Causality in Time Series - ClopiNet

Moneta Chlass Entner Hoyerto the ‘right’ rotation of the VAR model, that is the rotation compatible both with thecontemporaneous causal structure of the variable and the structure of the innovationterm. Let us consider a matrix B 0 = I − Γ 0 . If the system is normalized such that thematrix Γ 0 has all the elements of the principal diagonal equal to one (which can be donestraightforwardly), the diagonal elements of B 0 will be equal to zero. We can write:Y t = B 0 Y t + Γ 1 Y t−1 + ... + Γ p Y t−p + ε t (6)from which we see that B 0 (and thus Γ 0 ) determines in which form the values of a variableY i,t will be dependent on the contemporaneous value of another variable Y j,t . The‘right’ rotation will also be the one which makes ε t a vector of authentic innovationterms, which are expected to be independent (not only over time, but also contemporaneously)sources or shocks.In the literature, different methods have been proposed to identify the SVAR model(4) on the basis of the estimation of the VAR model (5). Notice that there are moreunobserved parameters in (4), whose number amounts to k 2 (p + 1), than parametersthat can be estimated from (5), which are k 2 p + k(k + 1)/2, so one has to impose atleast k(k − 1)/2 restrictions on the system. One solution to this problem is to get arotation of (5) such that the covariance matrix of the SVAR residuals Σ ε is diagonal,using the Cholesky factorization of the estimated residuals Σ u . That is, let P be thelower-triangular Cholesky factorization of Σ u (i.e. Σ u = PP ′ ), let D be a k × k diagonalmatrix with the same diagonal as P, and let Γ 0 = DP −1 . By pre-multiplying (5) byΓ 0 , it turns out that Σ ε = E[Γ 0 u t u ′ t Γ′ 0 ] = DD′ , which is diagonal. A problem withthis method is that P changes if the ordering of the variables (Y 1t ,...,Y kt ) ′ in Y t and,consequently, the order of residuals in Σ u , changes. Since researchers who estimate aSVAR are often exclusively interested on tracking down the effect of a structural shockε it on the variables Y 1,t ,...,Y k,t over time (impulse response functions), Sims (1981)suggested investigating to what extent the impulse response functions remain robustunder changes of the order of variables.Popular alternatives to the Cholesky identification scheme are based either on theuse of a priori, theory-based, restrictions or on the use of long-run restrictions. Theformer solution consists in imposing economically plausible constraints on the contemporaneousinteractions among variables (Blanchard and Watson, 1986; Bernanke,1986) and has the drawback of ultimately depending on the a priori reliability of economictheory, similarly to the Cowles Commission approach. The second solution isbased on the assumptions that certain economic shocks have long-run effect to othervariables, but do not influence in the long-run the level of other variables (see Shapiroand Watson, 1988; Blanchard and Quah, 1989; King et al., 1991). This approach hasbeen criticized as not being very reliable unless strong a priori restrictions are imposed(see Faust and Leeper, 1997).In the rest of the paper, we first present a method, based on the graphical causalmodel framework, to identify the SVAR (section 2). This method is based on conditionalindependence tests among the estimated residuals of the VAR estimated model.Such tests rely on the assumption that the shocks affecting the model are Gaussian.108