Identifying causal relationships in case of non-stationary time series

The standard linear Granger non-causality test is effective only when time series are stationary. In case of non-stationary data, a vector autoregressive model (VAR) in first differences should be used instead. However, if the examined time series are co-integrated, a VAR in first differences will also fail to capture the long-run relationships. The vector error-correction model (VECM) has been introduced to correct a disequilibrium that may shock the whole system. The VECM accounts for both short run and long run relationships, since it is fit to the first differences of the non-stationary variables, and a lagged error-correction term is also included. An alternative approach of estimating causality when time series are non-stationary, is to use a non-parametric information-based measure, such as the transfer entropy on rank vectors (TERV) and its multivariate extension partial TERV (PTERV). The two approaches, namely the VECM and the TERV / PTERV, are evaluated on simulated and real data. The advantage of the TERV / PTERV is that it can be applied directly to the non-stationary data, whereas no integration / co-integration test is required in advance. On the other hand, the VECM can discriminate between short run and long run causality.