## 4 Granger Causality  
Consider two discrete time random variables \(X\) and \(Y\) with autoregressive representation in the form:  
\[\begin{split} X_{t}=\sum\
olimits_{j=1}^{\infty}a_{j}X_{t-j}+\sum \
olimits_{j=1}^{\infty}b_{j}Y_{t-j}+\varepsilon_{t}\\ Y_{t}=\sum\
olimits_{j=1}^{\infty}c_{j}X_{t-j}+\sum\
olimits_{j=1 }^{\infty}d_{j}Y_{t-j}+\xi_{t}\end{split} \tag{4}\]  
where \(\varepsilon\) and \(\xi\) are white noise prediction errors with covariance matrix:  
\[\mathbf{\Sigma}=\begin{pmatrix}\sigma_{\varepsilon}^{2}&\text{cov}( \varepsilon,\xi)\\ \text{cov}(\varepsilon,\xi)&\sigma_{\xi}^{2}\end{pmatrix}=\begin{pmatrix} \Sigma_{xx}&\Sigma_{xy}\\ \Sigma_{yx}&\Sigma_{yy}\end{pmatrix} \tag{5}\]  
\(a\), \(b\), \(c\) and \(d\) are coefficients describing the linear interactions between the variables, with subscript \(j\) indicating time lags. When the above equation is compared with a univariate model \(X_{t}=\sum\
olimits_{j=1}^{\infty}a_{j}X_{t-j}+\eta_{t}\), and when the multivariate model outperforms the univariate case, (e.g., \(\sigma_{\varepsilon}^{2}<\sigma_{\eta}^{2}\)), \(Y\) is said to have a causal effect on \(X\) (and similarly for the effect of \(X\) on \(Y\)). This is the statistical interpretation of causality proposed by [PERSON] (1966) and is commonly referred to as Granger or G-causality.  
G-causality is a measure of coupling with explicit time directionality. For this reason, it is based on prediction errors rather than on linear interactions among coefficients. Traditionally, it is expressed as the logarithmic ratio between the residual variance of the bivariate and the null model (_i.e.,_ the univariate case).  
Let's consider a multivariate system of \(k+2\) stochastic variables (\(X\), \(Y\), \(Z_{l}\),..., \(Z_{k}\)) which admits spectral representation and factorization in the form ([PERSON] and [PERSON], 1995):  
\[\begin{split}\mathbf{S}(X,Y,Z_{3},...,Z_{k},\omega)& =\mathbf{H}(\omega)\mathbf{\Sigma}\mathbf{H}^{*}(\omega)\\ \mathbf{U}(X,Z_{3},...,Z_{k},\omega)&=\mathbf{G}( \omega)\mathbf{\Gamma}\mathbf{G}^{*}(\omega)\end{split} \tag{6}\]  
Figure 3: Time series of the four variables measured at control (AmeriFlux in blue) and treatment (FASET in green) sites.  
\(\mathbf{S}\) and \(\mathbf{U}\) are spectral matrices respectively, of the complete system and of a system which does not include the variable \(Y\), whose causality is tested, * indicates matrix adjoint.  
The conditional G-causality of \(Y\) on \(X\) given \(Z_{l}\), \(Z_{2}\),..., \(Z_{k}\) is computed as [48]:  
\[G(\omega)_{{}_{Y\to\mathrm{i}Z_{3},\ldots,Z_{k}}}=\ln\frac{\Gamma_{xx}}{ \left|\mathrm{Q}_{xx}(\omega)\Sigma_{xx}\mathrm{Q}_{xx}^{*}(\omega)\right|} \tag{7}\]  
where:  
\[\mathbf{Q}=\begin{pmatrix}\tilde{\mathrm{G}}_{\mathrm{XX}}&0&\tilde{\mathrm{ G}}_{\mathrm{XZ_{1}}}&...&\tilde{\mathrm{G}}_{\mathrm{1k}}\\ 0&1&0&...&0\\ \tilde{\mathrm{G}}_{\mathrm{3}1}&0&\tilde{\mathrm{G}}_{\mathrm{3}3}&...& \tilde{\mathrm{G}}_{\mathrm{3k}}\\...&...&...&...&...\\ \tilde{\mathrm{G}}_{\mathrm{k}1}&0&\tilde{\mathrm{G}}_{\mathrm{k}3}&...& \tilde{\mathrm{G}}_{\mathrm{kk}}\end{pmatrix}^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!spectral matrices were calculated as an ensemble. A significant interaction is found when both bivariate and conditional G-causalities are greater than the 99 th percentile of the results from the randomly permutated blocks.