# Exploring the Dynamics of Global Plate Motion Based on the Granger Causality Test
## 2 Methods  
The Granger causality test is mostly used to examine the causal relationships between two or more variables. If the accuracy of prediction of one time series can be improved by the incorporation of another time series, it can be said that the second series has a causal effect on the first one. In other words, according to Granger causality, a time series \(X\) can cause \(Y\) if the hysteresis value of \(X\) can help increase the accuracy of the prediction of \(Y\). The idea of causal effect, which is the Granger causality, was first introduced by [PERSON] [36] and then formulated by [PERSON] with linear regression models [37,38].  
There are three main steps in processing the Granger causality test: estimating a bivariate linear model, determining the optimal lag length, and testing the significance of the lags in the exogenous variables [39]. Vector autoregression (VAR) is the most popular model used in the economic field to capture the interdependencies and causality relationship among multiple time series. VAR has a unique advantage where it treats each variable in the system as potentially endogenous and relates each to its past values and the past values of all other variables included in the model [40].  
\[X_{\rm t}=\alpha+\sum_{i=1}^{m}\beta_{\rm i}X_{\rm t-i}+\sum_{j=1}^{n}\gamma_{ \rm j}Y_{\rm t-j}+u_{\rm t} \tag{1}\]  
\[Y_{\rm t}=\alpha+\sum_{i=1}^{q}b_{\rm i}Y_{\rm t-i}+\sum_{j=1}^{r}c_{\rm j}X_{ \rm t-j}+v_{\rm t} \tag{2}\]  
where \(u_{\rm t}\) and \(v_{\rm t}\) refer to random disturbances that may be correlated for various time series; \(t\) to the time period; \(i\) and \(j\) to the lag; and \(m\), \(n\), \(q\), and \(r\) to the optimum lag length, which are determined by the Akaike information criterion.  
Now, for the two equations there is one-way Granger causality running from \(Y\) to \(X\) if, in the first equation, not all \(\gamma_{\rm j}\) are zero but, in the second equation, all \(c_{\rm i}\) is zero; on the contrary, \(X\) Granger causes \(Y\) if, in the second equation, not all \(c_{\rm i}\) is zero but, in the first equation, all \(\gamma_{\rm j}\) are zero. In addition to these two cases, there is two-way Granger causality between \(X\) and \(Y\) if neither all \(\gamma_{\rm j}\) nor all \(c_{\rm i}\) is zero, and there is no Granger causality between \(X\) and \(Y\) if all \(\gamma_{\rm j}\) and \(c_{\rm i}\) is zero [41, 42, 43].  
However, spurious and misleading results may be obtained if the variables are non-stationary [44,45]. Therefore, before performing the Granger causality test, a unit root test, which is used to test time-series stability, should be performed first using augmented Dickey-Fuller (ADF) and/or nonparametric \(Z({\rm t}_{\alpha})\) statistics [46]. If the test is not passed, depending on the time series, an initial or some higher difference of the time series needs to be made. The next step is to test the possibility of cointegration among the variables, which is used to rule out the possibility of a \"spurious\" correlation with the help of the Engle-Granger technique. However, the cointegration test indicates only whether there is a Granger causality between the variables. It should be emphasized that the direction of Granger causality can be obtained by constructing the VAR model.