR model with Gaussian shocks. In this section we show how non-Gaussianand independent shocks can be exploited for model identification by using the statisticalmethod of ‘Independent Component Analysis’ (ICA, see Comon (1994); Hyvärinenet al. (2001)). The method is again based on the VAR-residuals u t which can be obtainedas in the Gaussian case by estimating the VAR model using for example ordinaryleast squares or least absolute deviations, and can be tested for non-Gaussianity usingany 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 in section 1.2 (with matrix PD −1 ), the VAR-disturbances u t and114